Exponential Dichotomies,the Fredholm Alternative,and Transverse Homoclinic Orbits in Partial Functional Differential Equations

In this paper we investigate the transversality of homoclinic orbits in partial functional differential equations. We first discuss the exponential dichotomies for linear operator equations. Then we show that the Fredholm Alternative holds if the homogeneous equation has exponential dichotomies on R. Transversality of homoclinic orbits for periodically perturbed partial functional differential equations is studied using the Liapunov-Schmidt method and the Melnikov integral. Ams Subject Classifications: 35R10; 58F14.     

Homoclinic Twist Bifurcation in a System of Two Coupled Oscillators

We analyse the dynamics of two identical Josephson junctions coupled through a purely capacitive load in the neighborhood of a degenerate symmetric homoclinic orbit. A bifurcation function is obtained applying Lin's version of the Lyapunov–Schmidt reduction. We locate in parameter space the region of existence of n-periodic orbits, and we prove the existence of n-homoclinic orbits and bounded nonperiodic orbits. A singular limit of the bifurcation function yields a one-dimensional mapping which is analyzed. Numerical computations of nonsymmetric homoclinic orbits have been performed, and we show the relevance of these computations by comparing the results with the analysis.     

Non-existence of Shilnikov chaos in continuous-time systems

In this paper,a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional,continuous-time,and smooth systems is obtained.Based on this result and an elementary example,it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation(ODE) systems characterized by the nonexistence of homoclinic and heteroclinic orbits.     

Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study

In earlier paper we have developed a numerical method for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in ⁿ . The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using linear approximation of the unstable and stable manifolds. In this paper we extend our algorithm to incorporate higher-order approximations of the unstable and stable manifolds. This approximation is especially useful if we want to compute center manifolds accurately. A procedure for switching between the periodic approximation of homoclinic orbits and the higher-order approximation of homoclinic orbits provides additional flexibility to the method. The algorithm is applied to a model problem: the DC Josephson Junction. Computations are done using the software package AUTO.     

Numerical Continuation of Homoclinic Orbits to Non-Hyperbolic Equilibria in Planar Systems

In this paper we develop numerical algorithms for thecontinuation of degenerate homoclinic orbits to non-hyperbolicequilibria in planar systems. The first situation corresponds to asaddle-node equilibrium (a zero eigenvalue) and the second one is theso-called cuspidal loop (double-zero eigenvalue). The methods proposedmay deal with codimension-two and -three homoclinic connections.Application of the algorithms to several examples supports its validityand demonstrates its usefulness.     

STUDIES OF MELNIKOV METHOD AND TRANSVERSAL HOMOCLINIC ORBITS IN THE CIRCULAR PLANAR RESTRICTED THREE－BODY PROBLEM

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Orbits Homoclinic to Exponentially Small Periodic Orbits for a Class of Reversible Systems. Application to Water Waves

In this paper a class of reversible analytic vector fields is investigated near an equilibrium. For these vector fields, the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues , . In the first part of this paper, we study the 4-dimensional problem. The existence of a family of solutions homoclinic to periodic orbits of size less than μ^N for any fixed N, where μ is the bifurcation parameter, is known for vector fields. Using the analyticity of the vector field, we prove here the existence of solutions homoclinic to a periodic orbit the size of which is exponentially small ( of order . This result receives its significance from the still unsolved question of whether there exist solutions that are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. In the second part of this paper we prove that the exponential estimates still hold in infinite dimensions. This result cannot be simply obtained from the study of the 4-dimensional analysis by a center-manifold reduction since this result is based on analyticity of the vector field. One example of such a vector field in infinite dimensions occurs when describing the irrotational flow of an inviscid fluid layer under the influence of gravity and small surface tension (Bond number ) for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalized solitary wave. Our work shows that there exist generalized solitary waves with exponentially small oscillations at infinity. More precisely, we prove that for each F close enough to 1, there exist two reversible solutions homoclinic to a periodic orbit, the size of which is less than , l being any number between 0 and π and satisfying . (Accepted October 2, 1995)     

Symmetry-Breaking Homoclinic Bifurcations in Diffusively Coupled Equations

In this study we examine a symmetry-breaking bifurcation of homoclinic orbits in diffusively coupled ordinary differential equations. We prove that asymmetric homoclinic orbits can bifurcate from a symmetric homoclinic orbit when the equilibria to which the latter is homoclinic undergoes a pitchfork bifurcation. A condition which defines the direction of the bifurcation in a parameter space is given. All hypotheses of the main theorem are verified for a diffusively coupled logistic system and the twistedness of the bifurcating homoclinic orbits is computed for a range of coupling strengths.     

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.     

Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in \mathbbR⁴{\mathbb{R}^4} unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in \mathbbR³{\mathbb{R}^3} unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.     

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.     

Solar sail dynamics in the three-body problem: Homoclinic paths of points and orbits

In this paper we consider the orbital dynamics of a solar sail in the Earth-Sun circular restricted three-body problem. The equations of motion of the sail are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the sail. We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described.     

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.     

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.     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.     

Homoclinic Tubes in Discrete Nonlinear Schrödinger Equation under Hamiltonian Perturbations

In this paper, we study the discrete cubic nonlinear Schrödinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In particular, Bäcklund–Darboux transformations are utilized to generate heteroclinic orbits and Melnikov vectors. Then we give coordinate-expressions for persistent invariant manifolds and Fenichel fibers for the perturbed lattice. Finally based upon the above machinery, existence of codimension 2 transversal homoclinic tubes is established through a Melnikov type calculation and an implicit function argument. We also discuss symbolic dynamics of invariant tubes each of which consists of a doubly infinite sequence of curve segments when the lattice is four dimensional. Structures inside the asymptotic manifolds of the transversal homoclinic tubes are studied, special orbits, in particular homoclinic orbits and heteroclinic orbits when the lattice is four dimensional, are studied.     

Homoclinic-Doubling Cascades

Cascades of period-doubling bifurcations have attracted much interest from researchers of dynamical systems in the past two decades as they are one of the routes to onset of chaos. In this paper we consider routes to onset of chaos involving homoclinic-doubling bifurcations. We show the existence of cascades of homoclinic-doubling bifurcations which occur persistently in two-parameter families of vector fields on ?³. The cascades are found in an unfolding of a codimension-three homoclinic bifurcation which occur an orbit-flip at resonant eigenvalues. We develop a continuation theory for homoclinic orbits in order to follow homoclinic orbits through infinitely many homoclinic-doubling bifurcations.     

Hyperbolic evolution groups and dichotomic evolution families

Recently, Ben-Artzi and Gohberg [2] used the concept ofC ₀-semigroups in order to characterize the existence of dichotomies for nonautonomous differential equations on _n. A similar task was performed by Latushkin and Stepin [11] for dichotomies of linear skew-product flows. In this paper we will useC _o-semigroups to characterize existence of dichotomies for strongly continuous evolution families (U(t,s)) _t.s on Hilbert and Banach spaces. Under an exponential growth condition we show that the concepts of hyperbolic evolution groups and exponentially dichotomic evolution families are equivalent.