Towards global models near homoclinic tangencies of dissipative diffeomorphisms

Abstract

A representative model of a return map near homoclinic bifurcation is studied. This model is the so-called fattened Arnold map, a diffeomorphism of the annulus. The dynamics is extremely rich, involving periodicity, quasiperiodicity and chaos.The method of study is a mixture of analytic perturbation theory, numerical continuation, iteration to an attractor and experiments, in which the guesses are inspired by the theory. In turn the results lead to fine-tuning of the theory. This approach is a natural paradigm for the study of complicated dynamical systems.By following generic bifurcations, both local and homoclinic, various routes to chaos and strange attractors are detected. Here, particularly, the 'large' strange attractors which wind around the annulus are of interest. Furthermore, a global phenomenon regarding Arnold tongues is important. This concerns the accumulation of tongues on lines of homoclinic bifurcation. This phenomenon sheds some new light on the occurrence of infinitely many sinks in certain cases, as predicted by the theory.

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

“…The torus T of X β breaks down when approaching the heteroclinic structure. This phenomenon is only partially understood from the theoretical viewpoint [1,4,5,14,27,50]. For parameters inside a resonance tongue, homoclinic tangency bifurcations of periodic orbits lying inside T are often related to the breakdown of the torus and to the creation of strange attractors [36,41,44].…”

“…Notice that map G (3) is slightly simplified with respect to (14): ε 1 can be taken real, since a transformation of the form (w, z) = R θ (w ′ , z ′ ) = (exp(iθ)w ′ , z ′ ) for suitable θ yields a system of coordinates where Im(ε 1 ) = 0. Moreover, the parameter ε 3 is fixed at zero in G: this is reasonable, since the term in ε 3 of (14) is of order γ 4 , while the ∂/∂z-component of G already contains a term in γz 2 .…”

Catastrophe theory in Dulac unfoldings

“…In the complement of this perturbed Cantor foliation of hypersurfaces we expect all the dynamical complexity regarding Cantori, strange attractors, etc., as described in [19,20,21,22,38,42]. …”

“…I n the complement of this perturbed Cantor foliation of hypersurfaces we expect all the dynamical complexity regarding Cantori, strange attractors, etc., as described in [19,20,21,22,38,42]. This program will be the subject of [16] where we aim to apply [9,11,52,53] to establish the occurrence of quasi-periodic cuspoid bifurcations in the three cases (a), (b) and (c) of Figure 1.…”

Weakly Dissipative Systems in Celestial Mechanics

“…In order to approach the analysis of the dissipative nearly-integrable systems, we start by investigating a simple discrete model known as the dissipative standard map (see [3], [4], [6], [8], [20], [29], [32]). Its dynamics is studied through frequency analysis ( [21], [22]) and by means of a quantity called the differential fast Lyapunov indicator as introduced in [8].…”