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… Show more
“…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].…”
Section: Dynamics Of Hopf-saddle-node Vector Fieldsmentioning
confidence: 99%
“…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 .…”
The dynamics near a Hopf-saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf-saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.
“…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].…”
Section: Dynamics Of Hopf-saddle-node Vector Fieldsmentioning
confidence: 99%
“…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 .…”
The dynamics near a Hopf-saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf-saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.
“…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]. …”
Section: Example 1 (Bogdanov-takensmentioning
confidence: 99%
“…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.…”
In this paper, we study a class of families of planar vector fields. Each member of the family consists of an autonomous vector field and a nonautonomous perturbation which is periodic in time. The autonomous part is the sum of a Hamiltonian vector field which does not depend on the parameters and a parameter dependent dissipative part. This setting, modulo suitable rescalings, covers for example the codimension ¢ Hopf bifurcation and several cases of (subordinate) homoclinic bifurcation. In this setting we consider the non-autonomous part as a perturbation and mainly focus on the 'unperturbed' autonomous family of planar vector fields. Our interest is with the geometry of the bifurcation set of limit cycles, in particular in the neighbourhood of certain homoclinic bifurcations. In our approach the corresponding Hamiltonian vector field has a homoclinic loop to a hyperbolic saddle point. It is to be noted that the full system has the solid torus
“…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].…”
Summary. We investigate the dynamics associated to nearly-integrable dissipative systems, with particular reference to some models of Celestial Mechanics which can be described in a weakly dissipative framework. We start by studying some paradigmatic models provided by the dissipative standard maps in 2 and 4 dimensions. The dynamical investigation is performed applying frequency analysis and computing the differential fast Lyapunov indicators. After recalling a few properties of adiabatic invariants, we provide some examples of nearly-integrable dissipative systems borrowed from Celestial Mechanics, and precisely the spin-orbit coupling and the 3-body problem. We conclude with a discussion on the existence of periodic orbits in dissipative autonomous and non-autonomous systems.
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