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January 2016, 36(1): 509-540. doi: 10.3934/dcds.2016.36.509
Behavior $0$ nonsingular Morse Smale flows on $S^3$
Bin Yu 1,
Department of Mathematics, Tongji University, Shanghai 200092
Received June 2014 Revised December 2014 Published June 2015
In this paper, we first develop the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as behavior $0$ NMS flows). It mainly contains the following several directions:
1. we use WLG to list behavior $0$ NMS flows on $S^3$;
2. with the help of WLG, comparing with Wada's algorithm, we provide a direct description about the (indexed) link of behavior $0$ NMS flows;
3. to overcome the weakness that WLG can't decide topologically equivalent class, we give a simplified Umanskii Theorem to decide when two behavior $0$ NMS flows on $S^3$ are topological equivalence;
4. under these theories, we classify (up to topological equivalence) all behavior 0 NMS flows on $S^3$ with periodic orbits number no more than 4.
Keywords: topologically equivalent classes., links, Behavior $0$ nonsingular Morse-Smale flows, weighted Lyapunov graphs.
Mathematics Subject Classification: Primary: 37D15, 37C15; Secondary: 37C15, 57M2.
Citation: Bin Yu. Behavior $0$ nonsingular Morse Smale flows on $S^3$. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 509-540. doi: 10.3934/dcds.2016.36.509
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