Diffeomorphisms of the plane with stable periodic points

We consider self-diffeomorphisms of the plane with a hyperbolic fixed point and a nontransversal homoclinic point. We show that a neighborhood of the homoclinic point may contain countably many stable periodic sets whose characteristic exponents are bounded away from zero.     

On bifurcations of multidimensional diffeomorphisms having a homoclinic tangency to a saddle-node

We study the main bifurcations of multidimensional diffeomorphisms having a nontransversal homoclinic orbit to a saddle-node fixed point. On a parameter plane we build a bifurcation diagram for single-round periodic orbits lying entirely in a small neighborhood of the homoclinic orbit. Also, a relation of our results to the well-known codimension one bifurcations of a saddle fixed point with a quadratic homoclinic tangency and a saddle-node fixed point with a transversal homoclinic orbit is discussed.     

Critical homoclinic orbits lead to snap-back repellers

When nondegenerate homoclinic orbits to an expanding fixed point of a map f:X→X,X⊆Rⁿ, exist, the point is called a snap-back repeller. It is known that the relevance of a snap-back repeller (in its original definition) is due to the fact that it implies the existence of an invariant set on which the map is chaotic. However, when does the first homoclinic orbit appear? When can other homoclinic explosions, i.e., appearance of infinitely many new homoclinic orbits, occur? As noticed by many authors, these problems are still open. In this work we characterize these bifurcations, for any kind of map, smooth or piecewise smooth, continuous or discontinuous, defined in a bounded or unbounded closed set. We define a noncritical homoclinic orbit and a homoclinic orbit of an expanding fixed point is structurally stable iff it is noncritical. That is, only critical homoclinic orbits are responsible for the homoclinic explosions. The possible kinds of critical homoclinic orbits will be also investigated, as well as their dynamic role.     

Slow Passage Through the Nonhyperbolic Homoclinic Orbit of the Saddle-Center Hamiltonian Bifurcation

Slowly varying Hamiltonian systems, for which action is a well-known adiabatic invariant, are considered in the case where the system undergoes a saddle center bifurcation. We analyze the situation in which the solution slowly passes through the nonhyperbolic homoclinic orbit created at the saddle-center bifurcation. The solution near this homoclinic orbit consists of a large sequence of homoclinic orbits surrounded by near approaches to the autonomous nonlinear nonhyperbolic saddle point. By matching this solution to the strongly nonlinear oscillations obtained by averaging before and after crossing the homoclinic orbit, we determine the change in the action. If one orbit comes sufficiently close to the nonlinear saddle point, then that one saddle approach instead satisfies the nonautonomous first Painlevé equation, whose stable manifold of the unstable saddle (created in the saddle-center bifurcation) separates solutions approaching the stable center from those involving sequences of nearly homoclinic orbits.     

To the question of stability of periodic points of three-dimensional diffeomorphisms

Self-diffeomorphisms of three-dimensional space with a hyperbolic fixed point at the origin and a nontransversal point homoclinic to it are considered. It is assumed that the Jacobian matrix of the initial diffeomorphism has complex eigenvalues at the origin. It is shown that, under certain conditions imposed mainly on the character of tangency of the stable and unstable manifolds, a neighborhood of the nontransversal homoclinic point contains an infinite set of stable periodic points whose characteristic exponents are bounded away from zero.     

Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point

It is known that branches of homoclinic orbits emanate froma singular point of a dynamical system with a double zero eigenvalue(Takens-Bogdanov point). We develop a robust numerical methodfor starting the computation of homoclinic branches near sucha point. It is shown that this starting procedure relates tobranch switching. In particular, for a certain transformed problemthe homoclinic predictor is guaranteed to converge to the trueorbit under a Newton iteration.     

Homoclinic Orbits for a Class of Fractional Hamiltonian Systems via Variational Methods

This paper is devoted to the existence and multiplicity of homoclinic orbits for a class of fractional-order Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Here, we present a new approach via variational methods and critical point theory to obtain sufficient conditions under which the Hamiltonian system has at least one homoclinic orbit or multiple homoclinic orbits. Some results are new even for second-order Hamiltonian systems.     

Birth of discrete Lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points

It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is nonsimple.     

Numerical approximation of homoclinic chaos

Transversal homoclinic orbits of maps are known to generate a Cantor set on which a power of the map conjugates to the Bernoulli shift on two symbols. This conjugacy may be regarded as a coding map, which for example assigns to a homoclinic symbol sequence a point in the Cantor set that lies on a homoclinic orbit of the map with a prescribed number of humps. In this paper we develop a numerical method for evaluating the conjugacy at periodic and homoclinic symbol sequences in a systematic way. The approach combines our previous method for computing the primary homoclinic orbit with the constructive proof of Smale's theorem given by Palmer. It is shown that the resulting nonlinear systems are well conditioned uniformly with respect to the characteristic length of the symbol sequence and that Newton's method converges uniformly too when started at a proper pseudo orbit. For the homoclinic symbol sequences an error analysis is given. The method works in arbitrary dimensions and it is illustrated by examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stability of Periodic Points of Diffeomorphisms of Multidimensional Space

We study the diffeomorphism of a multidimensional space into itself with a hyperbolic fixed point at the origin and a nontransversal homoclinic point. From the works of Sh. Newhouse, B.F. Ivanov, L.P. Shilnikov, and other authors, it follows that there is a method of tangency for the stable and unstable manifold such that the neighborhood of a nontransversal homoclinic point can contain an infinite set of stable periodic points, but at least one of the characteristic exponents of those points tends to zero as the period increases. In this paper, we study diffeomorphisms such that the method of tangency for the stable and unstable manifold differs from the case studied in the works of the abovementioned authors. This paper continues previous works of the author, where diffeomorphisms are studied such that their Jacobi matrices at the origin have only real eigenvalues. In those previous works, we find conditions such that the neighborhood of a nontransversal homoclinic point of the studied diffeomorphism contains an infinite set of stable periodic points with characteristic exponents separated from zero. In the present paper, it is assumed that the Jacobi matrix of the original diffeomorphism at the origin has real eigenvalues and several pairs of complex conjugate eigenvalues. Under this assumption, we find conditions guaranteeing that a neighborhood of a nontransversal homoclinic point contains an infinite set of stable periodic points with characteristic exponents separated from zero.     

An example of chaotic behaviour in presence of a sliding homoclinic orbit

We present an example on the chaotic behaviour of a 3-dimensional quasiperiodically perturbed discontinuous differential equation whose unperturbed part has a homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity plane. Melnikov type analysis is applied.     

Smooth diffeomorphisms of the plane with stable periodic points in a neighborhood of a homoclinic point

We consider self-diffeomorphisms of the plane of the class C ^r (1 ?? r < ??) with a fixed hyperbolic point and a nontransversal point homoclinic to it. We present a method for constructing a set of diffeomorphisms for which the neighborhood of a homoclinic point contains countably many stable periodic points with characteristic exponents bounded away from zero.     

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.     

Invariant manifolds of the Bonhoeffer–van der Pol oscillator

The stable and unstable manifolds of a saddle fixed point (SFP) of the Bonhoeffer–van der Pol oscillator are numerically studied. A correspondence between the existence of homoclinic tangencies (which are related to the creation or destruction of Smale horseshoes) and the chaos observed in the bifurcation diagram is described. It is observed that in the non-chaotic zones of the bifurcation diagram, there may or may not be Smale horseshoes, but there are no homoclinic tangencies.     

Numerical continuation of degenerate homoclinic orbits in planar systems

In this paper we develop numerical algorithms for the continuationof degenerate homoclinic connections in planar systems. We considerthe case where the equilibrium point has zero trace and twocases of higher-order degeneracies. The method we propose isable to continue homoclinic connections of order up to codimension-four.Application of the algorithm to four examples supports its validityand demonstrates its usefulness.     

Existence of homoclinic solutions for higher-order periodic difference equations with p-Laplacian

Using the critical point theory in combination with periodic approximations, we establish sufficient conditions on the existence of homoclinic solutions for higher-order periodic difference equations with p-Laplacian. Our results provide rather weaker conditions to guarantee the existence of homoclinic solutions and considerably improve some existing ones even for some special cases.     

Homoclinic Orbits of Nonlinear Functional Difference Equations

In this paper, by using the critical point theory, we obtain the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of nontrivial homoclinic orbits which decay exponentially at infinity is obtained.      

Homoclinic orbits for nonlinear difference equations containing both advance and retardation

In this paper we discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of homoclinic orbits is obtained.     

Homoclinic orbits for second order nonlinear <Emphasis Type="Italic">p</Emphasis>-Laplacian difference equations

The paper proves the existence of nontrivial homoclinic orbits for second order nonlinear p-Laplacian difference equations without assumptions on periodicity using the critical point theory. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of nontrivial homoclinic orbits is proved.     

Stable nonperiodic points of two-dimensional C<Superscript>1</Superscript>-diffeomorphisms

It is proved that there exist two-dimensional diffeomorphisms with countably many stable periodic points in a neighborhood of a homoclinic point. The characteristic exponents of these points are negative and bounded away from zero.