Transverse intersections between invariant manifolds of doubly hyperbolic invariant tori, via the Poincaré-Mel’nikov method

We consider a perturbation of an integrable Hamiltonian system having an equilibrium point of elliptic-hyperbolic type, having a homoclinic orbit. More precisely, we consider an (n + 2)-degree-of-freedom near integrable Hamiltonian with n centers and 2 saddles, and assume that the homoclinic orbit is preserved under the perturbation. On the center manifold near the equilibrium, there is a Cantorian family of hyperbolic KAM tori, and we study the homoclinic intersections between the stable and unstable manifolds associated to such tori. We establish that, in general, the manifolds intersect along transverse homoclinic orbits. In a more concrete model, such homoclinic orbits can be detected, in a first approximation, from nondegenerate critical points of a Mel’nikov potential. We provide bounds for the number of transverse homoclinic orbits using that, in general, the potential will be a Morse function (which gives a lower bound) and can be approximated by a trigonometric polynomial (which gives an upper bound).     

Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov?s method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg–Landau partial differential equations, and demonstrate the theoretical results by numerical ones.     

Splitting Potential and the Poincaré-Melnikov Method for Whiskered Tori in Hamiltonian Systems

Summary. We deal with a perturbation of a hyperbolic integrable Hamiltonian system with n+1 degrees of freedom. The integrable system is assumed to have n -dimensional hyperbolic invariant tori with coincident whiskers (separatrices). Following Eliasson, we use a geometric approach closely related to the Lagrangian properties of the whiskers, to show that the splitting distance between the perturbed stable and unstable whiskers is the gradient of a periodic scalar function of n phases, which we call splitting potential. This geometric approach works for both the singular (or weakly hyperbolic) case and the regular (or strongly hyperbolic) case, and provides the existence of at least n+1 homoclinic intersections between the perturbed whiskers. In the regular case, we also obtain a first-order approximation for the splitting potential, that we call Melnikov potential. Its gradient, the (vector) Melnikov function, provides a first-order approximation for the splitting distance. Then the nondegenerate critical points of the Melnikov potential give rise to transverse homoclinic intersections between the whiskers. Generically, when the Melnikov potential is a Morse function, there exist at least 2 ⁿ critical points. The first-order approximation relies on the n -dimensional Poincaré-Melnikov method, to which an important part of the paper is devoted. We develop the method in a general setting, giving the Melnikov potential and the Melnikov function in terms of absolutely convergent integrals, which take into account the phase drift along the separatrix and the first-order deformation of the perturbed hyperbolic tori. We provide formulas useful in several cases, and carry out explicit computations that show that the Melnikov potential is a Morse function, in different kinds of examples. Received January 18, 1999; final revision received October 25, 1999; accepted December 12, 1999     

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

We prove the existence of chaotic motions in an equilateral planar circular restricted four body problem (CRFBP), establishing that the system is not integrable. The proof works by verifying the hypotheses of a topological forcing theorem for Hamiltonian vector fields on ${\mathbb{R}}^4$ which hypothesizes the existence of a transverse homoclinic orbit in the energy manifold of a saddle focus equilibrium. We develop mathematically rigorous computer assisted arguments for verifying these hypotheses, and provide an implementation for CRFBP. Due to the Hamiltonian structure, this also establishes the existence of a “blue sky catastrophe”, and hence an analytic family of periodic orbits of arbitrarily long period at nearby energy levels.Our method works far from any perturbative regime and requires no mass symmetry. Additionally, the method is constructive and yields additional byproducts such as the locations of transverse connecting orbits, quantitative information about the invariant manifolds, and bounds on transport times.     

Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.     

Symbolic dynamics and Arnold diffusion

We consider hyperbolic tori of three degrees of freedom initially hyperbolic Hamiltonian systems. We prove that if the stable and unstable manifold of a hyperbolic torus intersect transversaly, then there exists a hyperbolic invariant set near a homoclinic orbit on which the dynamics is conjugated to a Bernoulli shift. The proof is based on a new geometrico-dynamical feature of partially hyperbolic systems, the transversality-torsion phenomenon, which produces complete hyperbolicity from partial hyperbolicity. We deduce the existence of infinitely many hyperbolic periodic orbits near the given torus. The relevance of these results for the instability of near-integrable Hamiltonian systems is then discussed. For a given transition chain, we construct chain of hyperbolic periodic orbits. Then we easily prove the existence of periodic orbits of arbitrarily high period close to such chain using standard results on hyperbolic sets.     

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach is based on following the “outer dynamics” along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined “scattering map.” We show that for every finite sequence of successive iterations of the scattering map, there exists a true orbit that follows that sequence, provided that the inner dynamics is recurrent. We apply this result to prove the existence of diffusing orbits that cross large gaps in a priori unstable models of arbitrary degrees of freedom, when the unperturbed Hamiltonian is not necessarily convex and the induced inner dynamics is not necessarily a twist map, and the perturbation satisfies explicit conditions that are generic. We also mention several other applications where this mechanism is easy to verify (analytically or numerically), such as the planar elliptic restricted three-body problem and the spatial circular restricted three-body problem. Our method differs, in several crucial aspects, from earlier works. Unlike the well-known “two-dynamics” approach, the method we present here relies on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, such as on existence of its invariant objects (e.g., primary and secondary tori, lower-dimensional hyperbolic tori, and their stable/unstable manifolds, Aubry-Mather sets), which are not used at all. © 2019 Wiley Periodicals, Inc.     

Quasiperiodic Perturbations of Two-Dimensional Hamiltonian Systems

Quasiperiodic nonconservative perturbations of two-dimensional Hamiltonian systems are studied. The behavior of solutions in a neighborhood of resonance and nonresonance levels is considered. Conditions for the existence of resonant quasiperiodic solutions (m-dimensional resonance tori) are found, and the global behavior of solutions in domains separated from the unperturbed separatrices is discussed. The results are illustrated by the example of the Duffing equation with the homoclinic figure eight of a saddle.     

Global uniqueness of homoclinic orbits for a class of fourth order equations

In this paper we show the global existence and uniqueness of certain orbits homoclinic to the zero stationary solution of the fourth order equation

Shadowing Homoclinic Chains to a Symplectic Critical Manifold

We prove the existence of trajectories shadowing chains of heteroclinic orbits to a symplectic normally hyperbolic critical manifold of a Hamiltonian system.The results are quite different for real and complex eigenvalues. General results are applied to Hamiltonian systems depending on a parameter which slowly changes with rate ε. If the frozen autonomous system has a hyperbolic equilibrium possessing transverse homoclinic orbits, we construct trajectories shadowing homoclinic chains with energy having quasirandom jumps of order ε and changing with average rate of orderε| ln ε|. This provides a partial multidimensional extension of the results of A. Neishtadt on the destruction of adiabatic invariants for systems with one degree of freedom and a figure 8 separatrix.     

Horseshoes for the nearly symmetric heavy top

We prove the existence of horseshoes in the nearly symmetric heavy top. This problem was previously addressed but treated inappropriately due to a singularity of the equations of motion. We introduce an (artificial) inclined plane to remove this singularity and use a Melnikov-type approach to show that there exist transverse homoclinic orbits to periodic orbits on four-dimensional level sets. The price we pay for removing the singularity is that the Hamiltonian system becomes a three-degree-of-freedom system with an additional first integral, unlike the two-degree-of-freedom formulation in the classical treatment. We therefore have to analyze three-dimensional stable and unstable manifolds of periodic orbits in a six-dimensional phase space. A new Melnikov-type technique is developed for this situation. Numerical evidence for the existence of transverse homoclinic orbits on a four-dimensional level set is also given.     

Breaking hyperbolicity for smooth symplectic toral diffeomorphisms

We study a smooth symplectic 2-parameter unfolding of an almost hyperbolic diffeomorphism on two-dimensional torus. This diffeomorphism has a fixed point of the type of the degenerate saddle. In the parameter plane there is a bifurcation curve corresponding to the transition from the degenerate saddle into a saddle and parabolic fixed point, separatrices of these latter points form a channel on the torus. We prove that a saddle period-2 point exists for all parameter values close to the co-dimension two point whose separatrices intersect transversely the boundary curves of the channel that implies the existence of a quadratic homoclinic tangency for this period-2 point which occurs along a sequence of parameter values accumulating at the co-dimension 2 point. This leads to the break of stable/unstable foliations existing for almost hyperbolic diffeomorphism. Using the results of Franks [1] on π ₁-diffeomorphisms, we discuss the possibility to get an invariant Cantor set of a positive measure being non-uniformly hyperbolic.     

n-Dimensional stable and unstable manifolds of hyperbolic singular point

Invariant manifold play an important role in the qualitative analysis of dynamical systems, such as in studying homoclinic orbit and heteroclinic orbit. This paper focuses on stable and unstable manifolds of hyperbolic singular points. For a type of n-dimensional quadratic system, such as Lorenz system, Chen system, Rossler system if n = 3, we provide the series expression of manifolds near the hyperbolic singular point, and proved its convergence using the proof of the formal power series. The expressions can be used to investigate the heteroclinic orbits and homoclinic orbits of hyperbolic singular points.     

Homoclinic orbits for Hamiltonian systems induced by impulses

In this paper, we consider a class of impulsive Hamiltonian systems with a p‐Laplacian operator. Under certain conditions, we establish the existence of homoclinic orbits by means of the mountain pass theorem and an approximation technique. In some special cases, the homoclinic orbits are induced by the impulses in the sense that the associated non‐impulsive systems admit no non‐trivial homoclinic orbits. Copyright © 2015 John Wiley & Sons, Ltd.     

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.     

Multi-bump orbits homoclinic to resonance bands

We establish the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The homoclinic orbits, which include multi-bump \v{S}ilnikov orbits, connect equilibria and periodic orbits in the resonance band. The main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands.

     

Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system

This paper deals with existence and exponential decay of homoclinic orbits in the first-order Hamiltonian system

     

Homoclinic orbits for a nonperiodic Hamiltonian system

In this paper we prove the existence and multiplicity of homoclinic orbits for first order Hamiltonian systems of the form

     

Hunting for homoclinic orbits in reversible systems: A shooting technique

A dynamical system is said to be reversible if there is an involution of phase space that reverses the direction of the flow. Examples are Hamiltonian systems with quadratic potential energy. In such systems, homoclinic orbits that are invariant under the reversible transformation are typically not destroyed as a parameter is varied. A strategy is proposed for the direct numerical approximation to paths of such homoclinic orbits, exploiting the special properties of reversible systems. This strategy incorporates continuation using a simplification of known methods and a shooting approach, based on Newton's method, to compute starting solutions for continuation. For Hamiltonian systems, the shooting uses symplectic numerical integration. Strategies are discussed for obtaining initial guesses for the unknown parameters in Newton's method. An example system, for which there is an infinity of symmetric homoclinic orbits, is used to test the numerical techniques. It is illustrated how the orbits can be systematically located and followed. Excellent agreement is found between theory and numerics.This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4–14 July 1992 with support from the SERC under grant reference number GR/H03964.