An example of a wild strange attractor

An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.

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“…Examples of higher-dimensional robust strange attractors that contain equilibria are described in [46,406].…”

Section: Theorem 43 ([333]mentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

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“…However, the situation has changed drastically after the paper by Turaev and Shilnikov [1], in which a class of wild-hyperbolic attractors was introduced. Such attractors are also genuine, however, unlike the hyperbolic and Lorenz ones, they possess homoclinic tangencies and, thus, contain Newhouse wild hyperbolic sets [2].…”

Section: Introductionmentioning

confidence: 99%

Lorenz-like attractors in a nonholonomic model of a rattleback

Gonchenko

Гонченко

2015

Nonlinearity

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“…Hence, the fixed point e x is a saddle focus and satisfies the inequality 1 0 s σ > > , which implies that there is a homoclinic chaos according to Shil'nikov theorem [47][48][49]. In addition, the chaotic attractor has a dissipative feature since the divergence of flows is described by…”

Section: Quadratic Nonlinearitiesmentioning

confidence: 99%

Chaos in PID Controlled Nonlinear Systems

Ablay¹

2015

Journal of Electrical Engineering and Technology

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-Controlling nonlinear systems with linear feedback control methods can lead to chaotic behaviors. Order increase in system dynamics due to integral control and control parameter variations in PID controlled nonlinear systems are studied for possible chaos regions in the closed-loop system dynamics. The Lur'e form of the feedback systems are analyzed with Routh's stability criterion and describing function analysis for chaos prediction. Several novel chaotic systems are generated from second-order nonlinear systems including the simplest continuous-time chaotic system. Analytical and numerical results are provided to verify the existence of the chaotic dynamics.

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An example of a wild strange attractor

Cited by 122 publications

References 13 publications

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Lorenz-like attractors in a nonholonomic model of a rattleback

Chaos in PID Controlled Nonlinear Systems

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