Dissipative Homoclinic Loops and Rank One Chaos
نویسنده
چکیده
We prove that when subjected to periodic forcing of the form pμ,ρ,ω(t) = μ(ρh(x, y) + sin(ωt)), certain second order systems of differential equations with dissipative homoclinic loops admit strange attractors with SRB measures for a set of forcing parameters (μ, ρ, ω) of positive measure. Our proof applies the recent theory of rank one maps, developed by Wang and Young [30, 34] based on the analysis of strongly dissipative Hénon maps by Benedicks and Carleson [4, 5].
منابع مشابه
Periodically Forced Double Homoclinic Loops to a Dissipative Saddle
In this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point. We prove the existence of many complicated dynamical objects for given equations, ranging from attractive quasi-periodic torus, to Newhouse sinks ...
متن کاملDissipative Homoclinic Loops of 2-dimensional Maps and Strange Attractors with 1 Direction of Instability
We prove that when subjected to periodic forcing of the form pμ,ρ,ω(t) = μ(ρh(x, y) + sin(ωt)), certain 2-dimensional vector fields with dissipative homoclinic loops admit strange attractors with SRB measures for a set of forcing parameters (μ, ρ, ω) of positive Lebesgue measure. The proof extends ideas of Afraimovich and Shilnikov [1] and applies the recent theory of rank 1 maps developed by W...
متن کاملA Phenomenological Approach to Normal Form Modeling: a Case Study in Laser Induced Nematodynamics
An experimental setting for the polarimetric study of optically induced dynamical behavior in nematic liquid crystal films presented by G. Cipparrone, G. Russo, C. Versace, G. Strangi and V. Carbone allowed to identify most notably some behavior which was recognized as gluing bifurcations leading to chaos. This analysis of the data used a comparison with a model for the transition to chaos via ...
متن کاملMelnikov method approach to control of homoclinic/heteroclinic chaos by weak harmonic excitations.
A review on the application of Melnikov's method to control homoclinic and heteroclinic chaos in low-dimensional, non-autonomous and dissipative oscillator systems by weak harmonic excitations is presented, including diverse applications, such as chaotic escape from a potential well, chaotic solitons in Frenkel-Kontorova chains and chaotic-charged particles in the field of an electrostatic wave...
متن کاملHomoclinic Spirals: Theory and Numerics
In this paper we examine spiral structures in bi-parametric diagrams of dissipative systems with strange attractors. First, we show that the organizing center for spiral structures in a model with the Shilnikov saddle-focus is related to the change of the structure of the attractor transitioning between the spiral and screw-like types located at the turning point of a homoclinic bifurcation cur...
متن کامل