arXiv subjects

A. Granados

Publications and source records attributed to A. Granados.

3 recordsLinked to original sources

arXiv2015

From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation

We study canard solutions of the forced van der Pol (fvdP) equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. For low-frequency forcing, we demonstrate the existence of primary maximal canards induced by folded saddle-nodes of type I, and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. We then turn to the intermediate- and high-frequency forcing regimes, and show that the fvdP equation possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms our numerical observation. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that their fold curves turn around in the intermediate-frequency regime. We identify the mechanism responsible for this turning. Finally, we show that the fvdP equation is a normal form for a class of single-frequency periodically-driven slow/fast systems with two fast variables and one slow variable which possess a nondegenerate fold of limit cycles.

math.DS↗

arXiv2013

The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

We consider a non-autonomous dynamical system formed by coupling two piecewise-smooth systems in $\RR^2$ through a non-autonomous periodic perturbation. We study the dynamics around one of the heteroclinic orbits of one of the piecewise-smooth systems. In the unperturbed case, the system possesses two $C^0$ normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them. These heteroclinic manifolds are foliated by heteroclinic connections between $C^0$ tori located at the same energy levels. By means of the {\em impact map} we prove the persistence of these objects under perturbation. In addition, we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of Melnikov-like functions. The heteroclinic manifolds allow us to define the {\em scattering map}, which links asymptotic dynamics in the invariant manifolds through heteroclinic connections. First order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds. Hence we have an essential tool for the construction of a heteroclinic skeleton which, when followed, can lead to the existence of Arnol'd diffusion: trajectories that, on large time scales, destabilize the system by further accumulating energy. We validate all the theoretical results with detailed numerical computations of a mechanical system with impacts, formed by the linkage of two rocking blocks with a spring.

math.DS↗

arXiv2012

The Melnikov method and subharmonic orbits in a piecewise smooth system

In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold $x=0$. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the system. We also suppose that the system possesses an invisible fold-fold at the origin and two heteroclinic orbits connecting two hyperbolic critical points on either side of $x=0$. Finally, we assume that the region closed by these heteroclinic connections is fully covered by periodic orbits surrounding the origin, whose periods monotonically increase as they approach the heteroclinic connection. When considering a non-autonomous ($T$-periodic) Hamiltonian perturbation of amplitude $\varepsilon$, using an impact map, we rigorously prove that, for every $n$ and $m$ relatively prime and $\varepsilon>0$ small enough, there exists a $nT$-periodic orbit impacting $2m$ times with the switching manifold at every period if a modified subharmonic Melnikov function possesses a simple zero. We also prove that, if the orbits are discontinuous when they cross $x=0$, then all these orbits exist if the relative size of $\varepsilon>0$ with respect to the magnitude of this jump is large enough. We also obtain similar conditions for the splitting of the heteroclinic connections.

math.DS↗