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Register a new userPeriodic forcing of a heteroclinic network
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We present a comprehensive mechanism for the emergence of rotational horseshoes and strange attractors in a class of two-parameter families of periodically-perturbed differential equations defining a flow on a three-dimensional manifold. When both parameters are zero, its flow exhibits an attracting heteroclinic network associated to two periodic solutions. After slightly increasing both parameters, while keeping a two-dimensional connection unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the periodic solutions do not intersect. We prove a wide range of dynamical behaviour, ranging from an attracting quasi-periodic torus to rotational horseshoes and Henon-like strange attractors. We illustrate our results with an explicit example.
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There are few examples of non-autonomous vector fields exhibiting complex dynamics that may be proven analytically. We analyse a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two-sphere. We derive the first return map near the heteroclinic cycle for small amplitude of the perturbing term, and we reduce the analysis of the non-autonomous system to that of a two-dimensional map on a cylinder. Interesting dynamical features arise from a discrete-time Bogdanov-Takens bifurcation. When the perturbation strength is small the first return map has an attracting invariant closed curve that is not contractible on the cylinder. Near the centre of frequency locking there are parameter values with bistability: the invariant curve coexists with an attracting fixed point. Increasing the perturbation strength there are periodic solutions that bifurcate into a closed contractible invariant curve and into a region where the dynamics is conjugate to a full shift on two symbols.
This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits a weakly attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the ghost of the attractor and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov-Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the classical Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory.
Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and Maslov indices. For chaotic systems, the relative actions of such orbits can be expressed in terms of phase space areas bounded by segments of stable and unstable manifolds, and Moser invariant curves. This also generates direct relations between periodic orbits and homoclinic (heteroclinic) orbit actions. Simpler, explicit approximate expressions following from the exact relations are given with error estimates. They arise from asymptotic scaling of certain bounded phase space areas. The actions of infinite subsets of periodic orbits are determined by their periods and the locations of the limiting homoclinic points on which they accumulate.
A cyclic permutation $pi:{1, dots, N}to {1, dots, N}$ has a emph{block structure} if there is a partition of ${1, dots, N}$ into $k otin{1,N}$ segments (emph{blocks}) permuted by $pi$; call $k$ the emph{period} of this block structure. Let $p_1<dots <p_s$ be periods of all possible block structures on $pi$. Call the finite string $(p_1/1,$ $p_2/p_1,$ $dots,$ $p_s/p_{s-1}, N/p_s)$ the {it renormalization tower of $pi$}. The same terminology can be used for emph{patterns}, i.e., for families of cycles of interval maps inducing the same (up to a flip) cyclic permutation. A renormalization tower $mathcal M$ emph{forces} a renormalization tower $mathcal N$ if every continuous interval map with a cycle of pattern with renormalization tower $mathcal M$ must have a cycle of pattern with renormalization tower $mathcal N$. We completely characterize the forcing relation among renormalization towers. Take the following order among natural numbers: $ 4gg 6gg 3gg dots gg 4ngg 4n+2gg 2n+1ggdots gg 2gg 1 $ understood in the strict sense. We show that the forcing relation among renormalization towers is given by the lexicographic extension of this order. Moreover, for any tail $T$ of this order there exists an interval map for which the set of renormalization towers of its cycles equals $T$.
Equilibrium, traveling wave, and periodic orbit solutions of pipe, channel, and plane Couette flows can now be computed precisely at Reynolds numbers above the onset of turbulence. These invariant solutions capture the complex dynamics of wall-bounded rolls and streaks and provide a framework for understanding low-Reynolds turbulent shear flows as dynamical systems. We present fluid dynamics videos of plane Couette flow illustrating periodic orbits, a close pass of turbulent flow to a periodic orbit, and heteroclinic connections between unstable equilibria.
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