text:
\section{Introduction}\label{Sec:Introduction}
Complex networks have largely enriched our understanding of a variety of complex dynamical systems in many fields, such as biology, ecology, climatology, sociology, and others~\cite{Newman2018,Yanchuk2021a}.
By modeling real-world systems as networks in which collections of dynamic nodes are connected by static or adaptive edges, one can study collective behaviors both analytically and numerically~\cite{Blekhman1988,arenas2008synchronization,NEW03,gross2008adaptive,Cabral2022a}.

Synchronization is a ubiquitous dynamical phenomenon that has been observed in many natural and engineering systems~\cite{arenas2008synchronization,pikovsky_rosenblum_kurths_2001,osipov2007synchronization,boda2013kuramototype,chhabria2018targeted,ikeguchi2019analysis}.
Different types of synchronous patterns have been identified involving complete synchronization \cite{PEC98} (oscillators' states become asymptotically the same with time), cluster synchronization (a network splits into groups of synchronous elements)~\cite{dahms2012cluster,Lucken2012a}, special types of spatial coexistence of coherent  and incoherent states~\cite{abrams2004chimera,omelchenko2011loss,kasatkin2017selforganized}, and many others.
Various patterns have been found in experimental contexts, such as optoelectronic networks~\cite{soriano2013complex}, chemical networks~\cite{tinsley2012chimera}, neural networks~\cite{hammond2007pathological},  ecological~\cite{blasius1999complex}, and climate systems~\cite{tsonis2004architecture}.

Apart from synchronization, multistability -- the coexistence of several asymptotic stable states (attractors) for a given set of parameters -- is another intriguing phenomenon which has been studied for decades in modern nonlinear science~\cite{feudel2008complex,pisarchik2014control,dudkowski2022multistabilityb}.
The final state of a system with multistability depends crucially on initial conditions.
Multistability has also been observed in many areas of science, such as nonlinear optics~\cite{arecchi1991rate},
neuroscience~\cite{foss1996multistability},
climate dynamics~\cite{power1993multiple},
laser physics~\cite{masoller2002noiseinduced}, electronic oscillators~\cite{borresen2002further},
and in different classes of systems,
such as weakly dissipative systems~\cite{feudel1996map}, systems with time delays~\cite{Yanchuk2010a,balanov2005delayed}, and coupled systems~\cite{feudel1998dynamical}.

Understanding the emergence of co-existing attractors is an important issue, and controlling multistability is an even more difficult task.
When the number of co-existing attractors increases infinitely, EM emerges.
In coupled systems, the presence of EM has been found to be closely related to partial synchrony~\cite{hens2015extreme}.
By designing a specific coupling scheme to achieve partial synchrony, one can obtain infinitely many coexisting stable states~\cite{hens2015extreme,sun1999uncertain,ngonghala2011extremea,hens2012how,pal2014multistablea}.
Apart from the conservative cases, a common reason for the occurrence of EM in networks is time-reversibility, a special type of spatio-temporal symmetry \cite{Politi1986,Pikovsky-Rosenau,Lamb1998,ASH16a,Burylko2018}.

Despite the extensive literature on multistable dynamical systems, the emergence of multistability or EM in networked dynamical systems remains a challenging problem due to a large number of possible routes to EM, some of which have yet to be discovered.
Analytical and numerical challenges arise from the diversity of coupling topologies and the complexity of individual models.

In this work, we address the multistability problem in a mathematical model of coupled clocks suspended on a rotating disc and additionally coupled with springs.
The interaction of the clocks with the disc provides the global coupling among all clocks and therefore influences their behavior, similar to the interaction of the pedestrians with the bridge in the famous effect of crowd synchrony on the Millennium Bridge~\cite{Strogatz2005}.
Such a global scheme has also proved useful in uncovering complex transient states~\cite{dudkowski2020transient}.
The oscillating clocks are also locally coupled via springs.
In Ref.~\onlinecite{dudkowski2020small}, a similar system of three coupled clocks was studied.

The following main results are obtained in this work:
\begin{itemize}
  \item We generalize the system of three coupled clocks \cite{dudkowski2020small} into a network-coupled scenario allowing arbitrary coupling configurations.
  \item We investigate how different coupling topologies affect the multistability in systems of three and four coupled clocks.
  We observe that more symmetric coupling topologies can lead to more complex dynamics with higher multistability.

  A particularly reach appears to be the ``cross-coupling'' structure with ``diagonal'' spring couplings in the system of four coupled clocks. In such a case, we observe EM that combines continuous family of stable attractors with different phase relations between the clocks. We provide an analytical and numerical description of this new phenomenon.
  \item
  Furthermore, we discuss how the discontinuity of the escapement mechanism affects the multistability in the system.
  The clocks within certain coupled groups (clusters) remain either silent or oscillating and in-phase synchronized due to the switching of the escapement mechanism. This leads to three qualitatively different discontinuity-induced types of attractors.
\end{itemize}

\begin{table*}[t]

    \caption{Parameters for the $N$-pendulums of system (\ref{Eq:general_model}).}\label{Tab:Symbol}
    \setlength{\tabcolsep}{10pt}
    \begin{tabularx}{\textwidth}{llX}
        \hline
        Parameter & Value & Definition  \\
        \hline
        $B_0$ & 5.115\ [kgm$^2$] & Support's moment of inertia \\
        $k_\varphi$ & 17.75\ [N/m] & Stiffness coefficient of springs \\
        $k_\theta$ & 34\ [Nm] & Stiffness of the spring connecting the base
        and the unmoving support\\
        $c_\theta$ & $\Delta=ln(2)$ &Damping of the supporting base \\
        $c_\varphi$ & 0.01\ [Nms] & Damping
        of the damper connecting the base and the unmoving
        support \\
        $m$ & 1\ [kg] & Mass of each pendulum clock\\
        $l$ & 0.24849\ [m] &  Lengths of the pendulums\\
        $g$ & 9.81\ [m/s$^2$] & Gravity acceleration\\
        $d$ & 1\ [m] & Distance between $O$ and $S_i$ $(i=1,2,...,N)$\\
        {$\alpha_i=\sphericalangle ({Ox},~\overline{OS_i})$}& $(\frac{360}{N})^\circ=\frac{2\pi}{N}$ & $Ox$ denotes the positive $x$ half axis\\
        $M$ & 0.075\ [Nm] & Fixed external momentum\\
        $\varepsilon_0$ & $5^\circ=5*\pi/180$ & Escapement threshold\\
        \hline
    \end{tabularx}
\end{table*}

\section{Model and measures}\label{Sec:Model and measures}
\subsection{General model}\label{Subsec:General model description}
We first present a mathematical model of the $N$ coupled pendulum clocks suspended on a rotating disc, see~Fig.~\ref{Fig:Model}(A).
The rotating disc provides a global coupling, while the springs allow for arbitrary local coupling structure.
Our model is a generalization of the system of three pendulums from Ref.~\onlinecite{dudkowski2020small}.

\begin{figure*}[!htbp] \centering
\includegraphics[width=0.95\textwidth]{Fig/Model.png}
    \caption{
    (A) The scheme of $N$ coupled identical pendulum clocks (shown as circles with different colors) suspended at evenly distributed black points $S_i$, $i = 1, 2,...,N$, on an oscillating supporting base (the $xy$ plane).
    The local coupling is realized using springs between the clocks.
    (B)--(D): For $N=3$, three types of coupling structures of springs include all-to-all, asymmetric, and symmetric topologies.
    (E)--(G): For $N=4$, three types of coupling structures of springs include all-to-all, asymmetric, and symmetric topologies.
    }
    \label{Fig:Model}
\end{figure*}

The supporting base is placed at the origin $O$ of the $xy$ plane and it can oscillate freely around the axis perpendicular to the plane of Fig.~\ref{Fig:Model}; the angular deviation of the base is $\theta$.
The properties of the base are described by the moment of inertia $B_0$ [kgm$^2$], the stiffness of the spring connecting the base and the static support $k_\theta$ [Nm], and the damping $c_\theta$ [Nms].
Identical pendulums (marked colored filled circles) are suspended at evenly distributed black points $S_i$, $i = 1, 2, 3,...,N$, i.e., the angles between ${OS}_i$ and ${OS}_{i+1}$ (index $i$ is considered mod $N$) are ${{2\pi}/{N}}$.
The angles $\alpha_i=\sphericalangle({Ox},~\overline{OS}_{i})$ characterize the angular position of the suspension points $S_i$, where $Ox$ is the positive $x$ half-axis.
The parameter $d=|\overline{OS}_i|$ $(i=1,2,...,n)$ is the distance between the origin $O$ and each suspension point $S_i$.
Each pendulum is described by the angle displacement $\varphi_i$, the mass $m$ [kg], the length $l$ [m], and the damping coefficient $c_\varphi$ [Nms].
The stiffness coefficients of the springs are $k_\varphi$ [N/m].
The description and the values for all parameters are summarized in Table~\ref{Tab:Symbol}.

The equations of motion of the $N$ coupled pendulums is given by the following system:
\begin{widetext}
    \begin{equation}\label{Eq:general_model}
    \begin{array}{ll}
    \displaystyle
    (B_0+n mr^2)\ddot{\theta}+k_\theta\theta+c_\theta\dot{\theta}+
    \sum\limits_{i=1}^nmr\{l[\ddot{\varphi}_i \sin(\varphi_i-\theta-\alpha_i)+\dot{\varphi}^2_i \cos(\varphi_i-\theta-\alpha_i)]+g\cos(\alpha_i+\theta)\}+\Delta V_{\theta}=0,\\
    ml^2\ddot{\varphi}_i+ mgl\sin{\varphi_i}+c_ \varphi{\dot{\varphi}_i}+mrl[\ddot{\theta} \sin(\varphi_i-\theta-\alpha_i)-
    \dot{\theta}^2\cos(\varphi_i-\theta-\alpha_i)]+\Delta V_{\varphi_i} =M_{E_i}, \\
    \end{array}
    \end{equation}
\end{widetext}
where $i=1,2,...,N$.

The build-in escapement mechanism produces the moment of force, which is modeled by the discontinuous functions $M_{E_i}$, $i = 1, 2,3,...,N$ \cite{czolczynski2011why,kapitaniak2012synchronization}.
These functions depend not only on the displacement
$\varphi_i(t)$, but also on the position of the $i$-th mechanism's cogwheel versus the mechanism's pallet $\sigma_i(t)$:
\begin{equation}\label{Eq:escapement_mechanism}
M_{E_i}=
\begin{cases}
M&: \sigma_i=1\wedge 0<\varphi_i<\varepsilon_0,\\
-M&: \sigma_i=2\wedge -\varepsilon_0<\varphi_i<0,\\
0&: \mathrm{otherwise.}
\end{cases}
\end{equation}
Here $M=0.075$ [Nm] represents the value of the external momentum, while $\varepsilon_0 = 5.0^\circ$ denotes the escapement's threshold (the mechanism turns off as the pendulum exceeds this threshold).
In fact, $\sigma_i(t)$ become additional discrete-valued variables in the system that are influencing the system's dynamics via the terms $M_{E_i}$ and which are changing discontinuously according to the following rules:
\begin{itemize}
    \item[(I)] When a pendulum $\varphi_i$ crosses the escapement threshold at some time moment $t^*$: $\varphi_i(t_*) = \varepsilon_0$ with increasing $\varphi_i$, i.e., $\dot \varphi_i(t_*)>0$, the variable $\sigma_i(t)$ is set to 2 for all $t\in [t^*,t_\mathrm{e})$, where $t_\mathrm{e}$ is the time of a next event.
    \item[(II)] When the pendulum $\varphi_i$ crosses the escapement threshold $\varphi_i(t_*) = - \varepsilon_0$ with decreasing $\varphi_i$, i.e., $\dot \varphi_i(t_*)<0$, the variable $\sigma_i(t)$ is set to 1 for all $t\in [t^*,~t_\mathrm{e})$, where $t_\mathrm{e}$ is the time moment of a next crossing event.
\end{itemize}
In this way, the variables $\sigma_i(t)$ are piece-wise constant with the possible discrete values 1 or 2, which change discontinuously when either event (I) or (II) occurs.

The terms $\Delta V_{\varphi_i}$ and $\Delta V_{\theta}$ in model (\ref{Eq:general_model}) describe the moments of forces from the coupling springs.
These two terms can be written explicitly using the following terms: $s_{ij}$, the constant distance between the $i$-th and $j$-th clocks when the system stays still,
and $\hat{s}_{ij}(t)$, the time-dependent distance between the $i$-th and $j$-th clocks for the moving system:
\begin{widetext}
\begin{equation}\label{Eq:springs_moments_of_forces_and_distance}
    \begin{array}{ll}
    \displaystyle
    s_{ij}=r\sqrt{2(1-\cos(\alpha_i-\alpha_j))},\\
    \hat{s}_{ij}=\sqrt{s_{ij}^2+2l^2(1-\cos(\varphi_i-\varphi_j))+8lr
    \sin\left(\frac{\varphi_i-\varphi_j}{2}\right)
    \sin\left(\frac{\alpha_i-\alpha_j}{2}\right)
    \sin\left(\frac{\varphi_i+\varphi_j-\alpha_i-\alpha_j}{2}-\theta\right)},\\
    \Delta V_{\theta}=2lrk_\varphi\sum\limits_{i=1}^n\sum\limits_{j=1}^na_{ij}
    \left(1-\frac{s_{ij}}{\hat{s}_{ij}}\right)
    \sin\left(\frac{\varphi_i-\varphi_j}{2}\right)\sin\left(\frac{\alpha_j-\alpha_i}{2}\right)\cos\left(\frac{\varphi_i+\varphi_j-\alpha_i-\alpha_j}{2}-\theta\right),\\
    \Delta V_{\varphi_i}=\sum\limits_{i=1}^na_{ij}k_\varphi l\left(1-\frac{s_{ij}}{\hat{s}_{ij}}\right)
    \left[l\sin(\varphi_i-\varphi_j)+2r
    \sin \left(\frac{\alpha_i-\alpha_j}{2} \right)
    \sin\left(\varphi_i-\frac{\alpha_i+\alpha_j}{2}-\theta\right)
    \right],
    \end{array}
\end{equation}
\end{widetext}
where $(a_{ij})$ is the coupling matrix via the springs, i.e., $a_{ij}=1$ if the pendulum $i$ is connected with the pendulum $j$ via a spring and $a_{ij}=0$ otherwise. $a_{ii}=0$ since there are no self-loops.

\begin{figure*}[!htbp]
\centering
\includegraphics[width=0.95\textwidth]{Fig/Figure_1.png}
\caption{
Distributions of the order parameter $r$ for different coupling topologies.
(A)-(C) show for 3-coupled pendulums ($N=3$), the identified multistabilities for all-to-all, asymmetric, and symmetric coupling structures (see right-upper corners), respectively.
For each coupling topology, 5000 order parameters are obtained from 5000 simulations using random initial conditions.
Parameters for $N=3$ are fixed as in Table \ref{Tab:Symbol}, in particular, $\alpha_1 = \frac{\pi}{2},$ $\alpha_2 = \frac{7\pi}{12}$, and $\alpha_3=\frac{11\pi}{12}$.
Each vertical bar corresponds to a potential attractor.
Similarly, (D)-(F) are multistabilities for 4-coupled pendulums ($N=4$), based on 1000 order parameters obtained from 1000 simulations.
Parameters for $N=4$ are also fixed as in Table \ref{Tab:Symbol}, in particular, $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$.
For $N=3$, compared with (A) and (C), the asymmetric coupling structure in (B) decreases the dynamical complexity; while for $N=4$, compared with (E), (F) shows that the symmetric coupling can increase the dynamical complexity due to the emergence of EM.
}
\label{Fig:Coupling_R}
\end{figure*}

The influence of different coupling structures on collective dynamics has not been systematically reported for this model.
In the remaining part of this paper, we consider the cases $N=3$ (Figs.~\ref{Fig:Model}(B)-(D)) and $N=4$ (Figs.~\ref{Fig:Model}(E)-(G)).
In particular, we focus on the following questions:
\begin{itemize}
  \item[\textbf{(1)}] How do different coupling topologies alter synchronization states and their basins of attraction?
  \item[\textbf{(2)}] Is there extreme multistability in the coupled pendulum model? If so, what is its origin?
\item[\textbf{(3)}] What are the effects of the discontinuity of the escapement mechanism on the dynamics and multistability?
\end{itemize}
We fix the parameters as in Table~\ref{Tab:Symbol}.
For $N=3$, the three considered types of coupling structures of springs include all-to-all (Fig.~\ref{Fig:Model}(B)), asymmetric (Fig.~\ref{Fig:Model}(C)), and symmetric (Fig.~\ref{Fig:Model}(D); mirror symmetry with respect to the vertical axis) topologies.
Identical pendulums are suspended at evenly distributed points $S_i$, $i = 1, 2, 3$ with $\sphericalangle(\overline{OS}_i,~\overline{OS}_{i+1})=120^\circ$.
For $N=4$, we also consider three types of coupling structures of springs: all-to-all (Fig.~\ref{Fig:Model}(D)), asymmetric (Fig.~\ref{Fig:Model}(E)), and symmetric (Fig.~\ref{Fig:Model}(F)) topologies.
Here also the identical pendulums are suspended at evenly distributed points $S_i$, $i = 1, 2, 3, 4$ with $\sphericalangle(\overline{OS}_i,~\overline{OS}_{i+1})=90^\circ$.

We use Monte Carlo sampling and two classical measures (order parameter and mean frequencies) for the numerical quantification of synchronization states and the analysis of their basins of attraction.

\subsection{Measures}\label{Subsec:Measures}
\textit{Order parameter.}
We visualize the dynamics of the synchronization transitions with the Kuramoto order parameter:
\begin{equation}\label{Eq:order_parameter}
R(t)=\frac{1}{N}\sum\limits_{i=1} ^{N}e^{i\varphi_i(t)},
\end{equation}
where $N$ is the number of oscillators.
When $|R(t)|=1$ ($|R(t)|\approx0$), oscillators are in the complete synchronization (disordered) state.
The degree of synchronization in numerical simulations is quantified using the averaged value of the order parameter:
\begin{equation}\label{Eq:order_parameter_used}
r = \frac{1}{T_\mathrm{av}}
\int_{T_\mathrm{tr}}^{T_\mathrm{tr}+T_\mathrm{av}}
|R(t)|dt.
\end{equation}
over the time interval $T_\mathrm{av}=50$ after a sufficiently long transient time $T_\mathrm{tr}$.

\textit{Mean frequency.}
The mean oscillation frequency of a pendulum is given as:
\begin{equation}\label{Eq:mean_frequency_used}
\langle\omega_i\rangle=\frac{2\pi n_i}{T_\mathrm{av}},
\end{equation}
where the same time interval of $T_\mathrm{av}=50$ is applied and $n_i$ represents the number of complete oscillations of $i$th clock within this interval.
The number of complete oscillations can be computed using the number of crossings of the Poincare map $\varphi_i=0$ or $\varphi_i=\varepsilon_0$.
The mean frequency is calculated using the last $T_\mathrm{av}=50$ time units after a sufficiently long transient time $T_\mathrm{tr}$.

\section{Collective dynamics for different coupling topologies}\label{Sec:Collective dynamics for different coupling topologies}
We first conduct various simulations of the system for random initial conditions. More specifically, we choose the following initial conditions $[\theta^0=0.01, \phi_1^0, \dots, \phi_n^0,  \dot{\theta^0}=0, \dot{\phi_1^0}=0, \dots, \dot{\phi_n^0}=0]$, where $\phi_1^0,\dots,\phi_n^0$, are chosen randomly from the interval $[-\pi, \pi)$. For the case $N=3$, simulations with 5,000 different  initial conditions are performed with the integration time 15,000, and the last 50 time units are used for the calculation of the order parameter and the mean frequency.
For $N=4$, we perform 1,000 runs with the integration interval 10,000, and the transient 9,950.
The $r$ and $\langle\omega_i\rangle$ from Eqs. (\ref{Eq:order_parameter_used}) and (\ref{Eq:mean_frequency_used}), respectively, are used to estimate the synchronization state (attractor) in each simulation.
We found that further increase of the number of runs and the integration interval does not affect the obtained results qualitatively.

Figure~\ref{Fig:Coupling_R} shows the distribution of the order parameter $r$ for different initial conditions.
This distribution reveals the possible number of different attractors.
Figures~\ref{Fig:Coupling_R}(A)-(C) correspond to the coupling structures of three clocks in the Figs.~\ref{Fig:Model}(B)-(D), respectively.
Here we see finitely many isolated lines indicating a relatively small  number of possible synchronization states.
Interestingly, the case of asymmetric coupling in Fig.~\ref{Fig:Model}(C) exhibits lower dynamical complexity as that shown in Fig.~\ref{Fig:Coupling_R}(B), since only one line of the distribution of $r$ is achieved for all initial conditions.
Figures~\ref{Fig:Coupling_R}(A) and \ref{Fig:Coupling_R}(C) imply finite multistability with different possible asymptotic values of $r$.

For four coupled clocks (Figs.~\ref{Fig:Coupling_R}(D)-(F)), we also uncover that different coupling topologies lead to diverse dynamical complexities.
Specifically, three different lines of $r$ are observed in Figs.~\ref{Fig:Coupling_R}(D) and \ref{Fig:Coupling_R}(E).
More importantly, if the structure of the coupled clocks maintains the symmetry as Fig.~\ref{Fig:Model}(G), the distribution of the asymptotic order parameters in Fig. \ref{Fig:Coupling_R}(F) is no longer discrete, but contains continuous parts.
Such a distribution indicates higher complexity and even EM.
In order to characterize deeper the emergence of EM, we focus on analytical and numerical explanations of this phenomenon in the following sections.

\section{Extreme multistability}\label{Sec:Extreme multistability}
We recall that EM is potentially observed for the cross-coupling structure of four coupled clocks (Fig.~\ref{Fig:Model}(G)), where the distribution of asymptotic order parameters seems to be continuous  (Fig.~\ref{Fig:Coupling_R}(F)).
In this scheme, the opposite pendulums are connected by springs.
The corresponding coupling matrix has four nonzero entries $a_{13}=a_{31}=a_{24}=a_{42}=1$, and the angle position parameters are $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$.
The system (\ref{Eq:general_model}) becomes:
\begin{widetext}
    \begin{eqnarray}\label{Eq:general_model_4}
    \begin{array}{ll}
    \displaystyle
    (B_0+4mr^2)\ddot{\theta}+k_\theta\theta+c_\theta\dot{\theta}
    +mrl[-\ddot{\varphi}_1 \cos(\theta-\varphi_1)-\dot{\varphi}^2_1 \sin(\theta-\varphi_1)]
    +mrl[\ddot{\varphi}_2 \sin(\theta-\varphi_2)-\dot{\varphi}^2_2 \cos(\theta-\varphi_2)] \\
    +mrl[\ddot{\varphi}_3 \cos(\theta-\varphi_3)+\dot{\varphi}^2_3 \sin(\theta-\varphi_3)]
    +mrl[-\ddot{\varphi}_4 \sin(\theta-\varphi_4)+\dot{\varphi}^2_4 \cos(\theta-\varphi_4)]
    +\Delta V_{\theta}=0,\\
    ml^2\ddot{\varphi}_1+ mgl\sin{\varphi_1}+c_ \varphi{\dot{\varphi}_1}+mrl[-\ddot{\theta}
    \cos(\theta-\varphi_1)+
    \dot{\theta}^2\sin(\theta-\varphi_1)]+\Delta V_{\varphi_1} =M_{E_1}, \\
    ml^2\ddot{\varphi}_2+ mgl\sin{\varphi_2}+c_ \varphi{\dot{\varphi}_2}+mrl[\ddot{\theta} \sin(\theta-\varphi_2)+
    \dot{\theta}^2\cos(\theta-\varphi_2)]+\Delta V_{\varphi_2} =M_{E_2}, \\
    ml^2\ddot{\varphi}_3+ mgl\sin{\varphi_3}+c_ \varphi{\dot{\varphi}_3}+mrl[\ddot{\theta} \cos(\theta-\varphi_3)-
    \dot{\theta}^2\sin(\theta-\varphi_3)]+\Delta V_{\varphi_3} =M_{E_3}, \\
    ml^2\ddot{\varphi}_4+ mgl\sin{\varphi_4}+c_ \varphi{\dot{\varphi}_4}+mrl[-\ddot{\theta}\sin(\theta-\varphi_4)-
    \dot{\theta}^2\cos(\theta-\varphi_4)]+\Delta V_{\varphi_4} =M_{E_4}, \\
    \end{array}
    \end{eqnarray}
where
    \begin{equation}\label{Eq:springs_moments_of_forces_4}
    \begin{array}{ll}
    \displaystyle
    \Delta V_{\theta}=4lrk_\varphi \left[
    \left(1-\frac{s_{24}}{\hat{s}_{24}}\right)\sin\left(\frac{\varphi_2-\varphi_4}{2}\right)\sin\left(\theta-\frac{\varphi_2+\varphi_4}{2}\right)
    - \left(1-\frac{s_{13}}{\hat{s}_{13}}\right)\sin\left(\frac{\varphi_1-\varphi_3}{2}\right)\cos\left(\theta-\frac{\varphi_1+\varphi_3}{2}\right)
    \right],\\
    \Delta V_{\varphi_1}=k_\varphi l \left(1 - \frac{2r}{\hat s_{13}}\right)
    [l\sin(\varphi_1-\varphi_3)-2r\sin(\theta-\varphi_1)],\\
    \Delta V_{\varphi_2}=k_\varphi l\left( 1-\frac{2r}{\hat s_{24}}\right)
    [l\sin(\varphi_2-\varphi_4)-2r\cos(\theta-\varphi_2)],\\
     \Delta V_{\varphi_3}=k_\varphi l\left(1-\frac{2r}{\hat s_{31}}\right)
     [l\sin(\varphi_3-\varphi_1)+2r\sin(\theta-\varphi_3)],\\
    \Delta V_{\varphi_4}=k_\varphi l\left(1-\frac{2r}{\hat s_{24}}\right)[l\sin(\varphi_4-\varphi_2)+2r\cos(\theta-\varphi_4)].
    \end{array}
    \end{equation}
The distances $s_{ij}$ and $\hat s_{ij}$ are:
\label{Eq:distance_4}
    \begin{equation}\label{4-8}
    \begin{array}{ll}
    \displaystyle
    s_{13}=s_{31}=s_{24}=s_{42}=2r,\\
    \hat{s}_{13} = \hat{s}_{31}=\sqrt{4r^2+2r^2(1-\cos(\varphi_1-\varphi_3))-8lr\sin\left(\frac{\varphi_1-\varphi_3}{2}\right)\sin\left(\theta-\frac{\varphi_1+\varphi_3}{2}\right)},\\
    \hat{s}_{24} = \hat{s}_{42}=\sqrt{4r^2+2r^2(1-\cos(\varphi_2-\varphi_4))-8lr\sin\left(\frac{\varphi_2-\varphi_4}{2}\right)\cos\left(\theta-\frac{\varphi_2+\varphi_4}{2}\right)}.\\
    \end{array}
    \end{equation}
\end{widetext}

As we will see later, the regime of EM is characterized by the emergence of two frequency synchronized clusters each containing two clocks.
The following phase relations are observed for the synchronized clusters:  ``in-phase-in-phase" (II), ``in-phase-anti-phase" (IA), ``anti-phase-in-phase" (AI) and ``anti-phase-anti-phase" (AA). The exact meaning of these relations are given in Table \ref{Tab:phase possibility}.
For example, IA means that the clocks in the first cluster are in-phase and  anti-phase in the second cluster.
Additionally, due to the discontinuity induced by the escapement mechanism, the mixed states are observed, when one or both of the clusters are not oscillating.
This is possible due to the fact that the clocks do not cross periodically the escapement threshold and, hence, do not gain energy.
The following clusters are observed: "silent-in-phase" (SI), "in-phase-silent" (IS), and "silent-silent" (SS).

\begin{figure*}[!htbp] \centering
\includegraphics[width=0.95\textwidth]{Fig/Figure_2.png}
\caption{
Distribution of the order parameter $r$ for 4 coupled clocks (\ref{Eq:general_model_4}) with the coupling topology as in Fig.~\ref{Fig:Model}(G).
(A) is same as Fig. \ref{Fig:Coupling_R}(F), obtained by Monte Carlo sampling with 1,000 random trials.
(B)-(E) represent the parts of the distribution (extracted from (A)) that correspond to specific cluster states: (B) counts only the order parameters for the trials ending in II (in-phase-in-phase) configuration, (C) stands for IA, (D) for AI, and (E) for AA, see Table \ref{Tab:phase possibility} explaining the cluster states.
The main observation is that only case (B) is related to the emergence of EM.
Parameters are fixed as in Table \ref{Tab:Symbol} with $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$ for the 4-coupled clocks ($N=4$).
}
\label{Fig:EM}
\end{figure*}

\begin{table}[!htbp]
    \caption{Different phase-clusters possibilities for 4-coupled clocks (\ref{Eq:general_model_4}) with cross-coupling structure (see Fig. \ref{Fig:Model}(G)).}
    \setlength{\tabcolsep}{2.5pt}
    \begin{tabularx}{0.48\textwidth}{ll}
    \hline
    Symbol & Description \\
    \hline
    II & In-phase-in-phase ($\varphi_1(t)=\varphi_3(t),\ \varphi_2(t)=\varphi_4(t)$)\\
    IA & In-phase-anti-phase ($\varphi_1(t)=\varphi_3(t), \ \varphi_2(t)=-\varphi_4(t)$)\\
    AI & Anti-phase-in-phase ($\varphi_1(t)=-\varphi_3(t), \ \varphi_2(t)=\varphi_4(t)$)\\
    AA & Anti-phase-anti-phase ($\varphi_1(t)=-\varphi_3(t), \ \varphi_2(t)=-\varphi_4(t)$)\\
    SI & Silent-in-phase ($\varphi_1(t)=\varphi_3(t)=0, \ \varphi_2(t)=\varphi_4(t)$)\\
    IS & In-phase-silent ($\varphi_1(t)=\varphi_3(t), \ \varphi_2(t)=\varphi_4(t)=0$)\\
    SS & Silent-silent ($\varphi_1(t)=\varphi_3(t)=0, \ \varphi_2(t)=\varphi_4(t)=0$)\\
    \hline
    \end{tabularx}
    \label{Tab:phase possibility}
\end{table}

In Figure \ref{Fig:EM}, we split the probability distribution of the order parameter $r$ accordingly to the cluster states observed.
Specifically, Fig.~\ref{Fig:EM}(A) gives the whole distributions, same as in Fig.~\ref{Fig:Coupling_R}(F)).
Figure~\ref{Fig:EM}(B) exacts from Fig.~\ref{Fig:EM}(A) only the order parameters that correspond to II phase clusters, Fig.~\ref{Fig:EM}(C) to IA, Fig.~\ref{Fig:EM}(D) to AI, and Fig.~\ref{Fig:EM}(E) to AA.
Only II clusters exhibit a continuous distribution of $r$, thus suggesting that EM appears due to such type of clusters.
In contrast, Figs.~\ref{Fig:EM}(C)-(E) shows only a finite number of lines of $r$.
In the following sections, we provide additional analytical and numerical evidences that confirm our observation and explain the phenomenon of EM.

\subsection{Family of stable cluster states} \label{Subsec:Family of stable cluster regimes}
\subsubsection{Theoretical analysis of EM}
As Fig.~\ref{Fig:EM}(B) indicates, coexistence of infinitely many stable states can be related to the emergence of II, the in-phase-in-phase clusters.
To study the existence of such clusters, we show that the following subspace of II solutions:
\begin{equation}
\label{Eq:II_manifold}
\varphi_1(t)=\varphi_3(t)=\psi_1(t), \quad
\varphi_2(t)=\varphi_4(t)=\psi_2(t),\quad \theta(t)=0.
\end{equation}
is invariant with respect to the solutions of system (\ref{Eq:general_model_4}).
Indeed, substituting $\varphi_1(t)=\varphi_3(t)=\psi_1(t)$ and $\varphi_2(t)=\varphi_4(t)=\psi_2(t)$ into (\ref{Eq:general_model_4}), we obtain $\Delta V_{\theta}=0$ and $\Delta V_{\varphi_i}=0$, and the equations for the new variables $\theta$, $\psi_1$, and $\psi_2$ read:
\begin{widetext}

    \begin{subequations}\label{Eq:general_model_4_EM}
    \begin{align}
    &(B_0+4mr^2)\ddot{\theta}+k_\theta\theta+c_\theta\dot{\theta}=0,\label{Eq:general_model_4_EM_0}\\
    &ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1}+mrl[-\ddot{\theta}
    \cos(\theta-\psi_1)+
    \dot{\theta}^2\sin(\theta-\psi_1)]=M_{E_1},\label{Eq:general_model_4_EM_1}\\
    &ml^2\ddot{\psi}_2+ mgl\sin{\psi_2}+c_ \varphi{\dot{\psi}_2}+mrl[\ddot{\theta} \sin(\theta-\psi_2)+
    \dot{\theta}^2\cos(\theta-\psi_2)] =M_{E_2},\label{Eq:general_model_4_EM_2}\\
    &ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1}+mrl[\ddot{\theta} \cos(\theta-\psi_1)-
    \dot{\theta}^2\sin(\theta-\psi_1)]=M_{E_1},\label{Eq:general_model_4_EM_3}\\
    &ml^2\ddot{\psi}_2+ mgl\sin{\psi_2}+c_ \varphi{\dot{\bar{\varphi}}_2}+mrl[-\ddot{\theta}\sin(\theta-\psi_2)-
    \dot{\theta}^2\cos(\theta-\psi_2)]=M_{E_2},\label{Eq:general_model_4_EM_4}
    \end{align}
    \end{subequations}
\end{widetext}
with
\begin{equation}\label{Eq:escapement_mechanism_4_EM}
M_{E_i}=
\begin{cases}
M&: \sigma_i=1\wedge 0<\psi_i<\varepsilon_0\\
-M&: \sigma_i=2\wedge -\varepsilon_0<\psi_i<0\\
0&: \mathrm{otherwise}
\end{cases}
\end{equation}
where $i=1,2$, $M = 0.075$ [Nm], and $\varepsilon = 5.0^\circ$.

Now, by setting $\theta=0$, we observe that Eq. (\ref{Eq:general_model_4_EM_0}) is satisfied, and the dynamical equations for $\psi_1$ (Eqs. (\ref{Eq:general_model_4_EM_1}) and (\ref{Eq:general_model_4_EM_3})) and for $\psi_2$ (Eqs. (\ref{Eq:general_model_4_EM_2}) and (\ref{Eq:general_model_4_EM_4})) become the same:
\begin{subequations}\label{Eq:general_model_4_EM_final}
\begin{align}
&ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1}=M_{E_1}, \label{Eq:general_model_4_EM_final_1} \\
&ml^2\ddot{\psi}_2+ mgl\sin{\psi_2}+c_ \varphi{\dot{\psi}_2}=M_{E_2}. \label{Eq:general_model_4_EM_final_2}
\end{align}
\end{subequations}
Thus, we have proven the following:

\newtheorem{prop}{Proposition}

\begin{prop}[II-cluster subspace]\label{II-cluster subspace}
    The subspace (\ref{Eq:II_manifold}) of the cluster II solutions is invariant with respect to the solutions (flow) of system (\ref{Eq:general_model_4}).
    This subspace is 4-dimensional ($S^2\times \mathrm{R}^2$), and the flow on this subspace is given by Eqs.~(\ref{Eq:general_model_4_EM_final_1}) and (\ref{Eq:general_model_4_EM_final_2}) which describe the relative motion of the clusters.
\end{prop}

Another important observation is that the dynamical equations  (\ref{Eq:general_model_4_EM_final_1}) for $\psi_1$ and (\ref{Eq:general_model_4_EM_final_2}) $\psi_2$ in the II-cluster subspace are (i) the same, and (ii) \textit{uncoupled from each other}, i.e., the equation for $\psi_1$ is independent on $\psi_2$ and vise-versa.
The latter property leads to the coexistence of infinitely many asymptotic states (and to EM eventually).
We formulate the corresponding result as a proposition.

\begin{prop}[EM of II-cluster states]\label{EM of II-cluster states}
    Assume that $c_\varphi,m,l,g,k_\theta,c_\theta$ and $B_0$ are positive parameters.
    Assume also that system (\ref{Eq:general_model_4_EM_final_1}) (or, equivalently, (\ref{Eq:general_model_4_EM_final_2})) possesses a stable nontrivial asymptotic state (attractor), and $\psi^*(t)$ is a solution on this attractor (i.e., a single clock has a stable oscillatory state).
    Then the system on the II-cluster subspace (\ref{Eq:general_model_4_EM_final_1})--(\ref{Eq:general_model_4_EM_final_2}) possesses the following asymptotic states:\\
    -- II clusters:
    \begin{equation}
        \label{Eq:II}
        \varphi_1=\varphi_3=\psi_1=\psi^*(t), \quad
        \varphi_2=\varphi_4=\psi_2=\psi^*(t+\gamma).
    \end{equation}
    -- SI clusters:
    \begin{equation}
        \label{Eq:SI}
        \varphi_1=\varphi_3=\psi_1 = 0,\quad
        \varphi_2=\varphi_4=\psi_2 =\psi^*(t).
    \end{equation}
    -- SS clusters:
    \begin{equation}
        \label{Eq:SS}
        \varphi_1=\varphi_3=\psi_1 =0,\quad
        \varphi_2=\varphi_4=\psi_2=0.
    \end{equation}
    -- IS clusters:
    \begin{equation}
        \label{Eq:IS}
        \varphi_1=\varphi_3=\psi_1 =\psi^*(t),\quad
        \varphi_2=\varphi_4=\psi_2=0.
    \end{equation}
    where $\gamma$ is an arbitrary real constant describing a phase shift between the clusters.
    Moreover, if $\gamma^*(t)$ is an orbitally asymptotically stable limit cycle (stable periodic oscillations of the clock), then the states (\ref{Eq:II}) build a stable invariant torus foliated by (infinitely many) periodic solutions of the form (\ref{Eq:II}).
\end{prop}

The main message of Proposition \ref{EM of II-cluster states} is that under "normal conditions" when the single clock oscillates periodically, the coupled system can have stable cluster II oscillations with an arbitrary phase-shift between the clusters.
If the phase-shift is zero, the order parameter $r$ is highest and equal 1, while it can achieve a continuous range of smaller values depending on the phase shift. The coexistence of such states leads to EM.
Note that the states SI, SS, and IS do not lead to EM, but correspond to isolated attractors in the coupled system.

\textbf{Proof of Proposition \ref{EM of II-cluster states}.}
Let us first mention that the equilibrium $\psi_1=\psi_2=0$ is asymptotically stable, and it corresponds to the stability of a silent state of a pendulum with damping and without external energy inflow.
Therefore, the two independent systems (\ref{Eq:general_model_4_EM_final_1}) and (\ref{Eq:general_model_4_EM_final_2}) can reach both attractors: zero equilibrium and non-trivial attractor corresponding to $\psi^*(t)$, depending on initial conditions.
Moreover, an arbitrary phase shift $\psi^*(t+\gamma)$ is clearly also possible and belong to the same nontrivial attractor.
This provides the existence of the states (\ref{Eq:II})--(\ref{Eq:IS}).

The invariant torus from the proposition corresponds to the direct product of the limit cycles $\mathcal{C}\times \mathcal{C}$, where $\mathcal{C}=\{(\psi,~\dot \psi)\in (S^1\times \mathbb{R}): \psi=\gamma^*(t),~t\in \mathbb{R}\}$.
The stability of this torus follows from the orbital stability of the limit cycle in each of the subsystem and the properties of the cross-product of the uncoupled system (\ref{Eq:general_model_4_EM_final_1})--(\ref{Eq:general_model_4_EM_final_2}). \textbf{End of proof.}

\begin{figure*}[!htbp] \centering
\includegraphics[width=0.95\textwidth]{Fig/Figure_3.png}
\caption{
Three exemplary dynamical patterns for the regime of EM (see Fig. \ref{Fig:EM}(B)) for 4-coupled pendulums with cross-coupling structure (\ref{Eq:general_model_4}).
(A) Almost complete synchronization with a small phase shift between the clusters of clocks.
(B) Phase synchronization with an intermediate phase shift between the clusters.
(C) Anti-phase synchronization between the clusters.
Details of column figures are as follows:
{\bf{a}}: the time series for variables $\theta$, $\varphi_1$, $\varphi_2$, $\varphi_3$, $\varphi_4$, respectively,\ {\bf{b}}: the phase-time plots of the pendula,\ {\bf{c}}: the mean frequencies of the clocks,\ {\bf{d}}: Projections on two phase variables form different clusters (orange lines) and Poincaré maps (black dots).
Parameters are fixed as in Table \ref{Tab:Symbol} with $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$ for the 4-coupled clocks ($N=4$).
}
\label{Fig:Dynamical_Patterns_Of_EM}
\end{figure*}

\begin{figure*}[!htbp] \centering
\includegraphics[width=0.95\textwidth]{Fig/Figure_4.png}
\caption{
Dynamical patterns of three types of isolated attractors with large-amplitude oscillations for 4 coupled pendulums with cross-coupling structure (\ref{Eq:general_model_4}).
(A)-(C): IA, AI, and AA synchronization patterns (see Table~\ref{Tab:phase possibility} for the explanation of the abbreviations). IA, AI, and AA correspond to the order parameters from Figs.~\ref{Fig:EM}(C), (D), and (E), respectively.
Information on the columns:
{\bf{a}}: the time series for variables $\theta$, $\varphi_1$, $\varphi_2$, $\varphi_3$, $\varphi_4$, respectively;\ {\bf{b}}: phase-time plots of the pendulum angles;\ {\bf{c}} mean frequencies;\ {\bf{d}}: projection of the solution on the phase variables from different clusters (orange lines) and Poincaré maps (black dots).
Parameters are fixed as in Table \ref{Tab:Symbol} with $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$ for the 4-coupled clocks ($N=4$).
}
\label{Fig:Attractor_IA_AI_AA}
\end{figure*}

\subsubsection{Numerical study of EM}
Figure~\ref{Fig:Dynamical_Patterns_Of_EM} shows three examples of different stable II synchronization patterns from the continuous family of solutions by Eq. (\ref{Eq:II}).
Figure~\ref{Fig:Dynamical_Patterns_Of_EM} provides (A) almost in-phase, (B) a phase-shifted, and (C) anti-phase relations between the clusters. The dynamics within the clusters is completely synchronized: $\varphi_1=\varphi_3$ and $ \varphi_2=\varphi_4$.
These three states are exemplary, and different phase shifts are obtained from different initial conditions.
In spite of the phase shift, all pendulums are (mean) frequency synchronized (see the third column of Fig. \ref{Fig:Dynamical_Patterns_Of_EM}), since they follow the same motion according to Eq. (\ref{Eq:II}), only phase-shifted.
The fourth column of Fig.~\ref{Fig:Dynamical_Patterns_Of_EM} illustrates the phase-shift between the clusters and their periodic motion (orange curve). The black point shows the Poincaré map defined by $\varphi_1$ = 0 and $\dot{\varphi}_1 > 0$.

As for the order parameters $r$, Fig.~\ref{Fig:Dynamical_Patterns_Of_EM}(A) presents the trial leading to a relatively high order parameter close to 1 (almost complete synchronization) in Fig. \ref{Fig:EM}(B); while Fig.~\ref{Fig:Dynamical_Patterns_Of_EM}(C) represents the trial which falls into the left side (the inter-group anti-phase synchronization) of the order parameter distribution in Fig.~\ref{Fig:EM}(B).
The more trials one draws from random initial conditions, the more likely one can fill the gap regarding the order parameter between the inter-group anti-phase synchronization and complete synchronization to generate EM.

\begin{figure*}[!htbp] \centering
\includegraphics[width=0.95\textwidth]{Fig/Figure_5.png}
\caption{
Dynamical patterns of three types of isolated attractors with small-amplitude oscillations for 4 coupled pendulums with cross-coupling structure (\ref{Eq:general_model_4}).
(A)-(C): SI, IS, and SS synchronization patterns respectively,  (see Table \ref{Tab:phase possibility} for the pattern explanations).
Information along the columns:
{\bf{a}}: time series for variables $\theta$, $\varphi_1$, $\varphi_2$, $\varphi_3$, and $\varphi_4$,\ {\bf{b}}: phase-time plots,\ {\bf{c}}:  mean frequencies (not available if a pendulum converges to 0),\ {\bf{d}}: projections on the $(\varphi_2,\varphi_3)$-plane (orange line) and  Poincaré maps (black dots, defined by $\varphi_2=0$ and $\dot{\varphi}_2 > 0$, $\varphi_1=0$ and $\dot{\varphi}_1 > 0$, and $\varphi_1=0$ and $\dot{\varphi}_1 > 0$, for (A), (B), and (C), respectively).
Parameters are fixed as in Table \ref{Tab:Symbol} with $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$ for the 4-coupled clocks ($N=4$).
}
\label{Fig:Attractor_Discontinuity}
\end{figure*}

\subsection{Isolated attractors}\label{Subsec:Isolated attractors}
In addition to infinitely many stable II states from EM described above, system (\ref{Eq:general_model_4}) possesses coexisting isolated attractors corresponding to other synchronization patterns.
These states can also be treated analytically and numerically in more detail.
However, since the main focus of this work is the EM phenomenon, we consider here exemplary only isolated attractors corresponding to IA patterns.

\subsubsection{Three types of isolated attractors with large-amplitude oscillations}
\paragraph*{Theoretical analysis of IA patterns.}
The IA solutions are characterized by the phase relations $\varphi_1(t)=\varphi_3(t)=\psi_1$ and $\varphi_2(t)=-\varphi_4(t)=\psi_2$.
Substituting this into system (\ref{Eq:general_model_4}), we obtain:
\begin{widetext}
    \begin{eqnarray}\label{Eq:general_model_4_IA}
    \begin{array}{ll}
    \displaystyle
    (B_0+4mr^2)\ddot{\theta}+k_\theta\theta+c_\theta\dot{\theta}+mrl[\ddot{\psi}_2 \sin(\theta-\psi_2)-\dot{\psi}^2_2 \cos(\theta-\psi_2)]
    +mrl[\ddot{\psi}_2 \sin(\theta+\psi_2)-\dot{\psi}^2_2 \cos(\theta+\psi_2)]
    +\Delta V_{\theta}=0,\\
    ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1}+mrl[-\ddot{\theta}
    \cos(\theta-\psi_1)+
    \dot{\theta}^2\sin(\theta-\psi_1)]+\Delta V_{\varphi_1} =M_{E_1}, \\
    ml^2\ddot{\psi}_2+ mgl\sin{\psi_2}+c_ \varphi{\dot{\psi}_2}+mrl[\ddot{\theta} \sin(\theta-\psi_2)+
    \dot{\theta}^2\cos(\theta-\psi_2)]+\Delta V_{\varphi_2} =M_{E_2}, \\
    ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1}+mrl[\ddot{\theta} \cos(\theta-\psi_1)-
    \dot{\theta}^2\sin(\theta-\psi_1)]+\Delta V_{\varphi_3} =M_{E_3}, \\
    -ml^2\ddot{\psi}_2-mgl\sin{\psi_2}-c_ \varphi{\dot{\psi}_2}+mrl[-\ddot{\theta}\sin(\theta+\psi_2)-
    \dot{\theta}^2\cos(\theta+\psi_2)]+\Delta V_{\varphi_4} =M_{E_4}. \\
    \end{array}
    \end{eqnarray}
\end{widetext}
In the following, we introduce the new dimensionless parameter $\varepsilon=\frac{mrl}{B_0+4mr^2}$. For the chosen setup as in Table~\ref{Tab:Symbol}, we have $\varepsilon=0.02726$, i.e., $\varepsilon$ is small, and we will employ it in our analysis.
In fact, the smallness of this parameter is one of the reasons for the emergence of IA patterns.

Defining further $\bar{k}_\theta=\frac{k_\theta}{B_0+4mr^2}=3.7301$, $\bar{c}_\theta=\frac{c_\theta}{B_0+4mr^2}=0.02407$, $F(\theta,~\psi_2)=-[\ddot{\psi}_2 \sin(\theta-\psi_2)-\dot{\psi}^2_2 \cos(\theta-\psi_2)]
+[\ddot{\psi}_2 \sin(\theta+\psi_2)-\dot{\psi}^2_2 \cos(\theta+\psi_2)]
-\frac{\Delta V_{\theta}}{mrl}$ and $\theta=\varepsilon\psi$, the system (\ref{Eq:general_model_4_IA}) can be rewritten in the following form:
\begin{widetext}
    \begin{equation}\label{Eq:general_model_4_IA_perturbation}
    \begin{array}{ll}
    \displaystyle
    \ddot{\psi}+\bar{c}_\theta\dot{\psi}+\bar{k}_\theta\psi= F(\varepsilon\psi,~\psi_2),\\
    ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1}+\varepsilon mrl[-\ddot{\psi}
        \cos(\varepsilon\psi-\psi_1)+\varepsilon\dot{\psi}^2\sin(\varepsilon\psi-\psi_1)]+\Delta V_{\varphi_1} =M_{E_1}, \\
    ml^2\ddot{\psi}_2+ mgl\sin{\psi_2}+c_ \varphi{\dot{\psi}_2}+\varepsilon mrl[\ddot{\psi} \sin(\varepsilon\psi-\psi_2)+
        \varepsilon\dot{\psi}^2\cos(\varepsilon\psi-\psi_2)]+\Delta V_{\varphi_2} =M_{E_2},\\
    ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1}+\varepsilon mrl[\ddot{\psi}\cos(\varepsilon{\psi}-\psi_1)-
        \varepsilon\dot{\psi}^2\sin(\varepsilon{\psi}-\psi_1)]+\Delta V_{\varphi_3} =M_{E_3},\\
    -ml^2\ddot{\psi}_2-mgl\sin{\psi_2}-c_ \varphi{\dot{\psi}_2}+\varepsilon mrl[-\ddot{\psi}\sin(\varepsilon{\psi}+\psi_2)-\varepsilon \dot{\psi}^2\cos(\varepsilon{\psi}+\psi_2)]+\Delta V_{\varphi_4} =M_{E_4},
    \end{array}
    \end{equation}
where
    \begin{equation}\label{Eq:springs_moments_of_forces_4_IA}
    \begin{array}{ll}
    \displaystyle
        \Delta V_{\theta}=4lrk_\varphi
        \left(1-\frac{2r}{\sqrt{4r^2+2r^2(1-\cos(2\psi_2))-8lr\sin\psi_2\cos(\varepsilon\psi)}}\right)
        \sin\psi_2\sin(\varepsilon\psi),\\
        \Delta V_{\varphi_1}= \Delta V_{\varphi_3} =0,\\
        \Delta V_{\varphi_2}=k_\varphi l
        \left(1-\frac{2r}{\hat{s}_{24}}\right)
        [l\sin(2\psi_2)-2r\cos(\varepsilon\psi-\psi_2)],\\
        \Delta V_{\varphi_4}=k_\varphi l
        \left(1-\frac{2r}{\hat{s}_{24}}\right)
        [-l\sin(2\psi_2)+2r\cos(\varepsilon\psi+\psi_2)],
    \end{array}
    \end{equation}
\end{widetext}
and the values for the distances satisfy:
\begin{equation}\label{Eq:distance_4_IA}
    \begin{array}{ll}
    \displaystyle
        s_{13}=s_{31}=s_{24}=s_{42}=2r,\\
        \hat{s}_{13}=\hat{s}_{31}=2r,\\
        \hat{s}_{24}=\hat{s}_{24}=\sqrt{4r^2+2r^2(1-\cos(2\psi_2))-8lr\sin\psi_2\cos(\varepsilon\psi)}.
    \end{array}
\end{equation}

In the zeroth-order in $\varepsilon$, system (\ref{Eq:general_model_4_IA_perturbation}) is reduced to
\begin{subequations}\label{Eq:general_model_4_IA_perturbation_0}
\begin{align}
&\ddot{\psi}+\bar{c}_\theta\dot{\psi}+\bar{k}_\theta\psi=F(0,~\psi_2),\label{eqn-11-1}\\
&ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1} =M_{E_1},\label{eqn-11-2}\\
&ml^2\ddot{\psi}_2+ mgl\sin{\psi_2}+c_ \varphi{\dot{\psi}_2}+\Delta V_{\varphi_2}=M_{E_2},\label{eqn-11-3}\\
&ml^2\ddot{\psi}_1+ mgl\sin{\psi_1}+c_ \varphi{\dot{\psi}_1} =M_{E_1},\label{eqn-11-4}\\
&-ml^2\ddot{\psi}_2-mgl\sin{\psi_2}-c_ \varphi{\dot{\psi}_2}-\Delta V_{\varphi_2} =-M_{E_2},\label{eqn-11-5}
\end{align}
\end{subequations}
from which one can see that Eq.~(\ref{eqn-11-2}) is equivalent to Eq.~(\ref{eqn-11-4}) and  Eq.~(\ref{eqn-11-3}) to Eq.~(\ref{eqn-11-5}). Hence, in this approximation, the subspace of the IA solutions:
\begin{equation}
\label{Eq:subspace_IA_solution}
\varphi_1(t)=\varphi_3(t)=\psi_1,\quad \varphi_2(t)=-\varphi_4(t)=\psi_2, \quad \theta(t)=0.
\end{equation}
is invariant.
For nonzero but small $\varepsilon$, we observe the perturbed solutions:
\begin{equation}
\label{Eq:subspace_IA_solution_perturb}
\varphi_1(t)\approx\varphi_3(t)=\psi_1,\quad
\varphi_2(t)\approx-\varphi_4(t)=\psi_2.
\end{equation}
Summarizing, the existence of IA patterns can be exactly proven for the limit $\varepsilon=0$. Since small $\varepsilon$ is a regular perturbation of system (\ref{Eq:general_model_4_IA_perturbation}), all asymptotically stable periodic attractors in this system will be only slightly perturbed by small $\varepsilon$-order terms, and one can observe patterns close to IA.

\paragraph*{Numerical study of IA, AI, and AA patterns.}
The patterns IA, AI, and AA correspond to single lines of the order parameter distribution in Fig. \ref{Fig:EM}(C)-(E) and, hence, to  isolated attractors in the phase space.
Figure~\ref{Fig:Attractor_IA_AI_AA} reports one example for each of these three patterns.
All of them exhibit partial synchronization with quasiperiodic dynamics.

The phase-time plots in the column \textbf{b} of Fig.~\ref{Fig:Attractor_IA_AI_AA}(A) show that the 1st and 3rd pendulums are in-phase and the 2nd and 4th are anti-phase with $\varphi_1=\varphi_3$ and $\varphi_2=-\varphi_4$; this is also confirmed by the analytical solutions (\ref{Eq:subspace_IA_solution}) and (\ref{Eq:subspace_IA_solution_perturb}).
Inside the coupled groups, the 1st and 3rd or 2nd and 4th pendulums share the same mean frequency, leading to the multifrequency-clusters \cite{Berner2019b,omelchenko2011loss}.
Furthermore, the phase space projection on the plane $(\varphi_1,~\varphi_2)$ (orange curve) and the corresponding Poincaré map (black points, when $\varphi_4$ = 0 and $\dot{\varphi}_4 > 0$) indicates that the motion is quasiperiodic.
In summary, Fig.~\ref{Fig:Attractor_IA_AI_AA} reports different partially synchronous behaviors corresponding to isolated attractors with large-amplitude osillations: (A) IA, (B) AI, and (C) AA patterns. All these patterns coexist with EM contributing to a complex multistability scenario for the cross coupling structure (Fig.~\ref{Fig:Model}(G)).

\subsubsection{Three types of isolated attractors with small-amplitude oscillations}
The above-mentioned isolated attractors and EM states possess "large-amplitude" oscillations in the sense that their oscillation amplitude exceeds the threshold of the escapement mechanism $\varepsilon = 5.0^\circ$.
This guarantees an inflow of energy into the system and the emergence of stable self-sustained oscillations of all clocks.

In this section, we investigate the case when some (or all) of the clocks in system (\ref{Eq:general_model_4}) do not reach this threshold.
As a result, the SI, IS, and SS synchronization patterns appear (see Table \ref{Tab:phase possibility}).
Figure~\ref{Fig:Attractor_Discontinuity} illustrates the numerically observed SI, IS, and SS patterns. For example, in Fig.~\ref{Fig:Attractor_Discontinuity}(B), both the time series and phase-time plots exhibit the in-phase synchronization between $\varphi_1$ and $\varphi_3$, while $\varphi_2$ and $\varphi_4$ converge to 0 and stop oscillating.
The periodic motions of $\varphi_1$ and $\varphi_3$ are illustrated by the phase trajectory (the orange line) and corresponding Poincaré map (black points, defined by $\varphi_1=0$ and $\dot{\varphi}_1 > 0$) in the column \textbf{d} of Fig.~\ref{Fig:Attractor_Discontinuity}(B).
Summarizing Fig.~\ref{Fig:Attractor_Discontinuity}, it shows the coexistence of three types of isolated attractors with small-amplitude oscillations, induced by the discontinuity of the escapement mechanism.

\begin{figure}[t] \centering
\includegraphics[width=0.31\textwidth]{Fig/Figure_6.png}
\caption{
One-dimensional dependence of the order parameter on initial conditions for 4 coupled pendulums with cross-coupling structure (\ref{Eq:general_model_4}).
The graph reveals the basins of different attractors along a one-dimensional line in the phase space.
We fix
[$\theta^0=0.01$, $\varphi_1^0=\frac{\pi}{4}$, $\varphi_3^0=\varphi_1^0+0.001$, $\varphi_4^0=\varphi_2^0+0.001$, $\dot\theta^0=\dot\varphi_1^0=\dot\varphi_2^0=\dot\varphi_3^0=\dot\varphi_4^0=0$],
and vary $\varphi_2^0$ to obtain 100 discretized values evenly distributed in the interval $[-\frac{\pi}{4},\frac{\pi}{4}]$.
The change of order parameter with respect to $\varphi_2^0$ indicates how synchronization patterns depend on initial conditions, including complete synchronization for $\varphi_2^0=\pi/4$.
Plateaus correspond to isolated attractors, continuously changing parts to EM states, and jumps to the basin boundaries.
Parameters are fixed as in Table \ref{Tab:Symbol} with $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$ for the 4-coupled clocks ($N=4$).
}
\label{Fig:Basin_1D}
\end{figure}

\begin{figure}[!htbp] \centering
\includegraphics[width=0.26\textwidth]{Fig/Figure_7.png}
\caption{
Two-dimensional basins of attraction of 4 coupled pendulums with cross-coupling structure (\ref{Eq:general_model_4}). The initial conditions are chosen as
[$\theta^0=0.01$, $\varphi_1^0=\varphi_2^0=\frac{\pi}{4}$, $\varphi_3^0=\varphi_1^0-A$, $\varphi_4^0=\varphi_2^0-B$, $\dot\theta^0=\dot\varphi_1^0=\dot\varphi_2^0=\dot\varphi_3^0=\dot\varphi_4^0=0$], with $A$ and $B$
changing in the interval $[0,\frac{\pi}{2}]$. The uniform discretization on the grid $100\times 100$ is used.
Four regions in the bottom-left, top-left, bottom-right, and top-right correspond to the II,  IA, AI, and AA phase patterns, respectively.
The bottom-left part with the visible color gradient correspond to EM.
Parameters are fixed as in Table \ref{Tab:Symbol} with $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$ for the 4-coupled clocks ($N=4$).
}
\label{Fig:Basin_2D}
\end{figure}

\begin{figure}[!htbp] \centering
\includegraphics[width=0.26\textwidth]{Fig/Figure_8.png}
\caption{
Two-dimensional basins of the attractors corresponding to the mixed-mode dynamics with one cluster staying silent and another oscillating.
The model for 4 coupled pendulums with cross-coupling structure (\ref{Eq:general_model_4}) are considered.
Initial conditions are chosen as follows: [$\theta^0=0.01$, $\varphi_1^0=\varphi_2^0=0$, $\varphi_3^0=A$, $\varphi_4^0=B$, $\dot\theta^0=\dot\varphi_1^0=\dot\varphi_2^0=\dot\varphi_3^0=\dot\varphi_4^0=0$], where $A$ and $B$ are taken from the interval $[-\frac{\pi}{4},\frac{\pi}{4}]$ discretized by $100\times 100$ evenly sampled points.
The white region corresponds to the trivial SS pattern (equilibrium at the origin); light green to SI and IS patterns.
Parameters are fixed as in Table \ref{Tab:Symbol} with $\alpha_1 = \frac{\pi}{2}$, $\alpha_2 = \pi$, $\alpha_3=\frac{3\pi}{2}$ and $\alpha_4=2\pi$ for the 4-coupled clocks ($N=4$).
}
\label{Fig:Basin_2D_Discontinuity}
\end{figure}

\section{Basins of attractions}\label{Sec:Basins of attractions}

Having clarified the different EM-related asymptotic phase patterns in 4 coupled clocks given by system (\ref{Eq:general_model_4}) in Sec.~\ref{Sec:Extreme multistability}, we discuss here their basins of attractions.
Instead of randomly choosing initial conditions for each pendulum, we fix some of them while the rest are initialized with discretized values distributed evenly in the given intervals.
The corresponding synchronization states are estimated using order parameter $r$ from Eq. (\ref{Eq:order_parameter_used}), as it effectively identifies different attractors, including those belonging to EM.
Numerical results are summarized in Figs.~\ref{Fig:Basin_1D}, \ref{Fig:Basin_2D}, and \ref{Fig:Basin_2D_Discontinuity}.

Figure~\ref{Fig:Basin_1D} shows the dependence of the order parameter $r$
on the initial conditions.
For this we fix
[$\theta^0=0.01$, $\varphi_1^0=\frac{\pi}{4}$, $\varphi_3^0=\varphi_1^0+0.001$, $\varphi_4^0=\varphi_2^0+0.001$, $\dot\theta^0=\dot\varphi_1^0=\dot\varphi_2^0=\dot\varphi_3^0=\dot\varphi_4^0=0$],
and initialize $\varphi_2^0$ with 100 discretized values evenly distributed in the interval $[-\frac{\pi}{4},\frac{\pi}{4}]$.
We can clearly observe both the isolated attractors and a part of the EM regime.
Isolated attractors correspond to the flat segments.
For example, when $\varphi_2^0$ is around 0 in Fig. (\ref{Fig:Basin_1D}), in spite of different initial values of $\varphi_2^0$, trials in this segment part have almost the same $r$.
The continuously changing parts of $r$ in Fig. (\ref{Fig:Basin_1D}) are in line with the EM family of II states.
The abrupt "jumps" therefore represent boundaries between either the basins of the isolated attractors or between the isolated attractors and the EM family.

Figure \ref{Fig:Basin_2D} shows a two-dimensional basin of attraction, where we fix
[$\theta^0=0.01$, $\varphi_1^0=\varphi_2^0=\frac{\pi}{4}$, $\varphi_3^0=\varphi_1^0-A$, $\varphi_4^0=\varphi_2^0-B$, $\dot\theta^0=\dot\varphi_1^0=\dot\varphi_2^0=\dot\varphi_3^0=\dot\varphi_4^0=0$], and vary $A$ and $B$ in the interval $[0,\frac{\pi}{2}]$.
The grid of $100 \times 100$ points is used.
One can observe four parts corresponding to II, IA, AI, and AA phase patterns.
In particular, the bottom-left part with the non-constant dependence of the order parameter $r$ in Fig.~\ref{Fig:Basin_2D} corresponds to a subset of an infinite number of stable II states from EM.
The other three parts with constant colors are related to isolated attractors of AI, IA, and AA patterns.
With changing $A$ and $B$, the system moves from complete synchronization to inter-group anti-phase synchronization (see Fig.~\ref{Fig:Dynamical_Patterns_Of_EM}), and further to the multifrequency-cluster state (see Fig.~\ref{Fig:Attractor_IA_AI_AA}).

To visualize the basin of the attractors with small-amplitude oscillations, we analyse a set of initial conditions close to the origin.
Figure~\ref{Fig:Basin_2D_Discontinuity} shows the corresponding basins of attraction, with the initial conditions
[$\theta^0=0.01$, $\varphi_1^0=\varphi_2^0=0$, $\varphi_3^0=A$, $\varphi_4^0=B$, $\dot\theta^0=\dot\varphi_1^0=\dot\varphi_2^0=\dot\varphi_3^0=\dot\varphi_4^0=0$], where $A$ and $B$ vary in the interval $[-\frac{\pi}{4},\frac{\pi}{4}]$. In addition to II patterns from EM set, we obtain here three new regions corresponding to the patterns SI, IS, and SS, for which some of the clocks are not oscillating.
The basin of the trivial solution SS with all clocks silent is observed in the central part (white); see Fig.~\ref{Fig:Attractor_Discontinuity}(C) for the illustration of the pattern.
The SI and IS patterns are mixed-mode oscillations, with one cluster silent and another cluster oscillating, corresponding to the light green basin of attraction.

\section{Conclusions}\label{Sec:Conclusion}
In summary, we investigate how different coupling topology affects the collective dynamics in coupled clocks.
The considered model includes global as well as local couplings represented by the rotating support disc and springs, respectively.
Also, the model contains a discontinuity due to the escapement clock mechanism.
We focus on the model of 4 coupled clocks where surprisingly an EM is observed between patterns with different synchronization levels.
The EM phenomenon reveals the coexistence of infinitely many stable asymptotic states.
The dependence on initial conditions is clarified using the analysis of the basins of attractions.
The main conclusions based on our work are as follows:
\begin{itemize}
\item For both three coupled and four coupled clocks, we use Monte Carlo sampling and find that the symmetric coupling structure can increase the dynamical complexity.
This can lead to diverse synchronization patterns and attractors.
\item We observe the emergence of EM for the case of four coupled clocks (see Fig. \ref{Fig:Model}(G) and model (\ref{Eq:general_model_4})).
This phenomenon is solely induced by the cross-coupling topological structure, and it is stable against variations of the system parameters, as far as the clocks remain identical.
\item We show both analytically and numerically that the emergence of EM is closely related to the II synchronization pattern (see Table \ref{Tab:phase possibility}), where the system splits into two antipodal clusters such that the clocks within the clusters are fully synchronized, but the intra-cluster dynamics can be shifted by an arbitrary phase.
Moreover, other three types of isolated attractors with large-amplitude oscillations coexist with the infinite family of states from the EM family.
They correspond to the IA, AI, and AA synchronization patterns.
\item We further uncover the effect of the discontinuity of the system induced by the escapement mechanism.
It induces the emergence and coexistence of further three types of isolated attractors with small-amplitude oscillations. These states  correspond to SI, IS, and SS patterns (see Table \ref{Tab:phase possibility}).
In particular, the IS and SI patterns are mixed states \cite{Ebrahimzadeh2022} where two clocks are oscillating and the other two stay silent.
\end{itemize}

As observed in our work, the emergence of EM is caused by a particularly designed symmetric coupling structure, rather than by introducing additional quantities into the coupling design.
The inclusion of more coupled clocks with different topological coupling structures is an open way to clarify the emergence of EM or chimera states of large coupled systems.
Another possible generalization is to include adaptivity in the coupling scheme.

\section*{Acknowledgments}
Z.S. and Y.-R.L. was funded by the China Scholarship Council (CSC) scholarship.
J.K. was supported by the Federal Ministry of Education and Research (BMBF) grant No. 01LP1902J (climXtreme).
S.Y. was supported by the German Research Foundation DFG, Project No. 411803875.

\section*{Author Contributions}
Z.S. and Y.S. contributed equally to this work.

\section*{Data Availability}
The code used for this work is available online \href{https://github.com/Zsstarry/EM\_Clocks}{here}.

redpajama_set_name: RedPajamaArXiv


This was the first endurance race I've done in the car and I was happy overall with the car - the driver and pitstop need a bit of work before next week's round though!!
The little 924 was the oldest car in the field, but didn't disgrace itself in the tighter corners. The straights were a different story though - anyone got a spare 200hp lying around?
that or ditch the 12A (really? who uses those things anymore? Let alone find parts?) for a 13B Turbo or a 20B 4-rotor.
All kidding aside, great job and thanks for sharing it with us so we can live vicariously through you.
I'm not a huge fan of turbos, either piston or rotary, but will probably have to look at it sooner or later. Could get up to 400hp out of the little 1146cc beast then!
Will try to get more video of the next race next Saturday, which is at a track that suits my car better (pretty much all corners).
We had round 2 last weekend and finished 31st out off 44. Managed to beat one of the 944S2s too. A couple of big safety car periods meant that almost a third of the race was under safety car - very frustrating.
Looking forward to the last round at the end of this month now.


text:
Operación Impacto Profundo es el 12.º episodio de la primera temporada de los Thunderbirds, serie de televisión de Gerry Anderson para Supermarionation fue el 8.º episodio producido. El episodio salió al aire primero en ATV Midlands el 16 de diciembre de 1965. Fue escrito por Martin Crump y dirigido por Desmond Saunders.

Sinopsis

La aeronave Fireflash está desapareciendo misteriosamente. El Rescate internacional debe darse prisa para salvar a una tripulación atrapada después de su choque y hundimiento en el océano y atrapados en el lecho marino. ¿El Thunderbird 2 volara junto a otro Fireflash, los hermanos Tracy descubrirán una falla técnica u otro sabotaje?

Argumento

En la torre de control del Aeropuerto de Londres el teniente da la orden de despegue al Fireflash 3 con destino a San Francisco, pero pide a la tripulación informar su posición al cruzar la costa. El comandante Norman observa como el avión cruza la barrera del sonido. Poco después cruza la costa, pero el Fireflash 3 empieza a agitarse alarmantemente. El controlador de la torre de control recibe un mensaje de mayday que explica que el avión está perdiendo altura y dando la posición antes de que pierdan comunicación.

El teniente hace sonar la alarma de la emergencia. Norman comienza la "Operación Mar-halcón", lanzando aviones de Rescate Aéreo y desviando todos los barcos en el área para una búsqueda. Sin embargo, nada se encuentra, y Norman se ve obligado a cancelar la operación.

Una reunión se celebra entre el Ministro Internacional del Aire, Norman, el teniente y Patterson, un ingeniero. El Ministro le dice Norman que cancele todos los vuelos del Fireflash hasta que Patterson repita sus extensas pruebas. Los periódicos narran la historia.

Los miembros de Rescate Internacional observan un informe del desastre en la televisión. Tin-Tin está escéptica acerca de que los saboteadores pudieran estar involucrados de nuevo con las demasiadas precauciones se toma ahora. Jeff quiere mirar las pruebas muy estrechamente y le dice a Alan en el Thunderbird 5 que supervise cualquier transmisión relacionada. Alan informa que Fireflash 3 estaba 50 millas fuera de su posición informada en el momento de su desaparición. Jeff también les dice a los otros muchachos que estén en el estado de espera.

Patterson está probando un Fireflash para descartar problemas. Él ha eliminado la fatiga metálica y ha estado probando radiación, que resulta también estar bien. Norman informa a una nueva tripulación de la prueba en la cual volarán otro Fireflash a lo largo de la misma ruta.

En la Isla Tracy, los muchachos miran un reportero de televisión diciendo que los vuelos de Fireflash regresarán la próxima semana. Gracias a la ayuda de Rescate Internacional, una banda criminal ha sido desenmascarada. De repente la televisión se apaga. La abuela les dice que un fusible se ha fundido y ahora Gordon es el experto, ¡él tendrá que arreglarlo!
Hay otro corte de corriente de la radio de repente y el Fireflash empieza a descender. Afortunadamente, la red de comunicaciones de Rescate Internacional todavía es operacional. Scott informa que los mandos no responden. Londres sugiere que él y Hansen salten fuera pero Virgil tiene un plan.
El Thunderbird 4 se pone en posición y empieza a cortar los motores. Una vez que el motor de estribor esta en el suelo, el Fireflash empieza a subir hasta que se nivela fuera en la superficie. Los mandos están ahora incendiándose. El Thunderbird 4 sale a la superficie y Gordon empieza a cortar la ventana. Los esfuerzos del copiloto por controlar el fuego con un extintor fallan. Una cápsula se baja del Thunderbird 2. Gordon abre la ventana y les dice a los pilotos que suban en él cuando llegue. Sin otra palabra él regresa al Thunderbird 4. Los pilotos suben en la cápsula y Virgil la sube a bordo. Los Thunderbirds 2 y 4 rápidamente salen del área cuando una explosión destruye el Fireflash completamente.
Cuando el Fireflash despega, Norman le dice al Teniente que mantenga contacto constante. Alan supervisa los mensajes y cree que el Fireflash está dando mal su posición de nuevo. El copiloto intenta verificar la posición pero comienza a fallar el poder elevador del avión. La tripulación intenta avisar a Londres pero falla la radio. Así como Norman temió, el mismo problema de nuevo. Un saboteador se lanza en paracaídas del fondo del avión. El Fireflash está cayendo ahora y los controles no responden.

Alan informa esto a la Isla Tracy y agrega que el Fireflash esta 180 millas al Noroeste de dónde dice estar. Jeff llama a Scott, Virgil y Gordon del cuarto de juegos. El Fireflash impacta con el agua, flotando y retardando el hundimiento. Los pilotos no pueden abrir la puerta y el avión se empieza a hundir. El saboteador está en un bote de caucho y lanza una señal luminosa. Un helijet lo recoge.

Jeff envía a Scott en el Thunderbird 1 a examinar el área electromagnéticamente. Virgil, Gordon y Brains lo siguen en el Thunderbird 2 con la Vaina 4 a bordo. Brains piensa que la tripulación todavía está viva, atrapada en el lecho marino. Efectivamente, ellos están allí y han perdido todo el contacto con Londres. Virgil deja caer la vaina y Gordon se lanza en el Thunderbird 4. Scott le da el curso a la posición de la caída y él empieza una búsqueda en el área.

Los pilotos ven el Thunderbird 4 y encienden todas las luces que pueden. Gordon ve el Fireflash. Brains le dice que él tendrá que cortar con el láser los motores para hacerlo flotar. Gordon nada a al parabrisas y usa una computadora portátil para explicar el plan a los pilotos. El piloto está de acuerdo encendiendo una luz.

En otra reunión, Norman, el teniente y Patterson defienden la causa de la caída. El Ministro Internacional del Aire concluye que el problema se ha aislado al sistema hidráulico en el ala de estribor. En la Isla Tracy, Virgil sugiere que el Rescate Internacional debe probar el Fireflash con el Thunderbird 2 volando junto a él. Jeff envía una carta a Londres y Norman está de acuerdo a condición de que Capitán Hansen este a bordo. El vuelo es de máximo secreto, sin otro avión volando cerca.

El Thunderbird 2 se prepara para aterrizar en Londres con Scott, Virgil y Gordon a bordo. Jeff le dice a Alan que supervise el vuelo estrechamente. Scott se une a Hansen en el Fireflash y ambos despegan. Cuando el Fireflash cruza la costa, Scott informa su posición a Alan, pero Alan cree que es incorrecta. Scott ajusta el curso a la dirección de Alan y se notifica a Londres y al Thunderbird 2. Un localizador defectuoso podría explicar muchas cosas.

Con 15 minutos para el impacto, Gordon lanza del Thunderbird 2 un dispositivo de levantamiento. El Thunderbird 2 vuela bajo el Fireflash y Hansen abre la compuerta. Gordon cree haber visto a alguien dentro pero cree que solo fue su imaginación. Un cable es disparado en el ala y Gordon es arrastrado a bordo por medio del cable. Scott dirige a Gordon a la EPU (Unidad de Poder Elevador, por sus siglas en inglés). Quedan solamente tres minutos ahora.

Gordon ve que algunos alambres han sido cortados. El saboteador todavía esta a bordo e intenta dispararle a Gordon. Se inicia un pequeño tiroteo. El saboteador hace una partida rápida, a pesar de las advertencias de Gordon que su paracaídas no abriría a tiempo. Scott exige saber lo que está pasando pero Gordon no tiene tiempo para explicar. Así que como el avión está a punto de caer en el mar, él une los alambres cortados. El Fireflash asciende en el último segundo.

Reparto

Reparto de voz regular
 Jeff Tracy — Peter Dyneley
 Scott Tracy — Shane Rimmer
 Virgil Tracy — David Holliday
 Alan Tracy — Matt Zimmerman
 Gordon Tracy — David Graham
 John Tracy — Ray Barrett
 Brains — David Graham
 Tin-Tin Kyrano — Christine Finn
 Abuela Tracy — Christine Finn

Reparto de voz invitado
 Comandante Norman — Peter Dyneley
 Comandante Burroughs — Ray Barrett
 Piloto del Fireflash 3 — David Graham
 Piloto del Seahawk — Matt Zimmerman
 Reportero de TV - Matt Zimmerman
 Ministro Internacional del Aire — Peter Dyneley
 Patterson — David Graham
 Piloto del Fireflash — David Graham
 Copiloto del Fireflash — Ray Barrett
 Granjero — David Graham
 Teniente — Shane Rimmer
 Capitán Hanson — David Graham
 Saboteador — Ray Barrett
 Reportero — David Holliday

Equipo principal
Los vehículos y equipos vistos en el episodio son:

 Thunderbird 1
 Thunderbird 2 (llevando la Vaina 4)
 Thunderbird 4
 Thunderbird 5
 Fireflash
 EJ2 Helijet

Errores
 Mientras Gordon busca el Fireflash en el lecho marino, su pelo constantemente cambia de lado.
 Cuando Gordon regresa al Thunderbird 4 después de notificar a la tripulación de su intención para cortar las turbinas del Fireflash, el diseño del Thunderbird 4 tamaño títere no coincide con el modelo de tamaño normal.
 En las escenas donde se ve el Thunderbird 2 y el Fireflash se nota una diferencia importante entre el Fireflash de este episodio y el de "Atrapado en el Cielo" ya que este modelo es más grande que el del primer capítulo de la serie.

Notas
 El título original para este episodio era "La Tripulación de Prueba".
 Lugarteniente Burroughs, el ayudante del Comandante Norman, fueron los mismos modelos del Solarnauta Asher en "La Sonda del Sol".
 Una escena de una colección de periódicos y revistas que informan la pérdida de los primeros Fireflash incluye las ediciones de The Daily Telegraph, el Daily Mail, Life y Mad muestran que estos todavía están publicándose en 2065.
 El reportero de televisión que relata los eventos también aparece en "Atrapado en el Cielo" y en "La Sonda del Sol".
 Después de que el segundo Fireflash choca, el saboteador es recogido por un EJ2 Helijet color plata que es después disfrazado como el Thunderbird 2 falso en "Los Impostores" después de que es repintado de verde y amarillo.

Enlaces externos
 Fanderson.org.uk - Thunderbirds Series One Episode Guide
 ThunderbirdsOnline.co.uk - Episode: Operation Crash-Dive

Episodios de Thunderbirds

redpajama_set_name: RedPajamaWikipedia


text:
Petra Therese Högl (* 14. April 1971 in Eichendorf) ist eine deutsche Politikerin der CSU.

Seit den 2000er ist Högl stellvertretende Vorsitzende im Kreisvorstand der CSU und seit 2014 Fraktionsvorsitzende im Kreistag. Bei der Landtagswahl am 14. Oktober 2018 wurde sie als Direktkandidatin im Stimmkreis Kelheim in den Bayerischen Landtag gewählt. Högl erreichte 35,5 Prozent der Erststimmen. Des Weiteren ist Högl Mitglied des Ausschusses für Arbeit und Soziales, Jugend und Familie und Mitglied des Ausschusses für Ernährung, Landwirtschaft und Forsten im Bayerischen Landtag.

Privates
Högl ist verheiratet und Mutter von vier Kindern. Sie ist römisch-katholischer Konfession.

Weblinks
 Eigene Homepage

Einzelnachweise

Landtagsabgeordneter (Bayern)
CSU-Mitglied
Deutscher
Geboren 1971
Frau


Week Commencing 3rd July 2017.
Writing weather forecasts when it is to be changeable is pretty easy, difficult to get wrong. Next week remains changeable, some rain on Monday evening into Tuesday morning with possibly the sunniest on Wednesday. Thursday may be breezy.
The tides peak around midnight on Monday and there will be no new water till next week.
Catches. Again it was quite a quiet week, most beats caught fish but not in huge numbers. There are some grilse in the catches but not in any great numbers.
Gordon Castle finished in the high thirties, Ian Tennant thought to slow rise on Saturday morning put the fish off with only two landed for the day. Thanks to Ian and his team for the photographs.
Orton, David Wood reported, Orton finished the week in double figures after a slow start. Highlight of the week was the Engagement of Will Haggas and wildlife artist Clare Shaw on Wednesday. Catches were evenly spread between the Haggas party from Lincolnshire (successful rods Andrew Read, 8 and 12lbs, Will Haggas 4, and 8lbs and Jenny Baxter, 9lbs) and the other party from Lancashire.
Thanks to David for the pictures.
Will Haggas, Clare Short, gillie Richard Hold and a salmon!
Delfur finished in the high teens. Again just one picture this week.
I was pleased to receive a photograph from Easter Elchies owner Mr. Willie Robertson, of his grandson James aged 7 with his first salmon, he out fished some of his granddad's friends.
Craigellachie had half a dozen as did Wester Elchies, the low water did not suit Aberlour Angling Club but they still managed a couple. I hear Carron had eight.
My Grantown correspondent is on holiday again but before leaving he told me that they were heading towards twenty for the week, the best in the 20lb class. Sea trout are again being caught with at least two over 5lb.
Kinchurdy continues to fish well, this week fifty sea trout were landed along with 4 salmon and a grilse.
The post Week Commencing 3rd July 2017. appeared first on Spey Fishery Board.
← Week Commencing 26th June 2017.


text:
Топка — река в России, протекает по Воронежской области. Устье реки находится в 15 км от устья реки Икорец по правому берегу. Длина реки составляет 18 км, площадь водосборного бассейна — 259 км².

Данные водного реестра
По данным государственного водного реестра России относится к Донскому бассейновому округу, водохозяйственный участок реки — Дон от города Лиски до города Павловск, без реки Битюг, речной подбассейн реки — бассейны притоков Дона до впадения Хопра. Речной бассейн реки — Дон (российская часть бассейна).

Код объекта в государственном водном реестре — 05010101012107000003732.

Примечания

Притоки Икорца
Реки Воронежской области


text:
Power Of One: 'Netaji' Vibhuti Chauhan
Published on: Tuesday, 12th April 2016
What made Vibhuti Chauhan find within him such a strong empathy for a community that was barely recognised as part of humanity that he waged a four-decade-long struggle for its rights? After all, neither were the Musahars politically significant, nor was he himself from the community. Perhaps it began with that first encounter with injustice in the raw, when wages due to some Musahars remained unpaid and Chauhan, as a young man inspired by social democratic principles, demonstrated for justice outside the SDM's office.
"The first thing that struck me about the Musahar community when I started interacting with them in the 1970s was that hardly anyone knew of its existence – certainly not the officials. As for the Musahars themselves, they did not know their rights, they did not even know the significance of Independence Day, or that a Vidhan Sabha (state assembly) existed. They lived in broken huts, with nobody bothering about them," recalled Chauhan, now in his mid-sixties.
The other aspect that became quickly apparent to him was the severe repression they faced, particularly from the upper castes and landowning sections. The initial empathy he felt for the Musahars now turned into something deeper: a commitment. Said Chauhan, "I began spending more time with the community, sleeping in their huts, eating their food, witnessing for myself the daily humiliation they faced. I decided at that point that this is the community for which I will work for the rest of my life."
The community responded to him with affection and trust. The fact that they began to call him 'Netaji', or leader, testified to their confidence in him. Said Ram Vilas, a Musahar who is also the programme coordinator of the Musahar Vikas Pahal, "Through Chauhan saab we came to realise many issues. He guided us throughout."
Land and dignity – these were what the community needed most of all. Chauhan assisted them in getting their loans written off and taught them how to petition the authorities for ownership of ceiling land. The process was slow and often accompanied by violence. Any attempt at organising the Musahars or demanding justice for them was met almost immediately by aggression. Hamlets were flattened and houses set on fire, as landlords deployed their goons to unleash intimidation and terror. In this already unequal battle, the police often openly sided with the powerful. Chauhan has himself faced police firing and still carries the marks of these encounters on his body. But the traumas and tensions, notwithstanding, he remained steadfast in his purpose.
Chauhan described the coming of ActionAid into the region for the first time in 2001 as a "strong wind of change". As the Musahar Vikas Pahal took shape in eastern Uttar Pradesh and began representing the Musahars, with Chauhan himself as one of anchors of the process, the critical issues were clear, and ranged from violations of Scheduled Castes and Tribes (Prevention of Atrocities) Act and social recognition of the community, to land rights for the dispossessed.
However, there was a change of orientation to be made and Chauhan recognised this, "Once ActionAid came into the picture we had to understand the government system and learn to work within the constitutional framework. Earlier, we had perceived the administration as our enemy. Now we had to learn to enlist its support and sometimes link our work with that of the government. Whereas earlier we would resort to taking up our lathis, we now had to ensure that the authorities took action. We had to understand the laws and how to make use of them."
The central issues of land and dignity, however, remained. "With ActionAid's close involvement, we worked with the community to strengthen their understanding of their own rights. We taught them how to speak as equals with officers and interact with them. We ensured that men and women could speak to a minister without fear. We were always there when they faced difficulties but they had to understand that it was their struggle."
Today, according to Chauhan, the fear of the powerful that once haunted the mind of the Musahars has ebbed. "Of course, change doesn't happen in a day. The process of gaining equality takes time. But now even women can speak to an SP on the phone. I, who have seen them before this process began, can clearly perceive the difference. But this process is far from over. Only in name have the rajas left us; it will take much longer to shake off their exploitative grip on the Musahar community."
No blog stories found

redpajama_set_name: RedPajamaCommonCrawl


text:
About CHS
Past Chairs
Foundation Past Chairs
Prevention of Harassment, Intimidation & Bullying
Career Opportunities(opens in new window/tab)
Middle School (Grade 6-7)
Creative Arts & Technology
Academic Handbook
Clubs and Programs
Enriching Learning
Clubs & Programs
CHS Experience
Ivy Compass
Inclusive Community
Why CHS?
Visit CHS
Class News & Reunions
Culture of Giving
Speaker Events & Resources
Winter Bazaar
My CHS
Crofton House is an extraordinary school with a long and proud history of leadership in all-girls' education
Developing curious and critical thinkers through a challenging and supportive all-girls' learning environment
More than anything, Crofton House School is a community that supports, challenges and encourages girls for life
Becoming a Crofton House girl means joining a global community for life
Crofton House School girls graduate with a strong sense of purpose, confident they can do anything!
Social Links via JavaScript
We all play a critical role in shaping the future for the girls at Crofton House School
Crofton House is built on a foundation of philanthropy. The strong tradition of giving has played a key role in the School's growth and in its ability to continue as a leader in girls' education.
Over the past decade, the incredible generosity of the CHS community made the rebuilding of our campus possible and furthered our deep culture of philanthropy.
That spirit of giving continues through our New Parents Appeal, Annual Fund and signature events such as the annual Golf Classic.
New Parents' Appeal
CHS Golf Classic
CHS 2019 Gala
Each year our New Parents Appeal reaches out to all new families entering the school to raise funds in support of the Campus Master Plan.
In the words of Mariangela Brutti, Director, Advancement, "this appeal is the backbone of fundraising at Crofton House – it provides context to our spirit of giving and helps to build important relationships between parents and the School. Every year, we meet with the approximately 100 families who are new to CHS, encouraging each to support the School in a way that is meaningful to them, with the goal of achieving 100% participation."
This unique approach of individual meetings, with parent volunteers attending to connect and share their experience, sets Crofton House apart. By choosing to maintain a donor-centred approach to fundraising, our parents are able to ask questions about the fundraising priorities of the School, how to make donations, what to expect in the long term and the advantage of supporting a not-for-profit.
Understanding that today's students benefit from the forward thinking and generosity of past parents, the New Parents Appeal supports the Campus Master Plan. Current donations help meet capital needs and ensure that the campus and facilities will continue to support innovative programs and the evolving needs of a progressive school.
Over the past decade, the incredible generosity of the CHS community allowed the rebuilding of the our campus and furthered the deep culture of giving. That now continues through our Annual Fund.
The CHS Annual Fund provides the opportunity to support CHS through:
CHS Bursary Program Fund – The Bursary Program helps remove financial barriers for girls applying to Grade 8 who would benefit from and contribute to the CHS experience. Supporting the Bursary Program is the first priority of the Annual Fund.
CHS General Fund – Alternatively, you may choose to direct your support to the more general needs of the School.
The Annual Fund was reintroduced in fall 2017 and received an excellent response. Over 60% of funds raised went to support the CHS Bursary Program, removing financial barriers for girls wanting to attend Crofton House. The remaining funds were used to refresh Chan Hall, the Junior School gym, and to support the Senior School music program through the purchase of new instruments.
Since its inception in 1997, the Golf Classic has steadily evolved and expanded to become a signature fundraising event for the School. The Golf Classic continues to succeed, due to the outstanding participation of parents, past parents, alumnae and friends of CHS and the organization and leadership of the Golf Committee.
In 2017, the Golf Committee committed to raise $1 million in support of the new bursary program by 2020, which will provide financial assistance to extraordinary girls entering Grade 8, who show the courage, creativity and citizenship embodied at Crofton House. To date, $496,437 has been raised toward this goal.
The Crofton House Gala is a special way for our community to come together and recognize important milestones. The School's fourth gala, 2019 Gala: Beyond Extraordinary, was held on March 2, 2019.
The evening was an opportunity to celebrate the work and legacy of Dr. Patricia Dawson and her 19 years as Head of School, while raising funds for the Dr. Patricia Dawson Bursary Fund. The event was an overwhelming success and we are pleased to announce that the Gala, including gifts to the Dr. Patricia Dawson Bursary Fund, raised $1,000,000. Thank you to everyone who participated and made it an evening to remember.
on Giving
The Report on Giving is an opportunity to recognize the generosity of the Crofton House School community.
Dr. Jessie Gordon: A bold vision
Over a century ago, Dr. Jessie Gordon founded Crofton House School with a bold vision of educating girls to become confident, capable and accomplished young women.
Today, Crofton House girls are a part of a welcoming and inclusive community. They come to a beautiful 10-acre campus that has been purposely designed to meet their needs. As we watch each girl reach her own unique potential, we can see how our tradition of giving has helped her realize her dreams.
Learn morE About OUr History
"I am continually inspired by the commitment and generosity of the Crofton House community. Without their support, time and expertise our school would not be what it is today."
Mariangela Brutti | Director, Advancement
The Crofton House Advancement team is responsible for fundraising to support the priorities set by the CHS Board of Governors. The team is also responsible for alumnae relations, events and School archives. If you have any questions, please feel free to reach out.
Director, Advancement
Mariangela Brutti
mbrutti@croftonhouse.ca
Assistant Director, Alumnae Relations
Lesley Campbell
604 263 3255 ext. 4203
lcampbell@croftonhouse.ca
Assistant Director, Annual Fund and Community Engagement
Lauren Konrad
lkonrad@croftonhouse.ca
Manager, Alumnae Relations and Events
Morgan Pulkinen
mpulkinen@croftonhouse.ca
Administrator, Advancement
Jill Plotnikoff
jplotnikoff@croftonhouse.ca
Coordinator, Advancement
Sarah Forrer
arivera@croftonhouse.ca
Manda Haligowski
archives@croftonhouse.ca
Vancouver, BC V6N 3E1
Apply to Crofton House
communications@croftonhouse.ca
CAIS(opens in new window/tab)
ISABC(opens in new window/tab)
NCGS(opens in new window/tab)
© 2020 Crofton House School. All Rights Reserved


Because you compared Wordpress and Unbounce, we think this app could fit your needs.
It's a great tool for making landing pages!
It's pretty simple to use and comes with quite a few templates that can be adapted to just about any type of content / offer you are promoting. It's also easy to update and it's mobile friendly as well. It integrates with some great marketing software like HubSpot for easy integrated marketing.
It's often hard to organize landing pages within the system. The WordPress plugin can also be problematic and pages break without you knowing.
I was using Wordpress, but wanted something "self-hosting".
If they did websites, instead of "just" landing pages. Our website is a bunch of landing pages joined together.
Unbounce helps I and my clients build sales funnels that boosts our conversions.
I like the fact that these companies are working very hard to make online business easy for the average people. Unbounded has proven to be a master at the game of sales funnels. With a huge library of mobile responsive templates, building sales funnels is a breeze.
I also wish I can download pages and publish to wordpress.
Launch landing pages fast and not bogging my design/development team down with requests.
I've only run into two issues: mobile versions of landing pages take a lot of time to edit (they don't just snap in place if you make changes to a template), and editing page elements is not always intuitive.
Just amazing, all you need to configure landing pages and pop ups you have it in this software, perfect and clean design.
The price, if you have a good wordpress template and a good knowledge to manage it, you may not use this because of it's hight price.
If you have a product you can sell online for a good margin, Unbounce is the way!!!
Their support and sales team are fantastic. It isn't good to sterotype, but I LOVE working with this Canadian company.
Most drag and drop editors on WordPress are poorly reviewed, but not Unbounce.
It is pricey and the AMP is not available for essential product.
Reviewed 24th of January, 2017 by Elaine S.
They have a WordPress plugin which makes it very easy to publish Unbounce pages and make sure your URLs do not conflict.
They offer deep discounts for educational and nonprofit customers, so be sure to ask if that applies to your organization!
Save time and effort comparing leading Landing Page Software tools for small businesses. The table below compares Unbounce and Wordpress. Check out how both product compares looking at product details such as features, pricing, target market and supported languages. Quickly access the latest reviews to compare actual user opinions and ratings. Do not hesitate to click on Visit Website to access a Free Demo or Trial immediately.


We examined each building for compliance with local zoning regulations, building code requirements, and the ADA (Americans with Disabilities Act). Our inspection included structural condition, utilities, building exterior and interior conditions, and condition of mechanical and electrical systems.
The report, produced by Stephen Abel in September 2006, included summaries of each building with recommended improvements and repairs, with specific cost estimates.


ge freezer temperature control wrt138fzdb whirlpool 30 wide top freezer refrigerator with electronic temperature control black ge freezer temperature setting 1 90.
ge freezer temperature control temperature control refrigerator whirlpool side by side refrigerator w temperature control refrigerator temp control not working ge refrigerator temperature kenmore freezer temperature settings.
ge freezer temperature control wr55x11084 for ge freezer temperature control board kenmore bottom freezer temperature control.
ge freezer temperature control california residents see prop 65 warnings kenmore freezer temperature settings.
ge freezer temperature control refrigerator temp control setting numbers ge top freezer refrigerator temperature controls.
ge freezer temperature control refrigerator temperature control thermostat universal electric freezer refrigerator thermostat switch temperature control mechanical ge top freezer temperature control setting.
ge freezer temperature control upfront electronic temperature controls whirlpool top freezer temperature control.
ge freezer temperature control ge gte21gthcc 33 inch top freezer refrigerator with 212 cu ft capacity ge freezer temperature setting.
ge freezer temperature control ge gie18ishss 30 top freezer refrigerator with 182 cu ft capacity factory ge freezer temperature setting 1 900.
ge freezer temperature control ge fcm11ph 106 cu ft chest freezer with adjustable temperature control white whirlpool chest freezer temperature control.
ge freezer temperature control gte21gshss ge top freezer refrigerator with upfront temperature controls 33 inch stainless steel ge freezer temperature setting 1 99.
ge freezer temperature control temperature ge freezer temperature setting.
ge freezer temperature control ge top freezer temperature control setting.
ge freezer temperature control ge profile refrigerator temperature control problems profile refrirator temperature control ge profile refrigerator temperature control whirlpool freezer temperature settings fast freeze.
ge freezer temperature control ge freezer temperature control thermostat part wr50x21296 kenmore refrigerator freezer temperature control.
ge freezer temperature control ge gde03gmked 19 top freezer compact refrigerator in slate whirlpool freezer temperature settings fast freeze.
ge freezer temperature control this ge top freezer refrigerator features upfront temperature controls helping you easily regulate both fresh and frozen food ge freezer temperature setting 1 9 forms.
ge freezer temperature control a product photo of the ge gss25gshss our pick for the best side by whirlpool chest freezer temperature control.
ge freezer temperature control fridge secondary image whirlpool bottom freezer temperature control.
ge freezer temperature control ge 67003000 freezer temperature control kenmore freezer temperature settings.
ge freezer temperature control ge gte18ethbb 182 cu ft ada compliant top freezer refrigerator with upfront temperature controls whirlpool freezer temperature settings.
ge freezer temperature control ge gte15cth 28 inch top freezer refrigerator with 146 cu ft capacity ge freezer temperature setting 1 9 2019.
ge freezer temperature control refrigerator temperature setting image titled prevent freezer crystals step 1 kic fridge temperature setting 1 7 whirlpool freezer temperature settings.
ge freezer temperature control ge refrigerator temperature settings refrirator temperature setting 1 9 refrirator settings fresh whirlpool refrirator temperature settings ge top freezer temperature control setting.
ge freezer temperature control ge fcm7dudww freezer temperature control thermostat genuine oem ge freezer temperature setting 1 9 forms.
ge freezer temperature control ge freezer temperature control.
ge freezer temperature control ge profile refrigerator temperature control profile refrirator ge profile performance refrigerator temperature control ge profile refrigerator ge freezer temperature setting 1 9 form.
ge freezer temperature control ft refrigerator from ge is a good fit for any home upfront temperature controls allow you to easily adjust both fresh food and freezer temperatures ge freezer temperature setting 1 900.
ge freezer temperature control ge refrigerator freezer temperature control knob photo of wg03f05006 photo of wg03f05006 photo of wg03f05006 photo of wg03f05006 photo of wg03f05006 ge freezer temperature setting 1 9 forms.
ge freezer temperature control ge refrigerator temp control refrirator dial on parts fast shipping refrirator temperature controls black ge profile ge refrigerator temp control ge freezer temperature setting 1 95.