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Homoclinical structure of the chaotic attractor. (English) Zbl 1221.34124
Summary: In earlier work [ibid. 14, No. 4, 1486–1493 (2009; Zbl 1221.37030)], a relay system was introduced, which admits a chaotic attractor with Devaney’s ingredients. Now, we prove that the attractor consists of homoclinic solutions. A simulation of the attractor is provided for a pendulum equation.
MSC:
34C37Homoclinic and heteroclinic solutions of ODE
37D45Strange attractors, chaotic dynamics
37C29Homoclinic and heteroclinic orbits
37C70Attractors and repellers, topological structure
References:
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[2]Akhmet MU. Hyperbolic sets of impact systems. Dyn Contin Discrete Impuls Syst Ser A Math Anal 2008;15:(Suppl. S1):1 – 2. Proceedings of the 5th international conference on impulsive and hybrid dynamical systems and applications, Beijin: Watan Press; 2008.
[3]Birkhoff GD. Dynamical systems. Providence: Amer Math Soc; 1966.
[4]Smale, S.: Diffeomorphisms with many periodic points, Differential and combinatorial topology, 63-80 (1963) · Zbl 0142.41103
[5]Akhmet, M. U.: On the reduction principle for differential equations with piecewise constant argument of generalized, J math anal appl 336, 646-663 (2007) · Zbl 1134.34048 · doi:10.1016/j.jmaa.2007.03.010
[6]Wiener, J.: Generalized solutions of functional differential equations, (1993)
[7]Devaney, R.: An introduction to chaotic dynamical systems, (1990)
[8]Wiggins, S.: Global bifurcation and chaos: analytical methods, (1988)
[9]Akhmet MU. Dynamical synthesis of quasi-minimal sets. Int J Bifurc Chaos [accepted]. · Zbl 1176.34009 · doi:10.1142/S0218127409024190