Commercial McDonalds "We All Make The Games"
Commercial Soreen Doreen "Call On Me"
After gaining a first class (HONS) degree in Drama in 2006, I have been steadily gaining experience in both stage and screen acting, which I enjoy tremendously.
Most recently I have been cast in a training film for the NSPCC in which I play the Welfare Officer.
I work regularly as a roleplayer and in Murder Mysteries. I also run my own Street Theatre Act called Shenanigans.


text:
platformance castle pain
Game Bias' 15 Greatest 2D Platformers List — Intro and Honorable Mentions
In video games, jumping is as ubiquitous as shooting, and it's often considered an essential part of the 2D platformer genre. But that's not exactly the case from my view. Although the overwhelmingly majority of platformers involve jumping, there are historically significant games where you must move from platform to platform without jumping at all. This list will include entries that fit this description.
Some might wonder why I have chosen to focus on 2D platformers. The short answer is I don't think 3D platformers have been that impressive on the whole over their roughly two-decade lifespan. I will consider putting together a list of the greatest 3D platformers, but it would be shorter than this one.
The honorable mentions below show that 2D platformers remain vibrant and fascinating. But before I reveal these selections, I do want to say that the 2D platformer, more so than any other video-game genre, is heavily associated with blind nostalgia. Fez, Shovel Knight, Celeste, and others bring shame to the art form by referencing or utilizing aspects of classics without surpassing or interrogating what came before them (see Fez's Tetris, Mario, and Zelda allusions; Shovel Knight's easygoing nods to Mega Man, Zelda II: The Adventure of Link, and Super Mario Bros. 3; and Celeste's pixelated sprites, which look like god-awful mush during the game's precious zoom-ins). We must look beyond what reminds us so much of the past.
As for why the following five games weren't simply included as part of a 20 greatest 2D platformers list, I echo what I said in the intro to my 15 greatest shooters list: there are other honorable mentions I could name, but I want to highlight these choices for their unique appeal.
Platformance: Castle Pain (2010)
Unfortunately, this game might be forever lost after Microsoft abandoned support of Xbox Live Indie Games for the Xbox 360, but in case a port pops up somewhere, I must mention Platformance: Castle Pain by Magiko Gaming. This gem is simple: you can walk left or right, jump, or zoom in or out so that you can better detect and avoid obstacles on your way to rescuing a damsel (yeah, that trope is more worn out than a pair of 1980s blue jeans). The zooming mechanic is brilliantly executed. Let's say you're at the section where you need to traverse a long platform while jumping over arrows that are being shot at your back. You may want to attempt this trek with the default zoomed-in camera, reacting to the sudden appearance of a projectile behind you, or more cleverly, you can zoom all the way out so that you can see the game's entire single stage — it resembles an elaborate living picture that one would hang on a wall — and thus the release of the arrows from their origin. Unlike Celeste's phoned-in visuals, the pixel art here is superb whether it's in your face or in the distance. The experience is brief like a children's storybook and accompanied by an uplifting medieval-themed soundtrack, but Platformance: Castle Pain requires perfect timing and spacing to conquer its challenges as you move from checkpoint to checkpoint.
Rock Bottom (2014)
Amy Dentata's Rock Bottom is a fantasy in which levels that represent a state of depression can be completed by counterintuitive means. The goal of Rock Bottom is to jump to higher platforms, but the only way to increase the power of your jump is to fall to your death. To further strengthen your legs, you must extend your fatal plunge by avoiding platforms as you fall from greater heights. If viewed cynically, Rock Bottom's concept could be linked to suicide ideation, but I interpret its madness as wry hope for convenient change. Ultimately, the game is an affirmation of life after struggle, as suggested by the ending that celebrates the fact that the protagonist can finally jump without having to worry about escaping a hole.
The Duck Game (2013)
This quirky title from James Earl Cox III, one of the most fascinating and prolific developers of the decade, might not fit the traditional definition of a 2D platformer, but it effectively utilizes platforms in its depiction of a downward spiral of addiction and obsession. Absurdly, the protagonist is preoccupied with the idea of holding the legs of a duck as the bird flies. Unless you elect to hit "Escape" on your keyboard, you get to see what happens when the hero indulges in this practice. In addition to the trippy premise, visuals, and audio, the amusing part of The Duck Game is that the platforms don't matter. When you're flying high with the duck, the platforms are unnecessary for vertical advancement, and when flying with the duck becomes a problem (the protagonist stops caring about hygiene and everyday chores as the duck's strength wanes), you can't leap well enough to reach your previous high. The implication is that if the duck weren't in the picture, you could go from platform to platform like a normal video-game character.
Iconoclasts (2018)
Because I played Joakim Sandberg's Iconoclasts for the first time only a few months ago, and because it's practically new, I don't have the critical distance to state that it deserves to be on the main list. That's what my head says. My heart says the game should go down as an all-time great. Iconoclasts' combination of combat and puzzle-solving makes for some wonderful platforming moments, but it's the storytelling I want to focus on here. Not only does this game have the most complex plot of any platformer I can recall, but it has the most conflicted depiction of faith and religion that I've seen in any video game, period. With a theatricality that recalls the interweaving dramas of Final Fantasy VI, Iconoclasts never lets you forget that it involves human beings with worldviews shaped by their individual experiences and convictions. This is the most ambitious 2D platformer ever made, and in almost every respect, it succeeds. (See my full review here.)
Octahedron (2018)
Yet another 2018 game that I will continue to keep in mind as I evaluate the history of 2D platformers, Octahedron's ever-changing mechanics share center stage with a beyond-thirsty electronica soundtrack and neon-infused graphics that recall wild night clubs. The smooth and slippery movements of the platform-creating protagonist complement the pulsing beats and blanket-like textures of the music. A sensual powerhouse from developer Demimonde, this game is so sexy that one can feel dirty exploring every last part of its tunnel-like stages. (See my full review here.)
Posted in Uncategorized and tagged amy dentata, best 2d platformers ever, demimonde, greatest platformers list, iconoclasts, james earl cox III, joakim sandberg, magiko gaming, octahedron, platformance castle pain, rock bottom, the duck game on July 13, 2018 by Jed Pressgrove. 3 Comments

redpajama_set_name: RedPajamaCommonCrawl


text:
The particular smell of rain-soaked concrete greeted Ember's nostrils. She followed the demon out the back door of Kathy's Korner Diner. Water gushed out of the downspout in a white froth, spattering her with droplets. The deluge hissed as it crashed into the back parking lot's graying blacktop. It was the same all over Cauchemar, even though the rain had ceased at last.
Ember watched the demon limp across the parking lot, one hand clutching his shoulder. She saw a brief glow of white light, and the agony etched across his handsome, thin featured face diminished.
"There," Ember said, crossing her arms over her chest and glaring at the demon B.L.C. Bubb. His white suit and red tie belied his arcane nature. Ember knew that a pair of discrete, pointed horns lurked beneath the bowler-style hat on his head. "I've let you heal your little boo boo. Now tell me what your favor is so we can get this over with."
Bubb eased himself onto an overturned pickle bucket, gasping in pain. "Not quite healed, just yet, but I am able to palaver."
"So, let's palaver. What's the favor? I'm not letting you leave without telling me."
Bubb leaned against the red-brick wall, seeming very tired. Ember was well used to his normal smirk of smug supercilious superiority. Seeing the frank, tired expression on his face now disconcerted her.
"I'm not sure I can tell you, just yet."
"Hey, do you want her to stick Celestial steel through your other shoulder, pal?" Kali, Ember's tiger-striped, tabby, cat familiar, glared at Bubb with her yellow eyes. "Or maybe she should stick it right up your demonic—"
"Kali," Ember said. "Back off. He doesn't look so hot."
"Is that a demon pun?" Kali asked. "If so, it's a pretty good one."
Bubb glanced at Ember, his usual cockiness missing in action. "When I say I can't tell you, just yet, I don't mean any malice, believe it or not."
"Oh, I don't believe you." Ember glared at the demon, hating herself feeling any sympathy for him at all. Normally she eschewed combat magic. Ember preferred trickery and defense to hurting people. Apparently, that applied to demons as well. "Look, do you need me to perform a healing spell on you? I honestly didn't think you'd be hurt this badly. I thought you were a much more powerful demon."
"He has to be. He used weather magic." Kail's tail twitched as she stared up at Bubb. "Weather magic takes enormous power."
"Did I?" Bubb's brows climbed high on his slender face. "I don't recall ever making the claim that I had used weather magic."
"You implied that you were responsible for the rain," Ember snarled. "We're not playing this game anymore."
"Oh, I was most certainly responsible for the rain."
Ember forced her anger down. It worked to Bubb's advantage. "You're talking in circles, and I have no patience for it."
"Ember, now is not the time for you to know," Bubb said. "I know you have no reason to trust me. That's good. You shouldn't trust anyone at this point. You shouldn't trust anything, either."
Ember growled. "Is this your latest mind game? Just tell me, Bubb. What's the favor?"
"I can't tell you, yet."
"Can't, or won't?" Kali brandished her claws and hissed. "You know cat scratch fever is lethal to demons, right?"
"Now is not the time," Bubb said. "But I promise you, that time is coming. Very soon."
"Is that another threat?"
Bubb shrugged. "Take it how you will."
Ember's eyes narrowed to slits. "I know how deals with demons work. I know my soul is forfeit if I don't agree to do your favor. But if your favor involves harming another living creature, I'm willing to break the contract. And, if I'm going to break the contract, I see no reason why I should go down alone…"
Bubb's eyes widened. "I'm not sure what I find more surprising, Ember. The fact that you're making threats…"
He stood up, as spry and dexterous as before he'd taken a sword through the shoulder. "…or the fact that I believe them. I'll take your words to heart, my good Madame. I'll be in touch."
"You're not going anywhere, jerk," Kali snapped.
"Let him go, Kali." Ember shook her head in disgust as Bubb disappeared around the corner of the diner. "We don't want to play into his hands. Again."
"Remember when we used to run a bar?" Kali asked.
"Yes. I miss those days. Sometimes, but life is about change." Ember laughed. "Bar owner, housewife, foster parent, deputy…I'm getting a little bit confused about which hat I'm wearing today."
"Well, your best bet is not to let Bubb get to you." Kali sniffed. "Whatever his game is, a big part of it is keeping you scared and off balance."
"I think, after today, he's learned that I can only be pushed so far," Ember growled. "Don't pretend like you didn't feel bad about hurting him," Kali rubbed her face on Ember's leg. "Don't let Bubb make you heartless. For all we know, that's what he's wanted all along."
Ember leaned over and picked her familiar up. She buried her nose in Kali's soft fur and sighed.
"Come on, Kali," Ember sighed. "Let's go tell Cedric what I've done."
"I'm not sure what he can do about it."
"Probably nothing, but…I don't want any more secrets between us." Ember set her jaw hard. "Not ever."
Ember walked in the sun as she made her way across Cauchemar's main drag. It was as if the mighty daystar wished to make up for its many days shrouded behind an iron-gray curtain of clouds. Sunlight dappled the waters of the swollen Cauchemar Lake near the center of town. Reflections tinged the art deco roofs of downtown and the creole cottages of the residential district, alike.
"I never thought I'd miss the sun so much," Kali purred. She leaped up onto the flat, stone railing of the bridge spanning the rain-swollen, rushing Cauchemar river. "What do you make of Bubb claiming he didn't use weather magic?"
"In the next breath he said he was responsible for the rain, though. I don't think we can trust a thing that comes out of his mouth." Ember glanced over the railing as a black shape darted through the swollen, muddy waters of the river.
For a moment, she thought it might be a hapless resident swept away by a flash flood. She soon realized it was a tree limb, black with moisture and gnarled like an arthritic finger.
"It really was close, wasn't it?" Kali stopped to peer over the railing. "I mean, the water is less than an inch from the bottom of the bridge."
"It was close. Too close for comfort."
They crossed the bridge and turned into the gravel lot of the Sheriff's Station. The humble, one-story structure consisted of a small lobby, four desks, and Cedric's office in the back. Only one of the desks was occupied at any given time, when Deputy Smothers was on duty.
Ember and Kali crossed the polished, hardwood floors to Cedric's office. She rapped on the door, then peered inside. Why do I feel like a kid at the principal's office?
Cedric turned toward her, his wide, sensuous mouth drawn into a smile. His blue eyes and golden-blonde hair, combined with his hardbody physique made Cedric appear more akin to a California surfer than the Sheriff of Cauchemar County, Louisiana. Ember's arcane sense could detect the lupine aura about her shifter husband's form.
"Good morning," he said. "How did your meeting go?"
Ember flopped into the chair on the opposite side of the desk from Cedric and rubbed her eyes with the palms of her hands. "Not well. I agreed to do the favor for Bubb."
Ember glanced up at the sound of Cedric's gentle declaration. She found no judgment in his eyes, only worry. "How did you know?"
"The rain stopped, for one. For another, I can just tell by the way you walked into my office."
Ember groaned. "Is it that obvious?"
"Pretty much," Kali said, licking her paw while sitting on a stack of papers.
"Kali, those are important reports you're sitting on," Cedric said.
"Yes, without them my furry bottom would not be properly cushioned." She stopped licking her paw. "Don't worry about me, I'll be fine. Focus on your wife, that's what's important right now. Carry on."
Cedric cocked an eyebrow and Ember couldn't help but laugh. In the years they'd known each other, Ember had yet to see a time where Kali didn't get the better of Cedric in their duels of wit. "What does Bubb want you to do?"
"He told me it's not time for me to know yet." Ember's jaw worked silently for a moment while she literally chewed her words over. "I don't think he was lying, oddly enough."
"He's a demon, they trade in lies."
"I know. It's just all…" Ember sighed. "Look, I think the rain really wore on me, not to mention that last case. Bubb probably just wants me to sweat. The best thing I can do is forget about him until he shows his face again."
Cedric thought for a long moment, then nodded. "That's probably for the best. Besides, we're going to be mighty busy with the Cauchemar Blues Festival coming up."
Ember slapped herself on the side of the head. "Oh my Goddess! I'd forgotten all about it. That's, like, this weekend, isn't it?"
"Yup." Cedric stood up and stretched, then snagged his hat off the wall hook. "In fact, we're late for a meeting with Major Petit at the station north of town."
"So we can coordinate with the state troopers for security," Ember nodded. "I remember, now. Sorry I made us late."
"Ah, Petit's probably not there, yet, himself. We used to joke he had a disease that destroyed punctuality in a hundred-foot radius all around him."
Ember grinned. "There's a curse that will do the same thing. Of course, that's dark magic."
Ember felt a tightness in her belly. Cedric glanced down at her as he held the glass exit door open, and she passed through.
"Are you all right, Dear?"
"I'm fine," Ember said, squinting in the sunlight. The clouds had scooted away to the horizon, baring Cauchemar to the fury off the Louisiana sun.
"Something's bothering you. No more secrets, remember?"
Ember turned to him and stopped in her tracks. A slight smile came to her lips. "No more secrets. All right. I'm worried, because, well—"
"She hurt Bubb pretty bad with a spell," Kali said helpfully. "He was in rough shape and she feels bad about it."
"With all Bubb's done to you, and this family, I'd say you shouldn't be worried."
Ember sucked in a deep breath through her nostrils, then let it out through her mouth. "Cedric, it's never that simple for a witch. Magic is shaped by your intentions. If your intentions are good, the effect will generally be good. If the intention is to be destructive, then the effect will generally be destructive."
"I follow you, so far. I did marry a witch after all. You're worried about being tempted to use dark magic, yes?"
Ember started. "Yes, you do understand. That's exactly the struggle. Dark Magic is kind of like steroids to a witch. You can channel way more power, but, the more you use it, the more it poisons your mind. Eventually, you lose the ability to feel empathy for anyone. Even yourself. From there on, your only desire is to increase your power."
"It's like a crack addict trying to get high," Kali said, her paws on the back seat of the cruiser. "They try to use the magic to fill the emptiness inside, but it's temporary, at best. And, like any drug, you build up a tolerance to it."
Cedric pulled out of the parking lot and onto Cauchemar's main drag. "Then I understand why you're worried, Ember. Don't worry. You have a family to keep you grounded."
Ember smiled and patted his arm, but on the inside she knew Bubb had already used her family against her once before. Her love for them might wind up being a curse rather than a boon.


text:
\section{Introduction}\label{sec_intro}

There is a number of reasons for chaotic behaviour of trajectories in a
vicinity of heteroclinic network. This can be due to the complexity of the
network as, e.g., when a network involves nodes which are chaotic sets
\cite{a03,d95} or is a union of periodic orbits connected by
an infinite number of heteroclinic and homoclinic orbits \cite{rla11}.
Chaotic behaviour can be caused by chaotic forcing, even when the
amplitude of the added noise is arbitrarily small \cite{am03}.
When eigenvalues of linearisations near some of the equilibria in the network
are complex, the return map near a heteroclinic cycle may possess a horseshoe,
thus resulting in emergence of a strange attractor \cite{acl06,lab}.
Chaos due to the presence of complex eigenvalues of linearisations was found
in \cite{mechanism} and \cite{rod}, where either a horseshoe does not
exist, or its existence has not been investigated.
Strange attractors may also exist in periodically-perturbed
dynamical systems with an attracting heteroclinic network \cite{rl21,lr20}.

Examples of more primitive heteroclinic networks comprised of a finite number
of equilibria and heteroclinic connections feature only real eigenvalues
of linearisation near equilibria; in their vicinity irregular switching
between connections takes place, indicating chaotic behaviour of nearby
trajectories \cite{ac10,pd05,pr21}. Complexity of a network can be measured
by the number of splitting nodes (for the definition see, e.g., \cite{AshCasLoh18}).
For the networks studied in \cite{ac10,pd05,pr21}, it is the number
of equilibria having more than one outgoing connections (six, nine and five,
respectively). In the present paper we consider such a primitive network
which has only a single splitting node. Our numerical simulations indicate
a possibility of irregular switching of nearby trajectories.

A heteroclinic cycle is a union of nodes and connecting trajectories,
where all nodes are required to be distinct. A heteroclinic network is a
connected union of heteroclinic cycles. In this paper we introduce the notion of
an omnicycle, an ordered sequence of nodes and heteroclinic connections, where
repeating nodes are allowed. Regarded as a subset of the
phase space, an omnicycle is either a heteroclinic cycle or a network, however
in the latter case the stability properties of these objects can be very different.
Omnicycles were implicitly employed in \cite{pd05} to describe the behaviour of trajectories
near a heteroclinic network, where existence of an attracting omnicycle was
indicated by so-called ``regular cycling''. However, the terminology introduced
{\it ibid} was closely related to the specific structure of the considered
network, while the definition introduced in the present paper is of a general type.

Omnicycles, similarly to heteroclinic cycles and network, do not exist in a
generic dynamical systems, because a connection between saddles can be destroyed
by a small perturbation. However, they may exist and be robust in a system
where some restrictions are imposed, e.g., symmetries \cite{km95a,km04},
natural constrains, as in population dynamics or evolutionary game theory
\cite{ac10,hs94,hs98} or due to prescribed patterns of interaction, as in a
system of coupled cells or oscillators \cite{bu00,as10}.

An invariant set of a dynamical system is called asymptotically stable if it
attracts all trajectories from its small neighbourhood.
A non-asymptotically stable set can be stable in a weaker sense
and attract a positive measure set of initial conditions from its small
neighdourhood. In \cite{op12} this property was called fragmentary asymptotic
stability. Conditions for asymptotic stability or fragmentary asymptotic
stability for certain types of heteroclinic cycles are known for a long time,
see, e.g., Podvigina \cite{op12} or \cite{pcl20} and references therein.
Typically, conditions for stability of a cycle are derived by following
nearby trajectories and constructing the respective transition maps.
Since it is irrelevant how many times each particular node or connection
is present in the sequence, the existing conditions for stability of
cycles can be straightforwardly employed for stability of omnicycles.
The study of stability of a heteroclinic network is more difficult, because
a trajectory can follow different sequences of connections along the network.

If a fragmentarily asymptotically stable (f.a.s.) cycle is a subset of a
heteroclinic network then the whole network also is f.a.s. For all networks in
${\mathbb R}^4$ that were studied in literature the converse is also true: if a
heteroclinic network is f.a.s. then at least one of it subcycles is f.a.s.
\cite{br94,ks94,cl14,cl16,pc15,pc16,pl19}. The same property holds true
for some networks in higher-dimensional subspaces, namely stability of
a network implies stability of at least one of its subcycles \cite{ac10,pcl18}.
The authors of \cite{pd05,pr21} presented examples of f.a.s. networks, where
all subcycles are completely unstable. To prove stability the authors have
shown that certain omnicycles, that are subsets of the network, are f.a.s.
(To be more precise, in the proofs of \cite{pd05} a different technique was
employed, which in fact is similar to our theorem \ref{th_4}).

\begin{figure}
\hspace*{12mm}\includegraphics[width=18cm]{fig1.ps}

{\Large
\vspace*{-53mm}\hspace*{72mm}$\xi_1$

\vspace*{-54mm}\hspace*{72mm}$\xi_2$

\vspace*{-7mm}\hspace*{44mm}$\xi_3$\hspace{51mm}$\xi_5$

\vspace*{39mm}\hspace*{44mm}$\xi_4$\hspace{51mm}$\xi_6$

\vspace*{-35mm}\hspace*{37mm}${\cal C}_L$\hspace*{63mm}${\cal C}_R$

\vspace*{-27mm}\hspace*{58mm}$H_L$\hspace*{17mm}$H_R$
}

\vspace{55mm}

\caption{The network studied in sections \ref{sec4} and \ref{sec5}. The
crossections $H_L$ and $H_R$ employed in the derivation of conditions of
stability are introduced in section \ref{sec3}.
\label{fig1}}
\end{figure}

In this paper we study stability of a network in ${\mathbb R}^6$ which is a union of
two cycles,
${\cal X}={\cal C}_L\cup{\cal C}_R\subset{\mathbb R}^6$, admitted by a ${\mathbb Z}_2^6$-equivariant dynamical
system
\begin{equation}\label{eq_ode6}
\dot{\bf x}=f({\bf x}),\quad f:{\mathbb R}^6\to{\mathbb R}^6,
\end{equation}
where ${\mathbb Z}_2^6$ is generated by symmetries reversing the sign of one of the
spatial coordinates. Six equilibria involved in the network belong to
coordinate axes and heteroclinic connection belongs to coordinate
planes. The cycles are:
$${\cal C}_L:\,(\xi_1\to\xi_2\to\xi_3\to\xi_4\to)\hbox{ and }
{\cal C}_R:\,(\xi_1\to\xi_2\to\xi_5\to\xi_6\to).$$

In terminology of \cite{op12}, ${\cal X}$ is a {\it type Z} heteroclinic network
and ${\cal C}_L$ and ${\cal C}_R$ are {\it type Z} heteroclinic cycles. Generalisation
of this classification to omnicycles implies that any omnicycle, which is a
subset of ${\cal X}$, is a {\it type Z} omnicycle.
Theorems in \cite{op12} that give necessary and sufficient
conditions for a type Z heteroclinic cycle to be f.a.s. are also applicable to
study stability of type Z heteroclinic omnicycles.

The conditions are inequalities that involve eigenvalues and eigenvectors of
transition matrices, whose entries depend on eigenvalues of Jacobians
${\rm d} f(\xi_j)$. For ${\cal C}_L$ and ${\cal C}_R$ the respective $4\times4$ transitions
matrices are sparse, hence allowing to calculate eigenvalues and eigenvectors.
Transition matrix of the omnicycle ${\cal C}_{LR}$, where
$${\cal C}_{LR}:\,
(\xi_1\to\xi_2\to\xi_3\to\xi_4\to\xi_1\to\xi_2\to\xi_5\to\xi_6\to),$$
is full, therefore eigenvalues and eigenvectors can not be found analytically
in a general case. In order to be able to calculate them, we assume that some
eigenvalues of ${\rm d} f(\xi_j)$ are much larger then the
others. For vanishing small eigenvalues we calculate eigenvalues
and eigenvectors of the respective sparse transition matrix. Because of
continuous dependence of eigenvalues and eigenvectors of a matrix
on its entries, the stability or instability conditions
are satisfied true for small
non-vanishing eigenvalues of ${\rm d} f(\xi_j)$ as well.

The paper is organised as follows: In Section \ref{sec2} we recall basic
definitions of robust heteroclinic cycles, networks and asymptotic stability
and in Section \ref{sec3} we discuss asymptotic stability of omnicycles.
In sections \ref{sec4} and \ref{sec5} we study analytically and numerically
asymptotic stability of network shown in fig.~\ref{fig1}. Namely, in section
\ref{sec4} we derive conditions of stability of the cycles ${\cal C}_L$ and ${\cal C}_R$
and show that the network can be f.a.s. while both cycles are completely
unstable. In section \ref{sec5} we present results of numerical simulations
of solutions of system (\ref{eq_ode6}), in particular, in the cases when
all ${\cal C}_L$, ${\cal C}_R$ and ${\cal C}_{LR}$ are completely unstable.
By varying the r.h.s. of (\ref{eq_ode6}) we obtain examples of f.a.s.
omnicycles, different from ${\cal C}_{LR}$, or of apparently chaotic
switching between ${\cal C}_L$ and ${\cal C}_R$.

\section{Background and definitions}\label{sec2}

In this section we introduce the notions of omnicycles and trail-stability,
and recall definitions that will be used in the following sections.

\subsection{Robust heteroclinic cycles, omnicycles and networks}\label{defhet}

Consider a smooth $\Gamma$-equivariant dynamical systems
\begin{equation}\label{eqsys}
\dot x=f(x),\quad f(\gamma {\bf x})=\gamma f({\bf x}),\quad\mbox{ for all }
\gamma\in\Gamma\hbox{ and }x\in{\mathbb R}^n,
\end{equation}
where we assume that $\Gamma\subset{\bf O}(n)$ is finite.
For a group $\Gamma$ acting on ${\mathbb R}^n$ the {\it isotropy group} of
the point $x\in{\mathbb R}^n$ is the subgroup
$\Sigma_x=\{\gamma\in\Gamma\ ~:~\ \gamma x=x\}$,
and a {\it fixed-point subspace} of a subgroup $\Sigma\subset\Gamma$
is the subspace
${\rm Fix}(\Sigma)=\{{\bf x}\in{\mathbb R}^n\ ~: \ \sigma {\bf x}={\bf x}\mbox{ for all } \sigma\in\Sigma\}$.
(For more details about equivariant dynamical systems see, e.g.,
Golubitsky and Stewart \cite{GSS}.)

Let $\xi_1,\ldots,\xi_m$ be hyperbolic equilibria of (\ref{eqsys}),
where $\xi_j\ne\xi_k$ for any $1\le j,k\le m$, $j\ne k$, and
$\kappa_{j,j+1}\ne\varnothing$ be a set of trajectories from $\xi_j$ to
$\xi_{j+1}$, where $\xi_{m+1}=\xi_1$ is assumed.
A {\it heteroclinic cycle} is an invariant set $X\subset{\mathbb R}^n$ which is a union
of equilibria $\{\xi_1,\ldots,\xi_m\}$ and
heteroclinic connections $\{\kappa_{12},\ldots,\kappa_{m1}\}$.
A {\it heteroclinic network} is a connected union of a finite
number of heteroclinic cycles.

\begin{definition}\label{def01}
An {\it omnicycle} is a union of equilibria $\{\xi_1,\ldots,\xi_m\}$ and
heteroclinic connections, $\{\kappa_{12},\ldots,\kappa_{m1}\}$, where
$\kappa_{ij}\subset W^u(\xi_i)\cap W^s(\xi_j)$ and $\kappa_{ij}\ne\varnothing$.
\end{definition}
I.e., the difference between a heteroclinic cycle and an omnicycle is that
for the omnicycle repeating equilibria are allowed.

By analogy with a building block of a heteroclinic cycle \cite{pc15}, we
introduce building block of an omnicycle.
\begin{definition}\label{bb}
Given an omnicycle ${\cal X}=\{\xi_1,\ldots,\xi_m;\kappa_{12},\ldots,\kappa_{m1}\}$,
we say that its subset
$\{\xi_1,\ldots,\xi_j;\kappa_{12},\ldots,\kappa_{j,j+1}\}$ and the symmetry
$\gamma\in\Gamma$ is a {\it building block} of ${\cal X}$ if
$\gamma\xi_i=\xi_{i+j}$ and $\gamma\kappa_{i,i+1}=\kappa_{i+j,i+j+1}$ for
any $\xi_i\in{\cal X}$ and $\kappa_{i,i+1}\subset{\cal X}$.
\end{definition}

A heteroclinic cycle (or an omnicycle, or a network) ${\cal X}$ is called
{\it robust}, if any connection $\kappa_{ij}\subset{\cal X}$ belongs to a flow-invariant
subspace $P_{ij}$. In system (\ref{eqsys}) the invariance of $P_{ij}$
typically follows from its equivariance, namely that
$P_{ij}={\rm Fix}(\Sigma_{ij})$
for some $\Sigma_{ij}\subset\Gamma$. By $\Delta_j$ we denote the isotropy
subgroup of $\xi_j$ and $L_j={\rm Fix}(\Delta_j)$.
Denote by $P_{ij}^{\perp}$ and by $L_j^{\perp}$ the orthogonal complement to
$P_{ij}$ and $L_j$, respectively, in ${\mathbb R}^n$.

Given an equilibrium $\xi_j\in L_j=P_{j-1,j}\cap P_{j,j+1}$ that belongs to a
robust heteroclinic cycle ${\cal C}=(\xi_1\to\xi_2\to...\to\xi_m\to)$, the
eigenvalues of the Jacobian ${\rm d} f(\xi_j)$ can be divided into {\it radial}
(the associated eigenvectors belong to $L_j$), {\it contracting} (the
eigenvectors belong to $P_{j-1,j}\ominus L_j$), {\it expanding} (the
eigenvectors belong to $P_{j,j+1}\ominus L_j$) and {\it transverse} (the remaining
ones), where $P\ominus L$ denotes the orthogonal complement to $L$ in $P$.

\begin{definition}\label{def02} (Adapted from \cite{op12})
A heteroclinic network ${\cal X}$ (cycle or omnicycle) is called of {\it type Z} if\\
$\bullet$ for any $\xi_j\in{\cal X}$ we have that $\dim P_{i,j}=\dim P_{j,k}$ for
any incoming connection $\kappa_{i,j}$ and any outgoing connection $\kappa_{j,k}$;\\
$\bullet$ for any $\kappa_{i,j}\subset{\cal X}$ the isotropy subgroup $\Sigma_{i,j}$
decomposes $P_{i,j}^{\perp}$ into one-dimensional isotypic components.
\end{definition}

\subsection{Stability}\label{sec_stab}

Denote by $\Phi({\bf x},\tau)$ a trajectory of system (\ref{eqsys}) through
the point ${\bf x}\in{\mathbb R}^n$ and
by $\Phi({\bf x},(\tau_1,\tau_2))=
\cup_{\tau\in(\tau_1,\tau_2)}\Phi({\bf x},\tau)$ the subset
of this trajectory corresponding to the time interval $(\tau_1,\tau_2)$.
For a set $X\subset{\mathbb R}^n$ and $\varepsilon>0$, the
$\varepsilon$-neighbourhood of $X$ is
\begin{equation}\label{ep_nei}
B_{\varepsilon}(X)=\{{\bf x}\in R^n:\ d({\bf x},X)<\varepsilon\}.
\end{equation}
Given a compact invariant set $X$ of (\ref{eqsys}), its
$\delta$-local basin of attraction defined as
\begin{equation}\label{del_bas}
{\cal B}_{\delta}(X)=\{{\bf x}\in{\mathbb R}^n:\ d(\Phi({\bf x},\tau),X)<\delta\hbox{ for any }
\tau\ge0\hbox{ and }\lim_{\tau\to\infty}d(\Phi({\bf x},\tau),X)=0\}.
\end{equation}

\begin{definition}\label{def1}
An invariant set $X$ is called {\it asymptotically stable}, if for any
$\delta>0$ there exists an $\varepsilon>0$ such that
$$B_{\varepsilon}(X)\subset{\cal B}_{\delta}(X).$$
\end{definition}

\begin{definition}\label{def2}
An invariant set $X$ is called {\it fragmentarily asymptotically
stable}, if for any $\delta>0$
$$\mu({\cal B}_{\delta}(X))>0.$$
(Here $\mu$ is the Lebesgue measure of a set in ${\mathbb R}^n$.)
\end{definition}

\begin{definition}\label{def3}
An invariant set $X$ is called {\it completely unstable}, if there exists
$\delta>0$ such that $\mu({\cal B}_{\delta}(X))=0$.
\end{definition}

\begin{definition}\label{defn1}
Given an omnicycle ${\cal X}=\{\xi_1,\ldots,\xi_m;\kappa_{12},\ldots,\kappa_{m1}\}$,
we say that a trajectory $\Phi({{\bf x}},(\tau^{in},\tau^{fin}))$
$\varepsilon$-{\it follows} this omnicycle if
$$\Phi({{\bf x}},(\tau^{in},\tau^{fin}))\subset N_{\varepsilon}({\cal X})$$
and there exist a number $1\le j_0\le m$ and time instances
$\{\tau_0',\tau_0'',\tau_1',\tau_1'',...,\tau_S',\tau_S''\}$, where
$\tau^{in}\le\tau_0'<\tau_0''<\tau_1'<...<\tau_S'<\tau_S''\le\tau^{fin}$,
such that
$$
\renewcommand{\arraystretch}{1.5}
\begin{array}{l}
\Phi({{\bf x}},(\tau_s',\tau_s''))\subset B_{\varepsilon}(\xi_j),
\hbox{ where }j=j_0+s({\rm mod}\, m),\hbox{ for all }0\le s\le S,\hbox{ and}\\
\biggl((\cup_{0\le s<S}\Phi({{\bf x}},(\tau_s'',\tau_{s+1}'))\cup
\Phi({{\bf x}},(\tau^{in},\tau_0))\cup
\Phi({{\bf x}},(\tau_S,\tau^{fin}))\biggr)\bigcap
\biggl(\cup_{1\le j\le m}B_{\varepsilon}(\xi_j)\biggr)=\varnothing.
\end{array}
$$
\end{definition}

\begin{definition}\label{defn2}
Given an omnicycle ${\cal X}$ of (\ref{eqsys}), its
$\delta$-local basin of attraction is defined as
\begin{equation}\label{del_omni}
{\cal B}^{\rm omni}_{\delta}({\cal X})=
\{{\bf x}\in{\mathbb R}^n:\ \Phi({\bf x},(0,\tau))\ \delta-\hbox{follows }{\cal X}
\hbox{ for any }
\tau>0\hbox{ and }\lim_{\tau\to\infty}d(\Phi({\bf x},\tau),{\cal X})=0\}.
\end{equation}
\end{definition}

\begin{definition}\label{defn3}
An omnicycle ${\cal X}$ is called {\it trail-stable}, if for any $\delta>0$
$$\mu({\cal B}^{\rm omni}_{\delta}({\cal X}))>0.$$
\end{definition}

\begin{remark}\label{rem00}
For an omnicycle which is a heteroclinic cycle the notions of
f.a.s. and trail-stability are equivalent, because trajectories near the cycle
visit the equilibria in the prescribed order specified in the definition of
the cycle. An omnicycle, which is not a cycle, can be f.a.s. according to
definition \ref{def2} without being trail-stable.
\end{remark}

\begin{remark}\label{rem01}
Suppose ${\cal X}$ is a robust type Z omnicycle, which is not a heteroclinic cycle,
in a $\Gamma$-equivariant system. Then ${\cal X}$ is not asymptotically stable.
This can be shown by arguments similar to the ones employed in the proof
of theorem 3.1 in \cite{pcl18}.
\end{remark}

\section{Stability of omnicycles}\label{sec3}

In this section we state conditions for trail-stability of type Z omnicycles.
Due to similarity between heteroclinic cycles and omnicycles, the theorem
of \cite{op12}, which gives conditions for fragmentary asymptotic stability of
heteroclinic cycles of type Z, is also applicable to omnicycles of that type.
The conditions involve eigenvalues and eigenvectors of transition matrices,
which are constructed from local and global maps defined below.

\subsection{Local and global maps}
\label{locstr}

To study stability of an omnicycle, similarly to heteroclinic cycles
and networks, the behaviour
of nearby trajectories is approximated by local and global maps. By
$N_{\tilde\delta}(\xi_j)$ we denote a neighbourhood of $\xi_j$ bounded by
$|a_s|=\tilde\delta$, where $a_s$ are coordinates in the basis comprised
of eigenvectors of ${\rm d} f(\xi_j)$. We assume $\tilde\delta$ to be fixed and
small, so that in $N_{\tilde\delta}(\xi_j)$ the equation (\ref{eqsys})
can be approximated by its linearisation ${\rm d} f(\xi_j)$, and
$\tilde\delta\gg\varepsilon$ and $\delta$, where $\varepsilon$ and $\delta$ are
those employed in the definitions of stability in subsection \ref{sec_stab}

Given an omnicycle ${\cal C}:\,(\xi_1\to...\to\xi_m\to)$, let
$({\bf u}_j,v_j,w_j,{\bf z}_j)$ be local coordinates
near $\xi_j$ in the basis, where radial eigenvectors come the first
(the respective coordinates are ${\bf u}_j$), followed by the contracting
and the expanding eigenvectors, the transverse eigenvectors being the last.
(If repeating equilibria are present in the sequence, i.e. $\xi_i=\xi_j$
for $i\ne j$, then such decompositions for eigenvectors of ${\rm d} f(\xi_i)$
and ${\rm d} f(\xi_j)$ can be different.)
Denote by $H^{(in)}_{j-1,j}$ the face of $N_{\tilde\delta}(\xi_j)$ that
intersects with the connection $\kappa_{j-1,j}$ and
by $H^{(out)}_{j,j+1}$ the face intersected by $\kappa_{j,j+1}$.
In literature, $H^{(in)}_{j-1,j}$ and $H^{(out)}_{j,j+1}$ are often
called crossections or Poincar\'e sections. For a trajectory close to the
cycle the local map relates coordinates of its intersection with
$H^{(in)}_{j-1,j}$ to the coordinates of its intersection with
$H^{(out)}_{j,j+1}$.

Let the superscripts $^{in}$ and $^{out}$ denote the coordinate in
$H^{(in)}_{j-1,j}$ and $H^{(out)}_{j,j+1}$, respectively.
The local map $\phi_j:H^{(in)}_{j-1,j}\to H^{(out)}_{j,j+1}$ (see, e.g.,
\cite{km95a} or \cite{op12}) is
\begin{equation}\label{mapl0}
v^{out}=K_{1j}(w^{in})^{-c_j/e_j}\hbox{ and }
{\bf z}^{out}={\bf K}_j{\bf z}^{in}(w^{in})^{-{\bf t}_j/e_j},
\end{equation}
where $K_{1j}$ and ${\bf K}_j$ are constants (for a fixed $\tilde\delta$), and
$c_j,e_j,{\bf t}_j$ are the contracting, expanding and transverse eigenvalues of
${\rm d} f(\xi_j)$, respectively.
The coordinates ${\bf u}^{in},v^{in},{\bf u}^{out}$ and $w^{out}$ are ignored, because
they are irrelevant in the study of stability.

Near the connection $\kappa_{j,j+1}$ the system (\ref{eqsys}) can be
approximated by a global map (also called a {\em connecting diffeomorphism})
$\psi_{j,j+1}:H^{(out)}_{j,j+1}\to H^{(in)}_{j,j+1}$,
which relates coordinates of the
point where a trajectory exits $N_{\tilde\delta}(\xi_j)$ to coordinates of
the point where it enters $N_{\tilde\delta}(\xi_{j+1})$.
The global map is predominantly linear
\begin{equation}\label{mapgl}
(w^{j+1,in},{\bf z}^{j+1,in})=A'_j(v^{j,out},{\bf z}^{j,out}),
\end{equation}
where $A'_j$ is an $(n_t+1)\times(n_t+1)$ matrix. For type Z omnicycles
the matrix $A'_j$ is a product of a diagonal matrix $A''_j$ and
a permutation matrix $A_j$.

Denote by $g_j$ the superpositions of the local and global maps:
$$g_{j}=
\phi_{j}\circ\psi_{j-1,j}:\, H^{(out)}_j\to H^{(out)}_{j+1}.$$
We call the set of maps $\{g_{1:m}\}=\{g_{1},\ldots,g_{m}\}$,
$g_{j}:{\mathbb R}^{n_t+1}\to{\mathbb R}^{n_t+1}$,
{\it the collection of maps associated with the omnicycle}
${\cal C}=\{\xi_1,\ldots,\xi_m\}$.
For a given a collection of maps $\{g_1^m\}$ we define
superpositions
\begin{equation}\label{mapg}
G_j=g_{j-1}\circ\ldots\circ g_{1}\circ g_{m}\circ\ldots
\circ g_{j+1}\circ g_{j},
\quad G_j:\,H^{(out)}_j\to H^{(out)}_j,
\end{equation}
which approximate the behaviour of trajectories near the cycle and are
called {\it return maps}.

\subsection{Transition maps and transition matrices}
\label{trmatr}

Let us introduce new coordinates:
\begin{equation}\label{newc}
{\eta}=(\ln|w|,\ln|z_1|,...,\ln|z_{n_t}|).
\end{equation}
Due to the smallness of $\varepsilon$, in these coordinates
the maps $g_{j}$ become approximately linear, ${\eta}^{j+1}=M_j{\eta}^j$,
where the transition matrix $M_j$ is a product of permutation matrix
$A_j$ that relates local coordinates near $\xi_{j+1}$ to the ones near $\xi_j$
and the matrix
\begin{equation}\label{esm}
B_j:=\left(
\begin{array}{ccccc}
b_{j,1}&0&0&\ldots&0\\
b_{j,2}&1&0&\ldots&0\\
b_{j,3}&0&1&\ldots&0\\
.&.&.&\ldots&.\\
b_{j,N}&0&0&\ldots&0
\end{array}
\right).
\end{equation}
The entries
$b_{j,l}$ of the matrix $B_j$ depend on the eigenvalues of the
linearisation $df(\xi_j)$ of (\ref{eqsys}) near $\xi_j$ as follows
\begin{equation}\label{coeB}
b_{j,1}=c_j/e_j\mbox{ and }b_{j,l+1}=-t_{j,l}/e_j,\ 1\le l\le n_t,\ 1\le j\le m.
\end{equation}
Transition matrices of the superposition $G_j$ defined in (\ref{mapg})
are the products\\
 ${\cal M}^{(j)}=M_{j-1}\ldots M_1M_m\ldots M_{j+1}M_j$.

Consider the matrix $M:={\cal M}^{(1)}=M_m\ldots M_1:{\mathbb R}^N\to{\mathbb R}^N$; it is
a product of the basic transition matrices of the form (\ref{esm}). We separate
the coordinate vectors ${\bf e}_l$, $1\le l\le N$, into two groups.
The first group is comprised of the vectors ${\bf e}_l$ for which there exist
such $k$ and $j$ that $(A^{(j)})^kA_{j-1,1}{\bf e}_l={\bf e}_1$ (recall that
$A_j$ are permutation matrices), the second one incorporates the remaining
vectors. Denote by $V^{\rm sig}$ and $V^{\rm ins}$ the subspaces spanned
by vectors from the first and second group, respectively (the superscripts
``ins'' and ``sig'' stand for significant and insignificant). In the basis where
significant vectors come the first, matrix $M$ takes the form
\begin{equation}\label{mM}
M=\left(
\begin{array}{c|c}
\\
M^{\rm sig} & 0 \\
\\
\hline
M^{\rm ins} &
\begin{array}{cccc}
1&0&\ldots&0\\
0&1&\ldots&0\\
.&.&\ldots&.\\
0&0&\ldots&1
\end{array}
\end{array}
\right).
\end{equation}

\begin{theorem}\label{th_22} (Adapted from \cite{op12}.)
Let $V^{\rm sig}$ and $V^{\rm ins}$ be the subspaces defined above.
\begin{itemize}
\item[(a)] The subspace $V^{\rm ins}$ is $M$-invariant and
all eigenvalues associated with the eigenvectors from this subspace are one.
\item[(b)] Generically all components of eigenvectors that do not
belong to $V^{\rm ins}$ are non-zero.
\end{itemize}
\end{theorem}

\begin{remark}\label{rem31}
In the study of stability instead of whole heteroclinic cycle or omnicycle one
can consider its building block, see remark \ref{bb}. In such a case the
transition matrix take the form $M^{\rm block}=AM$, where $A$ is the permutation
matrix relating bases $\{{\bf e}_l\}$ and $\{\gamma{\bf e}_l\}$ and
$M$ has the form (\ref{mM}). The statement (a) therefore becomes \cite{op12}
``The subspace $V^{\rm ins}$ is $M$-invariant and the absolute value
of all eigenvalues associated with the eigenvectors from this subspace is one.''
\end{remark}

We call {\it insignificant} the eigenvalues associated with eigenvectors
from $V^{\rm ins}$, and {\it significant} the rest ones. Generically
the absolute values of all significant eigenvalues differ from one.
Below $\lambda_{\max}\ne 1$ denotes the largest significant eigenvalue of
a transition matrix ${\cal M}^{(j)}$.

The construction of transition matrices is identical for heteroclinic cycles
and omnicycles. Whether or not a set of trajectories stays close to a
given sequence of connections is independent of presence or absence of
repeating equilibria or connections in this sequence. Therefore the
theorem of \cite{op12} that gives necessary and sufficient conditions
for a heteroclinic cycle to be f.a.s. can be extended to an omnicycle
as follows.

\begin{theorem}\label{th_4}(Adapted from \cite{op12}.)
Let $M_j$ be basic transition matrices of a collection of maps $\{g_1^m\}$
associated with an omnicycle of type Z.
(For type Z omnicycles the matrices are of the form (\ref{esm}).)
Denote by $j=j_1,\ldots j_L$ the indices, for which $M_j$ involves negative
entries; all entries are non-negative for all remaining $j$. Denote
by $\lambda_{\max}$ the dominant eigenvalue of ${\cal M}^{(j)}$ and
by ${\bf w}_{\max}$ the associated eigenvector.
\begin{itemize}
\item[(a)] If for at least one $j=j_l+1$
the matrix ${\cal M}^{(j)}$ does not satisfy the following conditions
\begin{itemize}
\item[(i)]$\lambda_{\max}$  is real;
\item[(ii)] $\lambda_{\max}>1$;
\item[(iii)] $w_l^{\max}w_q^{\max}>0$ for all $l$ and $q$, $1\le l,q\le N$.
\end{itemize}
then the omnicycle is trail-unstable.
\item[(b)] If the matrices ${\cal M}^{(j)}$ satisfy the above conditions (i)-(iii)
for all $j$ such that $j=j_l+1$, then the omnicycle is trail-stable.
\end{itemize}
\end{theorem}

\section{Fragmentary asymptotic stability: analytical results}\label{sec4}

In this section we apply theorem \ref{th_4} to study dynamics near the network
${\cal X}={\cal C}_L\cup{\cal C}_R$ shown in fig.~\ref{fig1}. The network exists in a
$\Gamma$-equivariant system (\ref{eqsys}) in ${\mathbb R}^6$, where the group
$\Gamma\cong{\mathbb Z}^6_2$ is generated by symmetries inverting the sign of one
of coordinates. The equilibria in the network belong to
coordinate axes, the connections are one-dimensional and belong to coordinate
planes, and the equilibria are stable in directions transverse to the network.

According to theorem \ref{th_4}, stability of a cycle or an omnicycle depends on
the dominant eigenvalue and the associated eigenvectors of the respective
transition matrices. Transition matrices of the cycles ${\cal C}_L$ and ${\cal C}_R$ have
two-dimensional significant subspace, therefore we can calculate the dominant
eigenvalues and the associated eigenvectors.

The significant subspace of the transition matrix of the omnicycle
$${\cal C}_{LR}=(\xi_1\to\xi_2\to\xi_3\to\xi_4\to\xi_1\to\xi_2\to\xi_5\to\xi_6\to)$$
has dimension four, therefore in a general case its eigenvalues can not be found
analytically. To show that the omnicycle can be trail-stable, we calculate
the dominant eigenvalue and the associated eigenvector in the case when
some eigenvalues of ${\rm d} f(\xi_j)$ vanish. Since the conditions for
stability are non-strict inequalities, the inequalities remains true
for small non-vanishing eigenvalues as well. Under our assumption that the chosen
eigenvalues of ${\rm d} f(\xi_j)$ vanish, the conditions for stability of ${\cal C}_{LR}$
are not compatible with conditions for stability of ${\cal C}_L$ and ${\cal C}_R$. So,
we prove that the network ${\cal X}={\cal C}_L\cup{\cal C}_R$ can be f.a.s. while both
subcycles are completely unstable.

\subsection{Stability of cycles ${\cal C}_L$ and ${\cal C}_R$}
\label{sec31}

Since the system under consideration is ${\mathbb Z}_2^6$-symmetric, the eigenvalues
of ${\rm d}(\xi_j)$ coincide with the Cartesian basis vectors ${\bf e}_j$, $1\le j\le6$.
In crossections to the connections comprising the cycle ${\cal C}_L$ we take
the following local bases:
\begin{equation}\label{bases}
H^{out}_{12}:\{{\bf e}_3,{\bf e}_4,{\bf e}_5,{\bf e}_6\},\
H^{out}_{23}:\{{\bf e}_4,{\bf e}_1,{\bf e}_5,{\bf e}_6\},\
H^{out}_{34}:\{{\bf e}_1,{\bf e}_2,{\bf e}_5,{\bf e}_6\},\
H^{out}_{41}:\{{\bf e}_2,{\bf e}_3,{\bf e}_5,{\bf e}_6\}.
\end{equation}
In agreement with (\ref{esm}) the basic transition matrices are:
\begin{equation}\label{m14}
\renewcommand{\arraystretch}{1.5}
\begin{array}{cc}
M_{412}=\left(
\begin{array}{cccc}
b_{132}&1&0&0\\
b_{142}&0&0&0\\
b_{152}&0&1&0\\
b_{162}&0&0&1
\end{array}
\right),
&
M_{123}=\left(
\begin{array}{cccc}
b_{243}&1&0&0\\
b_{213}&0&0&0\\
b_{253}&0&1&0\\
b_{263}&0&0&1
\end{array}
\right)\\
M_{234}=\left(
\begin{array}{cccc}
b_{314}&1&0&0\\
b_{324}&0&0&0\\
b_{354}&0&1&0\\
b_{364}&0&0&1
\end{array}
\right),
&
M_{341}=\left(
\begin{array}{cccc}
b_{421}&1&0&0\\
b_{431}&0&0&0\\
b_{451}&0&1&0\\
b_{461}&0&0&1
\end{array}
\right),
\end{array}
\end{equation}
where
\begin{equation}\label{bbr}
b_{ijk}=-\mu_{ij}/\mu_{ik}
\end{equation}
and $\mu_{ij}$ is the eigenvalue of ${\rm d} f(\xi_i)$ associated with the
eigenvector ${\bf e}_j$.

The cycle ${\cal C}_L$ has only one equilibrium, $\xi_2$, where a transverse
eigenvalue of linearisation is positive. Therefore, theorem \ref{th_4} implies
that the cycle is f.a.s. whenever the matrix $M_L=M_{123}M_{412}M_{341}M_{234}$
satisfies the conditions (i)-(iii).
Multiplying the matrices (\ref{m14}) we obtain that
\begin{equation}\label{maA}
M_L=\left(
\begin{array}{cccc}
a_{11}&a_{12}&0&0\\
a_{21}&a_{22}&0&0\\
a_{31}&a_{32}&1&0\\
a_{41}&a_{42}&0&1
\end{array}
\right),
\end{equation}
where
\begin{equation}\label{bbb}
\renewcommand{\arraystretch}{1.2}
\begin{array}{l}
a_{11}=(b_{314}b_{421}+b_{324})(b_{132}b_{243}+b_{142})+b_{314}b_{431}b_{243}\\
a_{21}=(b_{314}b_{421}+b_{324})b_{132}b_{213}+b_{314}b_{431}b_{213}\\
a_{31}=(b_{314}b_{421}+b_{324})(b_{132}b_{253}+b_{152})+b_{314}b_{431}b_{253}\\
a_{41}=(b_{314}b_{421}+b_{324})(b_{132}b_{263}+b_{162})+b_{314}b_{431}b_{263}\\
a_{21}=b_{421}(b_{132}b_{243}+b_{142})+b_{431}b_{243},\quad
a_{22}=b_{421}b_{132}b_{213}+b_{431}b_{213}\\
a_{32}=b_{421}(b_{132}b_{253}+b_{152})+b_{431}b_{253},\quad
a_{42}=b_{421}(b_{132}b_{263}+b_{162})+b_{431}b_{263}.
\end{array}
\end{equation}
Due to the assumption that the network ${\cal X}$ is stable in the transverse
directions, in (\ref{bbb}) the ratios $b_{ijk}$ are positive, except for
$b_{253}$. Therefore, the entries of matrix (\ref{maA}) are positive, except
possibly for $a_{31}$ and $a_{32}$.

The dominant eigenvalue of $M_L$, $\lambda_{\max}$, is the largest in absolute
value eigenvalue of its upper left $2\times2$ submatrix which we denote by
$M'_L$. The entries of $M'_L$ are positive, implying that
the discriminant of its characteristic polynomial $F(\lambda)$ is positive.
Hence, the eigenvalues of $M'_L$ are real, which implies that $\lambda_{\max}$
satisfies condition (i) of theorem \ref{th_4}. Since
$F(a_{11})=F(a_{22})=-a_{12}a_{21}<0$, the dominant eigenvalue satisfies
$\lambda_{\max}>\max(a_{11},a_{22})>0$. Condition (ii) is satisfied
whenever \cite{pa11}
\begin{equation}\label{cond1}
\max\biggl({a_{11}+a_{22}\over2},
a_{11}+a_{22}-a_{11}a_{22}+a_{12}a_{21}\biggr)>1.
\end{equation}
Let ${\bf v}=(v_1,v_2,v_3,v_4)$ be the eigenvector of $A$ associated
with $\lambda_{\max}$ and $v_1=1$. The component $v_2$ is positive because
\begin{equation}\label{cond21}
v_2=a_{21}v_1/(\lambda_{\max}-a_{22}).
\end{equation}
From (\ref{maA}) the latter two components satisfy
\begin{equation}\label{cond22}
v_3={a_{31}v_1+a_{32}v_2\over\lambda_{\max}-1},\quad
v_4={a_{41}v_1+a_{42}v_2\over\lambda_{\max}-1}.
\end{equation}
Since $a_{41}>0$ and $a_{42}>0$, the latter equality implies that $v_4>0$.
Substituting (\ref{cond21}) into the first inequality in (\ref{cond22}) we
obtain that (iii) is equivalent to
\begin{equation}\label{cond2}
a_{31}(\lambda_{\max}-a_{22})+a_{32}a_{21}>0.
\end{equation}

Therefore, the conditions for fragmentary asymptotic stability of the cycle
${\cal C}_L$ are (\ref{cond1}) and (\ref{cond2}), where $\lambda_{\max}$ is
the largest eigenvalue of the upper left $2\times2$ submatrix of (\ref{maA})
and the dependence of its entries on $\mu_{ij}$ is given by (\ref{coeB}) and
(\ref{bbb}). The permutation of coordinates
$(x_1,x_2,x_3,x_4,x_5,x_6)\to(x_1,x_2,x_5,x_6,x_3,x_4)$ maps ${\cal C}_L$ to
${\cal C}_R$. Hence, the condition for stability of ${\cal C}_R$ are obtained from
those for ${\cal C}_L$ by the permutation of subscripts $(123456)\to(125634)$
of $\mu_{ij}$ in (\ref{coeB}).

\subsection{Trail-stability of ${\cal C}_{LR}$.}\label{sec32}

In this subsection we show that the omnicycle ${\cal C}_{LR}$ can be
trail-stable. Stability of the cycle depends on eigenvalues and eigenvectors
of its $4\times4$ transition matrix. In order to be able to satisfy the
conditions for stability we make two assumptions.
Namely, we assume that the equation (\ref{eq_ode6}) has an additional symmetry
\begin{equation}\label{sym}
(x_1,x_2,x_3,x_4,x_5,x_6)\to(x_1,x_2,x_5,x_6,x_3,x_4)
\end{equation}
and that the eigenvalues
\begin{equation}\label{eig}
\mu_{24},\mu_{26},\mu_{31},\mu_{35},\mu_{36},
\mu_{42},\mu_{45},\mu_{46},
\mu_{52},\mu_{53},\mu_{54},\mu_{61},\mu_{63},\mu_{64}
\end{equation}
are much smaller than other eigenvalues of the Jacobians ${\rm d} f(\xi_j)$.
To prove existence of a system where ${\cal C}_{LR}$ is trail-stable, we first prove
that for vanishing eigenvalues (\ref{eig}) it is possible to identify other
eigenvalues $\mu_{ij}$ such that the transition matrix satisfies
conditions (i)-(iii) of theorem \ref{th_4}. The conditions are not strict and
eigenvalues and eigenvectors of a matrix depend continuously on its entries,
therefore (i)-(iii) remain true for sufficiently small eigenvalues in
(\ref{eig}) as well. For a given set of eigenvalues of ${\rm d} f(\xi_j)$, a system
that has such eigenvalues can be constructed by the procedure discussed in
subsection \ref{sec51}.

Existence of the symmetry (\ref{sym}) implies that the omnicycle is comprised
of two building blocks (see definition \ref{bb}), therefore to study its
trail-stability it is sufficient to consider stability of the return map
$H^{out}_{23}\to H^{out}_{25}$
with respective transition matrix $M_{LR}=M_{125}M_{412}M_{341}M_{234}$.
Let the bases in $H^{out}_{12}$, $H^{out}_{34}$ and $H^{out}_{41}$ be given by
(\ref{bases}) and the basis in $H^{out}_{25}$ be $\{{\bf e}_6,{\bf e}_1,{\bf e}_3,{\bf e}_4\}$.
The basic transition matrices of the cycle ${\cal C}_{LR}$ in a system with
the symmetry (\ref{sym}) and vanishing eigenvalues (\ref{eig}) are
\begin{equation}\label{mLR}
\renewcommand{\arraystretch}{1.7}
\begin{array}{cc}
M_{412}=\left(
\begin{array}{cccc}
b_{132}&1&0&0\\
b_{142}&0&0&0\\
b_{152}&0&1&0\\
b_{162}&0&0&1
\end{array}
\right),
&
M_{234}:=\left(
\begin{array}{cccc}
0&1&0&0\\
b_{324}&0&0&0\\
0&0&1&0\\
0&0&0&1
\end{array}
\right)\\
M_{341}=\left(
\begin{array}{cccc}
0&1&0&0\\
b_{431}&0&0&0\\
0&0&1&0\\
0&0&0&1
\end{array}
\right),
&
M_{125}=\left(
\begin{array}{cccc}
0&0&0&1\\
0&0&b_{215}&0\\
1&0&-1&0\\
0&1&0&0
\end{array}
\right)
\end{array}
\end{equation}
where $b_{ijk}$ are defined in (\ref{coeB}).

Multiplication of the matrices implies that
\begin{equation}\label{mLR1}
M_{LR}=\left(
\begin{array}{cccc}
\tilde a_{11}&0&0&1\\
\tilde a_{21}&0&\tilde a_{23}&0\\
0&\tilde a_{32}&-1&0\\
\tilde a_{41}&0&0&0
\end{array}
\right),
\end{equation}
where
\begin{equation}\label{mLR2}
\tilde a_{11}=b_{162}b_{324},\ \tilde a_{21}=b_{152}b_{324}b_{213},\
\tilde a_{23}=b_{213},\ \tilde a_{32}=b_{413},\ \tilde a_{41}=b_{142}b_{324}.
\end{equation}

Upon the permutation of bases in $H^{out}_{25}$:
$\{{\bf e}_6,{\bf e}_1,{\bf e}_3,{\bf e}_4\}\to\{{\bf e}_6,{\bf e}_4,{\bf e}_1,{\bf e}_3\}$, matrix
(\ref{mLR1}) takes the form
\begin{equation}\label{mLR3}
M_{LR}=\left(
\begin{array}{cccc}
\tilde a_{11}&1&0&0\\
\tilde a_{41}&0&0&0\\
\tilde a_{21}&0&0&\tilde a_{23}\\
0&0&\tilde a_{32}&\tilde a_{33}
\end{array}
\right).
\end{equation}
The eigenvalues of this matrix are
\begin{equation}\label{mLR4}
\lambda_{1,2}={\tilde a_{11}\pm(\tilde a_{11}^2+4\tilde a_{11})^{1/2}\over2},
\quad\lambda_{3,4}={-1\pm(1+4\tilde a_{23}\tilde a_{32})^{1/2}\over2},
\end{equation}
where the indices 1 and 3 correspond to positive signs in front of the square
roots. Therefore, the conditions (i) and (ii) of theorem \ref{th_4} for the
matrix $M_{LR}$ take the form
\begin{equation}\label{mLR5}
\lambda_1>\max(1,|\lambda_4|).
\end{equation}
From (\ref{mLR3}) the components of the eigenvector ${\bf v}=(v_1,v_2,v_3,v_4)$
of $M_{LR}$ associated with $\lambda_1$ satisfy
$$v_2={v_1\tilde a_{41}\over\lambda_1},\ v_3={v_4\tilde a_{23}\over\lambda_1},\
v_4=v_1\tilde a_{31}
\biggl(\lambda_1+1-{\tilde a_{32}\tilde a_{23}\over\lambda_1}\biggr)^{-1}.$$
Therefore, the condition (iii) of theorem \ref{th_4} holds true whenever
\begin{equation}\label{mLR6}
(\lambda_1+1)\lambda_1-\tilde a_{32}\tilde a_{23}>0.
\end{equation}

Finally, we show that conditions (\ref{mLR5}),(\ref{mLR6}) imply that the
cycle ${\cal C}_L$ is completely unstable. (Therefore, by the same arguments as
applied to show trail-stability of ${\cal C}_{LR}$, it remains unstable for a
slightly perturbed non-vanishing eigenvalues listed in (\ref{eig}).)
The transition matrix (\ref{maA}) under the assumptions
of symmetry (\ref{sym}) and for vanishing eigenvalues (\ref{eig}) takes the form
\begin{equation}\label{maAA}
M_L=\left(
\begin{array}{cccc}
a_{11}&0&0&0\\
a_{21}&a_{22}&0&0\\
0&a_{32}&1&0\\
a_{41}&0&0&1
\end{array}
\right),
\end{equation}
where $a_{32}=b_{235}b_{413}<0$. The components of the dominant eigenvector
satisfy $v_3=v_1a_{32}/(\lambda_{\max}-1)$. Hence,
$\lambda_{\max}>1$ implies that $v_1v_3<0$, i.e., for the matrix $M_L$
the conditions (ii) and (iii) of theorem \ref{th_4} can not be satisfied
simultaneously.

\begin{remark}\label{rem1}
To show that the cycle ${\cal C}_L$ can be stable in symmetric system (\ref{sym})
assume that the eigenvalues
\begin{equation}\label{list}
\mu_{13},\ \mu_{16},\ \mu_{24},\ \mu_{26},\ \mu_{3j},\ j=1,3,5;\
\mu_{41},\ \mu_{46}
\end{equation}
are much smaller that the other ones. Under the assumption of vanishing
eigenvalues (\ref{list}) the matrix (\ref{maA}) takes the form
\begin{equation}\label{maAAA}
M_L=\left(
\begin{array}{cccc}
b_{324}b_{142}&0&0&0\\
0&b_{213}b_{431}&0&0\\
0&b_{451}-b_{431}&1&0\\
b_{324}b_{142}&0&0&1
\end{array}
\right).
\end{equation}
The conditions (i)-(iii) are satisfied if
$$\max(b_{324}b_{142},b_{213}b_{431})>1\hbox{ and }b_{451}-b_{431}>0.$$
These conditions are also satisfied for sufficiently small eigenvalues
(\ref{list}).
\end{remark}

\section{Numerical examples}
\label{sec5}

In this section we study numerically behaviour of trajectories near the
heteroclinic networks ${\cal X}={\cal C}_L\cup{\cal C}_R$ shown in fig.~\ref{fig1}.
In subsection \ref{sec51} we discuss construction of a ${\mathbb Z}^6_2$-equivariant
dynamical system, possessing such a network with prescribed eigenvalues
of Jacobians ${\rm d} f(\xi_j)$, $1\le j\le 6$. Then, using results of section
\ref{sec4}, we give numerical examples of attracting ${\cal C}_L$ and ${\cal C}_{LR}$.
Our simulations indicate that when none of ${\cal C}_L$, ${\cal C}_R$ or ${\cal C}_{LR}$
is attracting, nearby trajectories nevertheless can be attracted to the network
displaying regular or irregular switching between ${\cal C}_L$ and ${\cal C}_R$.

\subsection{Construction of the system}\label{sec51}

Consider the system
\begin{equation}\label{sysf}
\dot x_i=x_i(-1+\sum_{1\le j\le 6} \beta_{ij}x_j^2),\quad
\hbox{ where }\beta_{ii}=1,\ 0\le i\le 6.
\end{equation}
By construction (\ref{sysf}) is ${\mathbb Z}^6_2$-equivariant with
the symmetry group generated by the inversions of Cartesian coordinates.
Any coordinate axis, plane or a hyperplane is an invariant subspace of
(\ref{sysf}). It has equilibria $\xi_i=\pm1$ on each of the coordinate axes,
which are attractive in the radial direction.

The restriction of (\ref{sysf}) to the coordinate plane $<{\bf e}_l,{\bf e}_k>$ is
\begin{equation}\label{syskl}
\dot x_l=x_l(-1+x_l^2+\beta_{lk}x_k^2),\ \dot x_k=x_k(-1+x_k^2+\beta_{kl}x_l^2).
\end{equation}
It is known (see, e.g., \cite{op09} and references therein) that if both
$\xi_l$ or $\xi_k$ are stable in the radial direction, one of them is unstable
in the orthogonal direction and the other equilibrium is stable, then there
exists a robust heteroclinic trajectory connecting these equilibria.

The eigenvalues of ${\rm d} f(\xi_l)$ and ${\rm d} f(\xi_k)$ in the directions of
${\bf e}_k$ and ${\bf e}_k$, respectively, are
\begin{equation}\label{lambb}
\mu_{lk}=-1+\beta_{kl},\ \mu_{kl}=-1+\beta_{lk}.
\end{equation}
Each of $\beta_{ij}$ enters into only one expression for $\mu_{kl}$, therefore
we can design system (\ref{sysf}) for any prescribed $\mu_{kl}$ compatibles
with the connections shown in fig.~\ref{fig1}. (Except for radial eigenvalues, which
do not enter into the expressions for stability. These eigenvalues can be
altered by modifying $b_{ii}$.)

\subsection{Numerical results: symmetric system}\label{sec42}

Here we present results of numerical simulations (see fig.~\ref{fig2}) of
solutions to equations (\ref{sysf}) in the case when the system has additional
symmetry (\ref{sym}).
In section \ref{sec3} we derived conditions for stability of heteroclinic cycles
${\cal C}_L$ and ${\cal C}_R$, and omnicycle ${\cal C}_{LR}$, which involve ratios
$b_{ijk}=-\mu_{ij}/\mu_{jk}$. Therefore, we can choose $\mu_{lk}$ to make
the cycles is stable or unstable. The values of $\mu_{lk}$ employed in
simulations shown in the figure are given in table~\ref{tab1}.

\begin{table}[ht]
\begin{tabular}{|c|c|}
\hline plate & eigenvalues \\ \hline
(a) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.2$\\
& $\mu_{21}=-1,\ \mu_{23}=\mu_{25}=1/1.4,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{43}=\mu_{65}=-1.1,\ \mu_{45}=\mu_{63}=-1.7$\\\hline
(b) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.2$\\
& $\mu_{21}=-1,\ \mu_{23}=\mu_{25}=1/1.4,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{35}=\mu_{53}=-0.4,\ \mu_{43}=\mu_{65}=-1.1$\\\hline
(c) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.2$\\
& $\mu_{21}=-1,\ \mu_{23}=\mu_{25}=1/1.4,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{35}=\mu_{53}=-0.4,\ \mu_{43}=\mu_{65}=-1.6$\\\hline
(d) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.2$\\
& $\mu_{21}=-1,\ \mu_{23}=\mu_{25}=1/1.4,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{35}=\mu_{53}=-0.4,\ \mu_{43}=\mu_{65}=-1.4$\\\hline
(e) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.2$\\
& $\mu_{21}=-1,\ \mu_{23}=\mu_{25}=1/1.4,\ \mu_{32}=\mu_{52}=-1,\
\mu_{15}=\mu_{13}=-0.4,\ \mu_{43}=\mu_{65}=-2$\\\hline
\end{tabular}
\caption{Eigenvalues of linearisations employed in numerical simulations
shown in fig.~\ref{fig2}. We do not list $\mu_{ii}=-1$ or eigenvalues that
are equal to -0.01.}\label{tab1}
\end{table}

We start from stable ${\cal C}_L$ and ${\cal C}_{LR}$ (plates (a) and (b)\,) where the
values of $\mu_{lk}$ given in table 1 are taken to satisfy conditions
(\ref{cond1}),(\ref{cond2}) and (\ref{mLR2}),(\ref{mLR4})-(\ref{mLR6}).
For all other considered variants, where $\mu_{lk}$ are taken so that both
${\cal C}_L$ and ${\cal C}_{LR}$ are unstable, the trajectories staying near the network
display regular or irregular behaviour: they can switch between ${\cal C}_L$ and
${\cal C}_R$ making a fixed number of turns (this number depends on parameters of
(\ref{sysf})\,) around one cycle before switching to the other one, or this
number can vary. Examples of such regular of irregular switching are shown in
plates (c)-(e).

For $\tau\to\infty$ the trajectories are attracted by the heteroclinic network,
which is indicated by exponentially increasing subsequent time intervals
between visiting $\xi_2$
\footnote{On the figures only the coordinates $x_3$ and $x_5$ are displayed:
$x_3$ close to one indicates that after $\xi_2$ a trajectory goes to ${\cal C}_L$,
while $x_5$ close to one corresponds to the trajectory going to ${\cal C}_R$.}
(i.e., time intervals needed to make a turn around
${\cal C}_L$ or ${\cal C}_R$). On the plots the horisontal axis (time) is scales
exponentially, implying that temporal behaviour is visually similar to periodic.

\begin{remark}\label{rem00}
In simulations the values of $\beta_{lk}$ are taken such that the eigenvalues of
transition matrices $M_L$ and $M_R$ satisfy $\lambda_L^{\max}>1$
and $\lambda_R^{\max}>1$. We can not prove that these inequalities imply
fragmentary asymptotic stability of the network, however numerical results
indicate that this might be the case, because for all performed runs
trajectories were attracted by the network.
\end{remark}

\begin{figure}
{\Large
\hspace*{-12mm}\includegraphics[width=18cm]{fig2a.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(a)

\vspace{-4mm}

\hspace*{-12mm}\includegraphics[width=18cm]{fig2b.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(b)

\vspace{-4mm}

\hspace*{-12mm}\includegraphics[width=18cm]{fig2c.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(c)

\vspace{-4mm}
\hspace*{-12mm}\includegraphics[width=18cm]{fig2d.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(d)

\vspace{-4mm}
\hspace*{-12mm}\includegraphics[width=18cm]{fig2e.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(e)

\vspace{-2mm}
}

\caption{The dependence on time of $x_3$ (solid line) and
$x_5$ (dashed line).
\label{fig2}}
\end{figure}

\subsection{Numerical results: non-symmetric system}\label{sec52}

In this section we present results of numerical integration of system
(\ref{sysf}) when additional symmetry (\ref{sym}) is not imposed. Typical
behaviour of trajectories near the network is shown in fig.\ref{fig3} and the
respective eigenvalues of ${\rm d} f(\xi_j)$ are given in table \ref{tab2}.
The eigenvalues are chosen such that none of ${\cal C}_L$ or ${\cal C}_R$ is
f.a.s.

\begin{table}[ht]
\begin{tabular}{|c|c|}
\hline plate & eigenvalues \\ \hline

(a) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.2,\ \mu_{21}=-1,\ \mu_{23}=1/1.4$\\
& $\mu_{25}=1/1.5,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{43}=1.8,\ \mu_{65}=-1.4,\ \mu_{35}=-0.4,\ \mu_{53}=-0.5$\\\hline
(b) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.4,\ \mu_{21}=-1$\\
& $\mu_{23}=\mu_{25}=1/1.4,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{15}=-0.4,\ \mu_{43}=2.9,\ \mu_{65}=-1.6$\\\hline
(c) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.4$\\
& $\mu_{21}=-1,\ \mu_{23}=\mu_{25}=1/1.4,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{43}=2.9,\ \mu_{65}=-1.6$\\\hline
(d) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.2,\ \mu_{21}=-1,\ \mu_{23}=1/1.4$\\
& $\mu_{25}=1/1.7,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{35}=\mu_{53}=-0.4,\ \mu_{43}=-1.6,\ \mu_{65}=-1.1$\\\hline
(e) & $\mu_{12}=\mu_{34}=\mu_{41}=\mu_{56}=\mu_{61}=1,\ \mu_{14}=\mu_{16}=-1.2,\ \mu_{21}=-1,\ \mu_{23}=1/1.4$\\
& $\mu_{25}=1/1.5,\ \mu_{32}=\mu_{52}=-1.5,\
\mu_{35}=\mu_{53}=-0.4,\ \mu_{43}=-1.7,\ \mu_{65}=-1.5$\\\hline
\end{tabular}
\caption{Eigenvalues of linearisations employed in numerical simulations
shown in fig.~\ref{fig3}. We do not list $\mu_{ii}=-1$ or eigenvalues that
are equal to -0.01.}\label{tab2}
\end{table}

A regular behaviour (examples are shown in plates (a) and (d)\,) can be
described as a repeating
pattern comprised of $n_L$ iterations along ${\cal C}_L$ followed by $n_R$
iterations along ${\cal C}_R$. Apparently, this indicates that the respective
omnicycle, which can be labelled as $(n_L,n_R)$, is f.a.s.
Often we observe less regular
behaviour (as shown in plates (b), (c) and (e)\,) which either indicate chaos
or a long omnicycle.
Due to the presence of numerical errors, it makes no sense to investigate
which kind of behaviour, chaotic or periodic, takes place: what we see in
simulations may be different from what really takes place in the system.

\begin{figure}
{\Large
\hspace*{-12mm}\includegraphics[width=18cm]{fig3a.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(a)

\vspace{-4mm}

\hspace*{-12mm}\includegraphics[width=18cm]{fig3b.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(b)

\vspace{-4mm}

\hspace*{-12mm}\includegraphics[width=18cm]{fig3c.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(c)

\vspace{-4mm}
\hspace*{-12mm}\includegraphics[width=18cm]{fig3d.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(d)

\vspace{-4mm}
\hspace*{-12mm}\includegraphics[width=18cm]{fig3e.ps}

\vspace*{-18mm}\hspace*{138mm}$\tau$

\vspace*{-5mm}\hspace*{65mm}(e)

\vspace{-2mm}
}

\caption{The dependence on time of $x_3$ (solid line) and
$x_5$ (dashed line).
\label{fig3}}
\end{figure}

\section{Conclusion}\label{sec6}

In this paper we studied behaviour of trajectories near a network in
${\mathbb R}^6$ comprised of two heteroclinic cycles. We derived condition for f.a.s.
of these heteroclinic cycles, proved that the network can be f.a.s. while
both cycles are completely unstable and presented results of numerical simulations
indicating that for a positive measure set of initial conditions the behaviour
of nearby trajectories is apparently chaotic.

To prove stability of the network we introduce the notions of an omnnicycle
and trail-stability, which were implicitly used in earlier studies of behaviour
of trajectories near a heteroclinic network. The definition of
omnicycle is similar to the one of heteroclinic cycle, except that the equilibria
and heteroclinic connections are not required to be distinct. Hence, the
conditions for trail-stability of an omnicycle are identical to the condition
of f.a.s. of a heteroclinic cycle, for type Z objects they
were proven in \cite{op12}.

To prove possibility of the existence of an attracting omnicycle we have assumed
that some
eigenvalues of linearisations are much smaller than the others, which allows
us to calculate eigenvalues and eigenvectors of transition matrices.
The assumption of smallness of some of the eigenvalues of linearisations
can be useful in other studies of stability of cycles, omnicycles or
heteroclinic networks in high-dimensional spaces, where direct calculation
of eigenvalues is not possible due to large dimensions of transition matrices.

\subsection*{Acknowledgements}

I am grateful to Alexander Lohse for a discussion of definition \ref{defn1}.

redpajama_set_name: RedPajamaArXiv


Large gender gaps exist in access, learning achievement and continuation in education in many settings, most often at the expense of girls, although in some regions boys are at a disadvantage. Despite progress, more girls than boys still remain out of school - 16 million girls will never set foot in a classroom (UNESCO Institute for Statistics) - and women account for two thirds of the 750 million adults without basic literacy skills.
Poverty, geographical isolation, minority status, disability, early marriage and pregnancy, gender-based violence, and traditional attitudes about the status and role of women, are among the many obstacles that stand in the way of women and girls fully exercising their right to participate in, complete and benefit from education.


In this song Nina Simone combined the Richard Rodgers song with the melody of the popular carol "Good King Wenceslas".
It was released on her 1958 debut album "Little Girl Blue".
My piano transcription contains two versions of the easy prelude. (Easier album version and concert version).
I am also working on a more difficult arrangement of this song including the voice melody.


text:
Search Ads Placement on Facebook – What Is It?
by: Gina Ingoglia January 8, 2020
Facebook has continually evolved, starting as another social media network and later becoming a modern news source. Now the technological giant safely claims the role of a marketplace as well. Introduced in 2016, the Marketplace function allows users to buy and sell products locally or shipped to/from your door. With 2.41 billion active users each month, it doesn't make much sense not to utilize the platform to generate interest and traffic to your business via ads.
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In 2016, Facebook revealed that it was nearing its ad load, or the capacity of ads it could incorporate without negatively impacting users. The news implied that this lack of space for ads could affect the company's earnings. But, with the reintroduction of search ads, not only does this introduce another source of revenue for Facebook, but it opens another avenue for businesses to entice consumers. And after explaining how this all works, you'll see why you should be leveraging this placement, especially in a multi-channel strategy.
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In-Article – Instant Articles on the app.
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Additional placement is just another way for advertisers to make contact with their audience and get their ads in front of these users. This marketing strategy is a news-worthy decision, because, as the Facebook landscape becomes saturated with ads, opening up new placements gives advertisers more options to reach their target audience. As a result of initiating access to search ads, the platform is giving advertisers a way to target users who are actively looking for their service or product.
Thus, an individual that's already shopping is more likely to become a buyer than a user that might be drawn into browsing. Through search ad placement, businesses can hone in on the demographic with a higher probability of completing a sale.
How Can I Best Leverage this Placement?
With the option to manually or automatically run advertisements on top of choosing between platforms, Facebook marketing can become overwhelming. For those that may be uncertain as to how to make the most of their Facebook ads, consider using the Power 5.
What is the Power5?
Instead of worrying about every detail yourself, Facebook has detailed five automated ad tactics for success. Most effective when employed concurrently, Facebook's self-proclaimed "playbook for online sales excellence" has the "ability to transform ad performance and scale across the Facebook Family of App." Focusing on automatic functions unquestionably reduces the margin for human error as well as aims to reduce the squandering of resources.
As defined by Facebook, the Power5 entails auto advanced matching, dynamic ads, account simplification, campaign budget optimization, and, most relevant to search ads, automatic placements.
Auto Advanced Matching
With automatic advanced matching, marketers don't need to worry about doing any coding. All that users have to do is adjust the Events Manager settings. The pixel will comb through the business's website for customer information and details about the action that took place. The data is only kept long enough to analyze how users reacted to your ad and is then readily disposed of. From this, the business will be able to trace conversions more accurately and expand the size of your audience.
Dynamic Ads
The days of manually refreshing creative are over. Facebook will find the customers for you and show them the continually updated price and availability. Dynamic ads show your products to anyone who has engaged with your company on your app, website, or anywhere on the internet.
All you have to do is upload your product catalog and start your campaign. The dynamic ads will keep running until whenever you choose to stop them. Personalizing ads enables you to show people products relevant or similar to their interests, regardless of whether they have visited your website. And you can remind them about items they didn't purchase to prompt them to follow through.
Account Simplification
Simplicity can be best. In the case of Facebook marketing, this especially rings true. By simplifying your account, machine learning will automatically run tests in real-time. In addition to eliminating any redundancy that could arise if done manually, it will optimize your ad placements and creative for the best results.
Campaign Budget Optimization
Save both time and money by setting a limit across ad sets. While campaign budget optimization can be used for any objective, it is particularly useful if your campaign has numerous ad sets. With campaign budget optimization, rather than having to monitor and shift the budget yourself, the automated process removes the need to adjust ad set funds frequently throughout the day.
Constantly running and in real-time, this feature shifts distributions to ads with better performance automatically. Moreover, if it discovers audience overlap, funds will be reallocated to another set. Overall, your total costs will be lower, and you'll obtain the maximum value. To truly get the most out of your budget, use little to no ad set spend limits.
Automatic Placements
The most efficient way for marketers to leverage their search ad placement is opting into Automatic Placements. Facebook even proclaims that automatic is the most cost-effective choice. The targeting places ads on every default placement, increasing your reach and getting results from the broadest possible range.
Search Term | Specific Product – With automatic placements, advertisers will have the ability to show their ads to people whose search terms match with a specific product. This is ideal for e-commerce clients and is supported by product sales, conversions, and traffic campaign objectives. Although Facebook doesn't allow the advertiser to determine which terms your product shows up for, this still gives advertisers another great option that simultaneously acts as a way to reach their target audience and improve performance.
A Higher Conversion Rate – Research conducted by Facebook showed that advertisements made visible on Instagram, Facebook, and Audience Network had eight times the conversion rate compared to those that were only seen on Facebook, suggesting that multi-platform advertisement is more memorable. Consumer perception of brand quality and favorability were not adversely affected by having to view ads on more than one platform.
Optimize Your Ad Campaigns
Presently only 24.6% of Facebook pages endeavor to go beyond the basic free tools and use paid advertising. Therefore, you should be taking advantage of this newfound access to an area of Facebook that has not yet become oversaturated with competition. Capitalize on the direct link to potential customers who are already seeking your service or product.
If your ad is eligible, you should be able to view "Facebook Search Results" as a placement option. You can visit Facebook's Help Center if you'd like to learn more.
Facebook for Business. About Advanced Matching for Web. https://www.facebook.com/business/help/611774685654668?ref=power5#
Facebook for Business. About the Delivery System: Placements. https://www.facebook.com/business/help/965529646866485?id=802745156580214
Facebook for Business. Ads. https://www.facebook.com/business/help/407108559393196?id=369787570424415
Facebook for Business. Campaign Budget Optimization. https://www.facebook.com/business/m/one-sheeters/campaign-budget-optimization?ref=power5#
Facebook for Business. Dynamic Ads. https://www.facebook.com/business/ads/dynamic-ads?ref=power5#
Facebook for Business. Power5. https://www.facebook.com/business/m/power-five
Marketing Land. Facebook Opens Search Ads to All Advertisers. https://marketingland.com/facebook-opens-search-ads-to-all-advertisers-269321
Social Media Today. Facebook's Adding Search as an Ad Placement Option, Providing New Opportunities. https://www.socialmediatoday.com/news/facebooks-adding-search-as-an-ad-placement-option-providing-new-opportuni/544132/


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Home News Briefs Barings backs Duke Street's investment in Great Rail Journeys
Barings backs Duke Street's investment in Great Rail Journeys
Barings has provided a credit facility to back Duke Street's investment in Great Rail Journeys, a UK-based provider of escorted rail holidays. No financial terms were disclosed.
CHARLOTTE, N.C., Aug. 15, 2018 /PRNewswire/ — Barings, one of the world's leading financial services firms, announced it served as lead agent and sole lender on a senior secured credit facility to support London-based Duke Street's investment in Great Rail Journeys. In addition to the initial term loan extended by Barings, the credit agreement also includes an accordion facility allowing the borrower to tap additional capital as needed.
Established in 1974 and headquartered in York, England, Great Rail Journeys is the world's leading provider of escorted rail holidays, catering primarily to a growing demographic of 55-plus travelers with nearly 400 tours offered across destinations in the U.K. and Ireland, Europe, the Americas, Asia-Pacific and Africa. The company booked vacation tours for more than 50,000 travelers last year.
"Duke Street is pleased to have the support of Barings' global private finance team on our investment in Great Rail Journeys," said Paul Adams, investment director at Duke Street. "Barings' understanding of Great Rail's business model and its ability to fund the entire credit facility helped expedite the process and provide certainty of closing in a timely manner."
"Barings appreciates the opportunity to partner with Duke Street on its investment in Great Rail Journeys and we look forward to additional opportunities to work together in the near future," said Matt Carty, director at Barings. "We are excited about Great Rail's prospects for continued growth globally and believe Duke Street's deep leisure sector experience will help the company execute on the expansion of its rail tour business in promising source markets like the U.S. and the launch of its unique rail-based river cruise offering."
About Duke Street
Duke Street Private Equity has been investing in mature, mid-market Western European businesses for over twenty years, concentrating its investment strategy on four sectors: Consumer, Healthcare, Industrials & Engineering and Services. Duke Street invests in companies with an enterprise value of between £50m – £250m. To learn more, visit www.dukestreet.com.
About Barings
Barings is a $306+ billion* global financial services firm dedicated to meeting the evolving investment and capital needs of our clients. We build lasting partnerships that leverage our distinctive expertise across traditional and alternative asset classes to deliver innovative solutions and exceptional service. Part of MassMutual, Barings maintains a strong global presence with over 1,800 professionals and offices in 16 countries. Learn more at www.barings.com.


Frozen Elsa Anna Olaf Do You Want To Build A Snowman Disney Cartoon Coloring Games. Good and funny play Frozen Elsa Anna Olaf Do You Want To Build A Snowman Disney Cartoon Coloring Games. Use the mouse to select colors and paint the picture.


Chicago: "ZOOM, Series II," 01/24/2005, WGBH Media Library & Archives, accessed April 23, 2019, http://openvault.wgbh.org/catalog/V_8D30654938634EFAB50F8F5CB331BB05.
MLA: "ZOOM, Series II." 01/24/2005. WGBH Media Library & Archives. Web. April 23, 2019. <http://openvault.wgbh.org/catalog/V_8D30654938634EFAB50F8F5CB331BB05>.


This discount catalogue is no longer in effect, although we still have plenty of hellebores for sale at the nursery. The catalogue remains on the website for informational purposes only. If you would like to get an email notification when the 2020 Discount Hellebore, Winter Interest, or Snowdrop Catalogues are posted, please send your full name and cell number (for back up contact use only) to carolyn@carolynsshadegardens.com and let us know if you are local or mail order.
Newly introduced 'Sally's Shell' has all the same features as 'Penny's Pink' but with gorgeous shell pink flowers and bright green leaves with pink veins.
Spring has arrived, so it's time for the annual Discounted Hellebore Offer. Local customers who pre-order hellebores from the selection below will get 10% off their hellebore order. Your plants must be picked up prior to the April 6 Hellebore Extravaganza. We will schedule group pick up times during the weekend of March 29 to 31, but other times will also be available. While here you can shop for all the plants stocked for the April 6 sale and avoid the open house crowds and parking issues.
How to Order (sorry no mail order): Please send an email to carolyn@carolynsshadegardens.com or reply to this email with the plant names, quantities, your name, and phone number. I will send you an acknowledgment of your order within a day. Supplies are limited so order early.
French Kiss– white flowers with raspberry pink edges, shading, and a dark raspberry ruffle around the center.


vitamin one or two months before trying to conceive (this includes undergoing in vitro fertilization). Keep all drugs in a safe place. Yes, you should be taking a prenatal viatmin. Some drugs may need to be taken with food or on an empty stomach. For a completely organic, food-based supplement, we like. These are not all of the side effects that may occur. The best way to know whats best for you is to talk to your doctor. Do not flush down a toilet or pour down a drain unless you are told to. Check Price, as our only fully food-based recommendation, Garden of Life is great for people who prioritize organics and food-sourced nutrients. This change put many of our previous picks above the recommended daily allowance, so we took a closer look at all our products, updated the numbers, and chose new vitamins.
Further information Always consult your healthcare provider to ensure the information displayed on this page applies to your personal circumstances. The most salient question most articles asked was this: If expecting or pregnant women eat healthy diets and take folic acid and vitamin D via daily supplements, do they really need to add additional nutrients in daily pill form? Food and Drug Administration. Never delay or disregard seeking professional medical advice from your doctor or other qualified healthcare provider because of something you have read on WebMD. There wasnt any new data or anything.


The house will stay crazy, but more honest, more balanced and more ... complex. Thus no version 1.whatever, but a version 2.0 !
And meanwhile I will correct a gold/silver HJS somewhere in the house.
, I know that this is important, because otherwise you can't load your saved games. But ... to reach some saving points there will be some work to do. And it feels more correct to add at least one or two pedestals earlier in the mansion.
I'm looking forward to your new version, Freddy.
Some changes are more or less forced! It seems that in HD it is possible to open coloured doors in dark rooms. So I need to change that room seriously.
I also detected a ladder bug somewhere. And at the end, with perfect play, there is a purple key with no use ... So reason enough to get in action!
It seems that in HD it is possible to open coloured doors in dark rooms.
I wonder whether it is deliberate or a mistake by Vern.


The Program Manager aims to ensure excellence in performance in all areas of program implementation in full compliance with SFCG and donor rules and regulations. By supporting the country team and measuring their performance using specific and agreed-upon program implementation and quality measurement tools, the Program Manager shares a diagnosis of country programs' strengths and weaknesses with the Regional Director. This diagnosis will be based on the efficacy and effort by the country team in utilizing organizational tools.
Based on this diagnosis, priorities for each country program are set with the Regional Director. This is done in a manner which consistently builds capacity, strengthens lines of accountability, and enables and inspires talent at all levels within the country program. Accountability of the performance in managing the awards remains with the Country Director, who is supervised by the Regional Director. The Regional Director thus focuses more so on strategic development, new business development, and management of the senior country team.
Ensure that basic SFCG program implementation tools are being used and shared, including start-up and close-out protocols, costed work plans, operational budgets, BvA meetings, and quarterly work plans.
Support programs to speak to results, encouraging an implementation logic that focuses on the objectives of a project and seeks to capture the change as per the theory of change. Facilitating collaboration with ILT when needed.
Based on a sharing of the reports between the PM and the Business Development team (including in-country Program Associates), review and edit reports of country programs prior to submission. Use Quarterly Report Scorecard to track ongoing improvements in staff capacity in reporting; design coaching and capacity building to improve capacity.
Quickly identify obstacles to timely delivery of quality programs and work with country teams and Regional Director and Regional Finance Manager (RFM) to solve obstacles creatively and rapidly. This includes assessing the risks related to overall program performance, as well as efficacy of SFCG procedures, tools and guidelines.
Mobilize and monitor short term consultants when appropriate and delegated by the Regional Director.
Travel to field offices to provide in-country support as needed. Participate in key project and program launch and close out meetings, key program events and workshops.
Track the performance of each country management teams in their implementation of grants, achievement of program and spending targets, and meeting of reporting deadlines for feedback to the RD. Ongoing coaching of the teams to improve performance in all of these areas.
In collaboration with Country Directors, Finance Managers, and Project Managers actively monitor burn rates to ensure that grants are spent out to 90% according to grant agreements and SFCG's grant management policy.
Work in collaboration with specific units in HQ to ensure complete grant documentation, agreement review and signing, budget realignments, cost extensions, etc. as well as grant closure.
Engage support of other departments as needed to solve problems and provide support to country teams in addressing grant management challenges.
Contribute to ongoing improvement of grant management tools, including grant status tracking, donor requirements and regulations, burn rate monitoring, report compilation and collection, and SFCG policy and donor regulations.
Assume interim leadership roles in country programs when appropriate and timely.
Although the Program Manager is not the lead in the business development, the Program Manager will at times be delegated to take on specific pieces of large and strategic proposals, such as engaging and negotiating with local or international partners in the development of a consortium.
Liaise with ILT, GAP, and Communications to enhance the visibility of the region, through back-stopping on success stories and other data reflecting the change and transformation of our work.


Freshman 6'6-305. Playing DL, but his future will be right tackle. Good initial get off. Explosive. Good lateral range. Needs to maintain a good base for balance. Strong. Powerful. Uses his hands. Can play low, but needs to be a better bender. Runs okay, but will get better. Needs to work on feet, flexibility. and quickness in off season. Like to see him at 285. Chance to a very good right tackle. MSR watchlist as a top player in Class of 2020.


Market analysts have been talking about greens shoots for many months now. And although optimism may be good in general, it should be based on something. Some proof. Transactions evidencing the confidence of market parties, for instance. We have not seen too many of those yet.
People talking about green shoots of recovery have been spending too much time looking at the bulbs in their garden.
With transactions lacking for many months now, real estate markets have been looking for new price equilibriums. For direct real estate investments (bricks & mortar) this turned out to become a search for new bottoms. Such new price lows appeared both in terms of yield shift and in terms of rental value. The latter effect, I believe, is still ongoing or may be starting in some markets now. Once rental levels have stabilised with confidence, the investment climate becomes healthy again. Investment yields took a hike of 100 basis points -and much more- from their previous levels and I believe they have stabilised by now.
Stability is key for recovery. The market needs some stability in order to gain confidence. And confidence is needed for transactions to pick up. Nothing's new so far. But quite a few recent news items back my belief that dinner is being served right now. Anyone with some appetite is invited to pick a table that is stable enough to carry loads of flavourous courses. Markets are anticipating on their recovery and that the first movers have finished their starter dishes already and are looking for a tasty main course.
Confident that European property markets are recovering, investors are pouring money into Germany's open-ended real-estate funds, which are flush with cash and on the prowl for acquisitions.
The relative stabilisation of prime yields has reflected an improvement in investor sentiment. London and Paris have seen high levels of investor interest for the best assets in prime locations and with secure income in terms of lease length and covenant strength.
Take-up levels continue to be down on historic averages, headline rents are under continued downward pressure and incentives are increasing. Investors' focus on long leases and strong covenants means that competitive bidding for these assets may not fully recognise the extent of weakness in the occupier base.
After two years of dramatic decline, capital values are now stabilising for UK property, according to the UK Quarterly Snapshot from Aberdeen Property Investors. Investor demand is returning, aided by sterling's fall over the past year. The high income return now available from UK property relative to other asset classes also looks very attractive, despite continuing falls in rental values.
In the occupier market, the pace of rental decline has begun to show signs of stabilising. Rents are currently falling by an annualised 9% and 6% respectively in the retail and industrial sectors. The office sector is experiencing a sharper pace of decline, at an annualised 15%, although this is down from the 20% seen last quarter.
CBRE confirms that rents are falling across European property markets, but yields stabilizing in some markets.
The downturn in real estate values across Europe has so far been predominantly driven by increases in yields. However, we have now clearly entered a period in which rents are the driving force behind the property value corrections.
It is too early to signal the end of the period of rising yields, but equally the scale and incidence of yield increases are now a lot more patchy than they have been for a while. While investment turnover remains relatively low, there appears to be growing interest in several areas of the market, supported by the repricing that has already taken place.
Listed property funds, such as Dutch Wereldhave, are anticipating on market recovery. Wereldhave eyes buys after a successful EUR 230M bond issue.
In the coming months or at the start of 2010 we are going to look at buying opportunities. […] The extra proceeds from the bond issue would give the company a chance to act quickly when buying opportunities arose.
So, this dinner may be about opportunities ("All you can eat, for just over 8%!"). In that case, it doesn't tell you so much about the underlying (economic) drivers that will affect the value of the assets one way or another. The question whether economic recovery is underway, is still hard to answer. However, I do believe that low interest rates, combined with surging stock prices (e.g.: S&P 500 is up 50% from their March 2009 low) are fueling consumers' and investors' confidence alike. As long as some stability comes to the markets and remains, these are interesting times to grab a bite.
Next EXPO real 2009: Crisis? What crisis?
Rates will stay low for a while. Read the FED comments that were published this very week!
Real Estate started to correlate with other listed equity as investors were doubting cash flows would stay strong and keep growing. Economic turmoil raised risk premia to levels not heard of in decades. Liquid assets were slaughtered as cash became the new bubble. Correlations were high meaning stable strong cash flow generative "money machines" such as real state and telecoms were sold (almost) just as much.
The question to ask is: will economic growth falter? Answer, it will not be any of the consensus letter soup: V, W, -reverse-J, or the bears' favorite: the dreaded "L", rather it would be something more mundane and boring: a square root. No depression, but sub-trend growth for some time to come. Is such an environment bad for real estate?
Anyone with an opinion is invited to respond!