Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields

We study bifurcations of two types of homoclinic orbits—a homoclinic orbit with resonant eigenvalues and an inclination-flip homoclinic orbit. For the former, we prove thatN-homoclinic orbits (N3) never bifurcate from the original homoclinic orbit. This answers a problem raised by Chow-Deng-Fiedler (J. Dynam. Diff. Eq. 2, 177–244, 1990). For the latter, we investigate mainlyN-homoclinic orbits andN-periodic orbits forN=1, 2 and determine whether they bifurcate or not under an additional condition on the eigenvalues of the linearized vector field around the equilibrium point.     

Homoclinic bifurcation at resonant eigenvalues

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.Partially supported by DARPA and NSF.Partially supported by the Deutsche Forschungsgemeinschaft and by Konrad-Zuse-Zentrum für Informationstechnik Berlin.     

N-Homoclinic bifurcations for homoclinic orbits changing their twisting

We study bifurcations, calledN-homoclinic bifurcations, which produce homoclinic orbits roundingN times (N2) in some tubular neighborhood of original homoclinic orbit. A family of vector fields undergoes such a bifurcation when it is a perturbation of a vector field with a homoclinic orbit.N-Homoclinic bifurcations are divided into two cases; one is that the linearization at the equilibrium has only real principal eigenvalues, and the other is that it has complex principal eigenvalues. We treat the former case, espcially that linearization has only one unstable eigenvalue. As main tools we use a topological method, namely, Conley index theory, which enables us to treat more degenerate cases than those studied by analytical methods.     

Reversible Bifurcation of Homoclinic Solutions in Presence of an Essential Spectrum

We consider bifurcations of a class of infinite dimensional reversible dynamical systems which possess a family of symmetric equilibria near the origin. We also assume that the linearized operator at the origin L_ɛ has an essential spectrum filling the entire real line, in addition to the simple eigenvalue at 0. Moreover, for parameter values ɛ < 0 there is a pair of imaginary eigenvalues which meet in 0 for ɛ = 0, and which disappear for ɛ > 0. The above situation occurs for example when one looks for travelling waves in a system of superposed perfect fluid layers, one being infinitely deep. We give quite general assumptions which apply in such physical examples, under which one obtains a family of bifurcating solutions homoclinic to every equilibrium near the origin. These homoclinics are symmetric and decay algebraically at infinity, being approximated at main order by the Benjamin–Ono homoclinic. For the water wave example, this corresponds to a family of solitary waves, such that at infinity the upper layer slides with a uniform velocity, over the bottom layer (at rest).     

Homoclinic Shadowing

A new method for rigorously establishing the existence of a transversal homoclinic orbit to a periodic orbit (or a fixed point) of diffeomorphisms in Rⁿ is presented. It is a computer-assisted technique with two main components. First, a global Newton’s method is devised to compute a suitable pseudo (approximate) homoclinic orbit to a pseudo periodic orbit. Then, a homoclinic shadowing theorem, which is proved herein, is invoked to establish the existence of a true transversal homoclinic orbit to a true periodic orbit near these pseudo orbits.     

Minimizing Periodic Orbits with Regularizable Collisions in the <Emphasis Type="Italic">n</Emphasis>-Body Problem

In the Newtonian n-body problem, there are various subsystems with two degrees of freedom, such as the collinear three-body problem and the isosceles three-body problem. After we determine a normal form of the Lagrangians of these subsystems, we prove the existence of periodic solutions with regularizable collisions for these systems. Our result includes several examples, such as Schubart’s orbit with or without equal masses, among others.     

Normalizations with Exponentially Small Remainders for Nonautonomous Analytic Periodic Vector Fields

In this paper we deal with analytic nonautonomous vector fields with a periodic time-dependency, that we study near an equilibrium point. In a first part, we assume that the linearized system is split in two invariant subspaces E ₀ and E ₁. Under light diophantine conditions on the eigenvalues of the linear part, we prove that there is a polynomial change of coordinates in E ₁ allowing to eliminate up to a finite polynomial order all terms depending only on the coordinate u₀ ? E₀{u_0 \in E_0} in the E ₁ component of the vector field. We moreover show that, optimizing the choice of the degree of the polynomial change of coordinates, we get an exponentially small remainder. In the second part, we prove a normal form theorem with exponentially small remainder. Similar theorems have been proved before in the autonomous case: this paper generalizes those results to the nonautonomous periodic case.     

On a Model of a Currency Exchange Rate – Local Stability and Periodic Solutions

A delay differential equation is presented which models how the behavior of traders influences the short time price movements of an asset. Sensitivity to price changes is measured by a parameter a. There is a single equilibrium solution, which is non-hyperbolic for all a>0. We prove that for a< 1 the equilibrium is asymptotically stable, and that for a>1 a 2-dimensional global center-unstable manifold connects the equilibrium to a periodic orbit. Its birth at a=1 is not of Hopf type and seems part of a Takens–Bogdanov scenario.     

Travelling Breathers in Klein-Gordon Lattices as Homoclinic Orbits to <Emphasis Type="BoldItalic">p</Emphasis>-Tori

Klein-Gordon chains are one-dimensional lattices of nonlinear oscillators in an anharmonic on-site potential, linearly coupled with their first neighbors. In this paper, we study the existence in such networks of spatially localized solutions, which appear time periodic in a referential in translation at constant velocity. These solutions are called travelling breathers. In the case of travelling wave solutions, the existence of exact solutions has been obtained by Iooss and Kirchgässner. Formal multiscale expansions have been used by Remoissenet to derive approximate solutions of travelling breathers in the form of modulated plane waves. James and Sire have studied the existence of specific travelling breather solutions, consisting in pulsating travelling waves which are exactly translated of 2 lattice sites after a fixed propagation time T. In this paper, we generalize this approach to pulsating travelling waves which are exactly translated of p≥ 3 sites after a given time T p being arbitrary. By formulating the problem as a dynamical system, one is able to reduce the system locally to a finite dimensional set of ordinary differential equations (ODE), whose dimension depends on the parameter values of the problem. We prove that the principal part of this system of ODE admits homoclinic connections to p-tori under general conditions on the potential. One can obtain leading order approximations of these homoclinic connections and these orbits should correspond, for the oscillator chain, to small amplitude travelling breather solutions superposed on an exponentially small quasi-periodic tail.     

Asymptotic Two-Soliton Solutions in the Fermi-Pasta-Ulam Model

We prove the existence of asymptotic two-soliton states in the Fermi-Pasta-Ulam model with general interaction potential. That is, we exhibit solutions whose difference in ℓ ² from the linear superposition of two solitary waves goes to zero as time goes to infinity.     

Continuation of the Periodic Orbits for the Differential Equation with Discontinuous Right Hand Side

In this paper the discontinuous system with one parameter perturbation is considered. Here the unperturbed system is supposed to have at least either one periodic orbit or a limit cycle. The aim is to prove the continuation of the periodic orbits under perturbation by means of the bifurcation map and the zeroes of this map imply the periodic orbits for the perturbed system. The tools for this problem are jumps of fundamental matrix solutions and the Poincare map for discontinuous systems. Therefore, we develop the Diliberto theorem and variation lemma for the system with discontinuous right hand side. At the end, as application of our method, the effect of discontinuous damping on Van der pol equation, and the effect of small force on the discontinuous linear oscillator with add a ·sgn(x) are considered.     

The existence of infinitely many homoclinic doubling bifurcations from some codimension 3 homoclinic orbits

An inclination-flip homoclinic orbit of weak type on ³ is a homoclinic orbit given as the intersection of a special one-dimensionalC ²-weak stable manifold and the one-dimensional unstable manifold of a hyperbolic singularity with three real eigenvalues. In this paper, we show that in a generic unfolding of such a homoclinic orbit, there appear curves in the parameter space that correspond to ordinary inclination-flip homoclinic orbit of orderN for any integerN. As a consequence, there exist infinitely many homoclinic doubling bifurcation curves emanating from the codimension three degenerate point corresponding to the inclination flip homoclinic orbit of weak type.     

Bifurcations of Heteroclinic Orbits

The search for traveling wave solutions of a semilinear diffusion partial differential equation can be reduced to the search for heteroclinic solutions of the ordinary differential equation ü − cu̇ + f(u) = 0, where c is a positive constant and f is a nonlinear function. A heteroclinic orbit is a solution u(t) such that u(t) → γ ₁ as t → −∞ and u(t) → γ ₂ as t → ∞ where γ ₁, γ ₂ are zeros of f. We study the existence of heteroclinic orbits under various assumptions on the nonlinear function f and their bifurcations as c is varied. Our arguments are geometric in nature and so we make only minimal smoothness assumptions. We only assume that f is continuous and that the equation has a unique solution to the initial value problem. Under these weaker smoothness conditions we reprove the classical result that for large c there is a unique positive heteroclinic orbit from 0 to 1 when f(0) = f(1) = 0 and f(u) > 0 for 0 < u < 1. When there are more zeros of f, there is the possibility of bifurcations of the heteroclinic orbit as c varies. We give a detailed analysis of the bifurcation of the heteroclinic orbits when f is zero at the five points −1 < −θ < 0 < θ < 1 and f is odd. The heteroclinic orbit that tends to 1 as t → ∞ starts at one of the three zeros, −θ, 0, θ as t → −∞. It hops back and forth among these three zeros an infinite number of times in a predictable sequence as c is varied.     

Periodic Orbits in Hamiltonian Systems with Involutory Symmetries

We study the existence of families of periodic solutions in a neighbourhood of a symmetric equilibrium point in two classes of Hamiltonian systems with involutory symmetry. In both classes, the involution reverses the sign of the Hamiltonian function, and the system is in 1:1 resonance. In the first class we study a Hamiltonian system with a reversing involution R acting symplectically. We first recover a result of Buzzi and Lamb showing that the equilibrium point is contained in a three dimensional conical subspace which consists of a two parameter family of periodic solutions with symmetry R, and furthermore that there may or may not exist two families of non-symmetric periodic solutions, depending on the coefficients of the Hamiltonian (correcting a minor error in their paper). In the second problem we study an equivariant Hamiltonian system with a symmetry S that acts anti-symplectically. Generically, there is no S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we prove the existence of at least 2 and at most 12 families of non-symmetric periodic solutions. We conclude with a brief study of systems with both forms of symmetry, showing they have very similar structure to the system with symmetry R.     

Center Manifolds for Homoclinic Solutions

In this article, center-manifold theory is developed for homoclinic solutions of ordinary differential equations or semilinear parabolic equations. A center manifold along a homoclinic solution is a locally invariant manifold containing all solutions which stay close to the homoclinic orbit in phase space for all times. Therefore, as usual, the low-dimensional center manifold contains the interesting recurrent dynamics near the homoclinic orbit, and a considerable reduction of dimension is achieved. The manifold is of class C ^1, for some >0. As an application, results of Shilnikov about the occurrence of complicated dynamics near homoclinic solutions approaching saddle-foci equilibria are generalized to semilinear parabolic equations.     

On the Homoclinic Orbits in a Class of Two-Degree-of-Freedom Systems under the Resonance Conditions

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Bifurcations of double homoclinic flip orbits with resonant eigenvalues

Concerns double homoclinic loops with orbit flips and two resonant eigen- values in a four-dimensional system.We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium,and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops,then compose them to get the important Poincarémap.A simple cal- culation gives explicitly an expression of the associated successor function.By a delicate analysis of the bifurcation equation,we obtain the condition that the original double homoclinic loops are kept,and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface,2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions.We also locate the large periodic orbits and large homoclinic orbits and their number.     

Singular Perturbations, Transversality, and Sil'nikov Saddle-Focus Homoclinic Orbits

We consider the singularly perturbed system $\dot x$ =εf(x,y,ε,λ), $\dot y$ =g(x,y,ε,λ). We assume that for small (ε,λ), (0,0) is a hyperbolic equilibrium on the normally hyperbolic centre manifold y=0 and that y ₀(t) is a homoclinic solution of $\dot y$ =g(0,y,0,0). Under an additional condition, we show that there is a curve in the (ε,λ) parameter space on which the perturbed system has a homoclinic orbit also. We investigate the transversality properties of this orbit and use our results to give examples of 4 dimensional systems with Sil'nikov saddle-focus homoclinic orbits.     

Study of Two-Dimensional Axisymmetric Breathers Using Padé Approximants

We analyze axisymmetric, spatially localized standing wave solutions with periodic time dependence (breathers) of a nonlinear partial differential equation. This equation is derived in the 'continuum approximation' of the equations of motion governing the anti-phase vibrations of a two-dimensional array of weakly coupled nonlinear oscillators. Following an asymptotic analysis, the leading order approximation of the spatial distribution of the breather is shown to be governed by a two-dimensional nonlinear Schrödinger (NLS) equation with cubic nonlinearities. The homoclinic orbit of the NLS equation is analytically approximated by constructing [2N × 2N] Padé approximants, expressing the Padé coefficients in terms of an initial amplitude condition, and imposing a necessary and sufficient condition to ensure decay of the Padé approximations as the independent variable (radius) tends to infinity. In addition, a convergence study is performed to eliminate 'spurious' solutions of the problem. Computation of this homoclinic orbit enables the analytic approximation of the breather solution.