Bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps

We study bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps. We distinguish two types of cubic homoclinic tangencies, and each type gives different first return maps derived to diverse conservative cubic H\'enon maps with quite different bifurcation diagrams. In this way, we establish the structure of bifurcations of periodic orbits in two parameter general unfoldings generalizing to the conservative case the results previously obtained for the dissipative case. We also consider the problem of 1:4 resonance for the conservative cubic H\'enon maps.Comment: 20 pages, 12 figure

Reversible perturbations of conservative Hénon-like maps

For area-preserving Hénon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, a new method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic orbits in reversible families containing quadratic conservative orientable and nonorientable Hénon maps as well as a product of two Hénon maps whose Jacobians are mutually inverse.Peer ReviewedPostprint (published version

Homoclinic phenomena in conservative systems

The goal of this thesis is the study of homoclinic orbits in conservative systems (area-preserving maps and Hamiltonian systems). We consider homoclinic (bi-asymptotic) orbits either to saddle periodic orbits or to whiskered tori. Such type orbits, called homoclinic by Poincaré, are of great interest in the theory of dynamical systems since their presence implies complicated dynamics. The thesis is divided in two parts according to two quite different topics considered. In the first part, we study area-preserving maps (APMs) with a nontransversal homoclinic orbit (homoclinic tangency) to a saddle fixed point in order to know the behavior of orbits near the given homoclinic trajectory. To this end, we construct first return maps, for which we use finitely-smooth normal forms of the saddle maps and introduce cross-coordinates. The fixed points of the first return maps correspond to single-round periodic orbits of the maps under consideration. Applying rescaling methods we derive the first return maps to the Hénon-like maps whose bifurcations are well known. Thus, translating the results obtained for the fixed points of the return maps to the periodic orbits, we prove the existence of cascades of elliptic periodic points. We also study the phenomenon of the coexistence of infinitely many single-round periodic orbits of different large periods (called global resonance). We consider the related problems in different types of APMs (symplectic maps and non-orientable APMs) with quadratic or cubic tangencies. We also establish the structure of 1:4 resonance for some conservative Hénon-like maps. The second part of the thesis is dedicated to the study of exponentially small splitting of separatrices arising from a perturbation of a Hamiltonian system with a homoclinic connection (separatrix). We consider a perturbation of an integrable Hamiltonian system having whiskered tori with coincident stable and unstable whiskers. Generally, in the perturbed system, the whiskers do not coincide anymore and our goal is to detect the transverse homoclinic orbits associated to the persistent whiskered tori. The perturbed system turns out to be not integrable due to the presence of these homoclinic trajectories and, consequently, there is chaotic dynamics near them. We give a suitable parametrization to the whiskers to determine the distance between them. This distance is given by the splitting function, and the simple zeros of this function give rise to transverse homoclinic orbits. We use the classical Poincaré-Melnikov approach to measure the splitting, although in the case of exponential smallness we have to ensure that the first order approximation overcome the error term. We consider Hamiltonian systems possessing two-dimensional whiskered tori with quadratic frequencies and three-dimensional whiskered tori with cubic golden frequency. In the two-dimensional case, we find 23 new quadratic numbers for which the Poincaré-Melnikov method can be applied and establish the existence of 4 transverse homoclinic orbits. We also study the continuation of the homoclinic orbits for all values of the parameter of perturbation in the case of the silver number sqrt(2)-1. For the three-dimensional whiskered torus with frequency vector given by the so-called "cubic golden number", we establish the existence of exponentially small splitting of separatrices and detect the transversality of 8 homoclinic orbits

Homoclinic phenomena in conservative systems

The goal of this thesis is the study of homoclinic orbits in conservative systems (area-preserving maps and Hamiltonian systems). We consider homoclinic (bi-asymptotic) orbits either to saddle periodic orbits or to whiskered tori. Such type orbits, called homoclinic by Poincaré, are of great interest in the theory of dynamical systems since their presence implies complicated dynamics. The thesis is divided in two parts according to two quite different topics considered. In the first part, we study area-preserving maps (APMs) with a nontransversal homoclinic orbit (homoclinic tangency) to a saddle fixed point in order to know the behavior of orbits near the given homoclinic trajectory. To this end, we construct first return maps, for which we use finitely-smooth normal forms of the saddle maps and introduce cross-coordinates. The fixed points of the first return maps correspond to single-round periodic orbits of the maps under consideration. Applying rescaling methods we derive the first return maps to the Hénon-like maps whose bifurcations are well known. Thus, translating the results obtained for the fixed points of the return maps to the periodic orbits, we prove the existence of cascades of elliptic periodic points. We also study the phenomenon of the coexistence of infinitely many single-round periodic orbits of different large periods (called global resonance). We consider the related problems in different types of APMs (symplectic maps and non-orientable APMs) with quadratic or cubic tangencies. We also establish the structure of 1:4 resonance for some conservative Hénon-like maps. The second part of the thesis is dedicated to the study of exponentially small splitting of separatrices arising from a perturbation of a Hamiltonian system with a homoclinic connection (separatrix). We consider a perturbation of an integrable Hamiltonian system having whiskered tori with coincident stable and unstable whiskers. Generally, in the perturbed system, the whiskers do not coincide anymore and our goal is to detect the transverse homoclinic orbits associated to the persistent whiskered tori. The perturbed system turns out to be not integrable due to the presence of these homoclinic trajectories and, consequently, there is chaotic dynamics near them. We give a suitable parametrization to the whiskers to determine the distance between them. This distance is given by the splitting function, and the simple zeros of this function give rise to transverse homoclinic orbits. We use the classical Poincaré-Melnikov approach to measure the splitting, although in the case of exponential smallness we have to ensure that the first order approximation overcome the error term. We consider Hamiltonian systems possessing two-dimensional whiskered tori with quadratic frequencies and three-dimensional whiskered tori with cubic golden frequency. In the two-dimensional case, we find 23 new quadratic numbers for which the Poincaré-Melnikov method can be applied and establish the existence of 4 transverse homoclinic orbits. We also study the continuation of the homoclinic orbits for all values of the parameter of perturbation in the case of the silver number sqrt(2)-1. For the three-dimensional whiskered torus with frequency vector given by the so-called "cubic golden number", we establish the existence of exponentially small splitting of separatrices and detect the transversality of 8 homoclinic orbits.Postprint (published version

A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies

The aim of this work is to provide asymptotic estimates for the splitting of separatrices in a perturbed 3-degree-of-freedom Hamiltonian system, associated to a 2-dimensional whiskered torus (invariant hyperbolic torus) whose frequency ratio is a quadratic irrational number. We show that the dependence of the asymptotic estimates on the perturbation parameter is described by some functions which satisfy a periodicity property, and whose behavior depends strongly on the arithmetic properties of the frequencies.Comment: 5 pages, 1 figur

Mixed dynamics of two-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.Peer ReviewedPostprint (author's final draft

On bifurcations of area-preserving and nonorientable maps with quadratic homoclinic tangencies

We study bifurcations of nonorientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable two-dimensional surfaces. We consider one- and two-parameter general unfoldings and establish results related to the emergence of elliptic periodic orbitPostprint (author’s final draft

Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector ¿/ev, with ¿=(1,O,O˜) where O is a cubic irrational number whose two conjugates are complex, and the components of ¿ generate the field Q(O). A paradigmatic case is the cubic golden vector, given by the (real) number O satisfying O3=1-O, and O˜=O2. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors ¿k,¿¿, k¿Z3. Applying the Poincaré–Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on e) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in e, and valid for all sufficiently small values of e. This estimate behaves like exp{-h1(e)/e1/6} and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function h1(e) in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector ¿, and proving that it is a quasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.Peer ReviewedPreprin