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Consider the geodesic flow on a real-analytic closed hypersurface M of R n , equipped with the induced metric. How commonly can we expect such flows to have a transverse homoclinic orbit? In this paper, we give the following two partial answers to this question: If M is a real-analytic closed hypersurface in R n (with n = 3) on which the geodesic flow with respect to the induced metric has a nonhyperbolic periodic orbit, then C ¿ -generically the geodesic flow on M with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit; and There is a C ¿ -open and dense set of real-analytic, closed, and strictly convex surfaces M in R 3 on which the geodesic flow with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit. These are among the first perturbation-theoretic results for real-analytic geodesic flows., Peer Reviewed, Postprint (published version)

In Chapters 1 and 2 we provide the theoretical underpinning of the parameterization method for center manifolds. In Chapter 1 -- which is published in Journal of Differential Equations -- we show how the classical parameterization method can be generalized to center manifolds associated with fixed points in discrete dynamical systems. We extend this new method in Chapter 2 to center manifolds for fixed points of ODEs. Besides showing that the parameterization method can be generalized, we also provide a way to obtain approximations of the center manifolds and explicit error bounds on those approximations. Furthermore, we simultaneously obtain similar bounds for the conjugate dynamics on the center manifold. In particular, we obtain qualitative information about the dynamical behaviour on the center manifold In the remainder of the thesis we use the theory from the first two chapters to compute orbits in two applications. In Chapter 3 -- which is published in Celestial Mechanics and Dynamical Astronomy -- we apply the method to numerically find or rule out homoclinic orbits at bifurcations in the Circular Restricted Four Body Problem. Finally, in Chapter 4, we use the parameterization method to prove the existence of spiral waves in the complex Ginzburg-Landau equation. In particular, in the last chapter we rigorously compute an approximation of the center manifold of the spiral wave equation, obtain an explicit error bound on this approximation, and find (part of) the basin of attraction on the center manifold.

In Chapters 1 and 2 we provide the theoretical underpinning of the parameterization method for center manifolds. In Chapter 1 -- which is published in Journal of Differential Equations -- we show how the classical parameterization method can be generalized to center manifolds associated with fixed points in discrete dynamical systems. We extend this new method in Chapter 2 to center manifolds for fixed points of ODEs. Besides showing that the parameterization method can be generalized, we also provide a way to obtain approximations of the center manifolds and explicit error bounds on those approximations. Furthermore, we simultaneously obtain similar bounds for the conjugate dynamics on the center manifold. In particular, we obtain qualitative information about the dynamical behaviour on the center manifold In the remainder of the thesis we use the theory from the first two chapters to compute orbits in two applications. In Chapter 3 -- which is published in Celestial Mechanics and Dynamical Astronomy -- we apply the method to numerically find or rule out homoclinic orbits at bifurcations in the Circular Restricted Four Body Problem. Finally, in Chapter 4, we use the parameterization method to prove the existence of spiral waves in the complex Ginzburg-Landau equation. In particular, in the last chapter we rigorously compute an approximation of the center manifold of the spiral wave equation, obtain an explicit error bound on this approximation, and find (part of) the basin of attraction on the center manifold.

To continue a branch of homoclinic solutions starting from a Bogdanov--Takens (BT) point in parameter and state space, one needs a predictor based on asymptotics for the bifurcation parameter values and the corresponding small homoclinic orbits in the phase space. We derive two explicit asymptotics for the homoclinic orbits near a generic BT point. A recent generalization of the Lindstedt--Poincaré (L-P) method is applied to approximate a homoclinic solution of a strongly nonlinear autonomous system that results from blowing up the BT normal form. This solution allows us to derive an accurate second-order homoclinic predictor to the homoclinic branch rooted at a generic BT point of an $n$-dimensional ordinary differential equation (ODE). We prove that the method leads to the same homoclinicity conditions as the classical Melnikov technique, the branching method, and the regular perturbation (R-P) method. However, it is known that the R-P method leads to a “parasitic turn” near the saddle point. The new asymptotics based on the L-P method do not have this turn, making them more suitable for numerical implementation. We show how to use these asymptotics to calculate the initial data to continue homoclinic orbits in two free parameters. The new homoclinic predictors are implemented in the MATLAB continuation package MatCont to initialize the continuation of homoclinic orbits from a BT point. Two examples with multidimensional state spaces are included.

Three-dimensional gravity-capillary steady waves on water of finite-depth, which are uniformly translating in a horizontal propagation direction and periodic in a transverse direction, are considered. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is the time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants: the Bond number b and λ (the inverse of the square of the Froude number). The property of Sobolev spaces and the spectral analysis show that the spectrum of the linear part consists of isolated eigenvalues of finite algebraic multiplicity and the number of purely imaginary eigenvalues are finite. The distribution of eigenvalues is described by b and λ. Assume that C1 is the curve in (b,λ)-plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and that the intersection point of the curve C1 with the line λ=1 is (b0,1) where b0>0. Two cases (b0,1) and (b,λ) â C1 where 0< b< b0 are investigated. A center-manifold reduction technique and a normal form analysis are applied to show that for each case the dynamical system can be reduced to a system of ordinary differential equations with finite dimensions. The dominant system for the case (b0,1) is coupled Schrödinger-KdV equations while it is a Schrödinger equation for another case (b,λ) â C1. Then, from the existence of the homoclinic orbit connecting to the two-dimensional periodic solution (called generalized solitary wave) for the dominant system, it is obtained that such generalized solitary wave solution persists for the original system by using the perturbation method and adjusting some appropriate constants.

We consider the Restricted Three Body Problem (RTBP), and we restrict our attention to the equilibrium point L3. Our aim is centered in the description, as global as possible, of the dynamics around this equilibrium point. In this communication, we initially consider small values of µ, for which homoclinic connections to the equilibrium point L3 are horseshoe-shaped, and then, other values of µ are considered. We compute the objects in the center manifold of L3, including the invariant manifolds associated with them. They are computed by purely numerical procedures, in order to avoid the convergence restrictions of the semi-analytical ones (typically used around L1 or L2). We deal with homoclinic connections of periodic orbits and develop some numerical tools in order to compute them. These tools can be extended to compute also homoclinic connections to invariant tori.

We consider the Restricted Three Body Problem (RTBP), and we restrict our attention to the equilibrium point L3. Our aim is centered in the description, as global as possible, of the dynamics around this equilibrium point. In this communication, we initially consider small values of µ, for which homoclinic connections to the equilibrium point L3 are horseshoe-shaped, and then, other values of µ are considered. We compute the objects in the center manifold of L3, including the invariant manifolds associated with them. They are computed by purely numerical procedures, in order to avoid the convergence restrictions of the semi-analytical ones (typically used around L1 or L2). We deal with homoclinic connections of periodic orbits and develop some numerical tools in order to compute them. These tools can be extended to compute also homoclinic connections to invariant tori.

We consider the Restricted Three Body Problem (RTBP), and we restrict our attention to the equilibrium point L3. Our aim is centered in the description, as global as possible, of the dynamics around this equilibrium point. In this communication, we initially consider small values of µ, for which homoclinic connections to the equilibrium point L3 are horseshoe-shaped, and then, other values of µ are considered. We compute the objects in the center manifold of L3, including the invariant manifolds associated with them. They are computed by purely numerical procedures, in order to avoid the convergence restrictions of the semi-analytical ones (typically used around L1 or L2). We deal with homoclinic connections of periodic orbits and develop some numerical tools in order to compute them. These tools can be extended to compute also homoclinic connections to invariant tori.

We consider the Restricted Three Body Problem (RTBP), and we restrict our attention to the equilibrium point L3. Our aim is centered in the description, as global as possible, of the dynamics around this equilibrium point. In this communication, we initially consider small values of µ, for which homoclinic connections to the equilibrium point L3 are horseshoe-shaped, and then, other values of µ are considered. We compute the objects in the center manifold of L3, including the invariant manifolds associated with them. They are computed by purely numerical procedures, in order to avoid the convergence restrictions of the semi-analytical ones (typically used around L1 or L2). We deal with homoclinic connections of periodic orbits and develop some numerical tools in order to compute them. These tools can be extended to compute also homoclinic connections to invariant tori.

We consider the Restricted Three Body Problem (RTBP), and we restrict our attention to the equilibrium point L3. Our aim is centered in the description, as global as possible, of the dynamics around this equilibrium point. In this communication, we initially consider small values of µ, for which homoclinic connections to the equilibrium point L3 are horseshoe-shaped, and then, other values of µ are considered. We compute the objects in the center manifold of L3, including the invariant manifolds associated with them. They are computed by purely numerical procedures, in order to avoid the convergence restrictions of the semi-analytical ones (typically used around L1 or L2). We deal with homoclinic connections of periodic orbits and develop some numerical tools in order to compute them. These tools can be extended to compute also homoclinic connections to invariant tori.

We consider the Restricted Three Body Problem (RTBP), and we restrict our attention to the equilibrium point L3. Our aim is centered in the description, as global as possible, of the dynamics around this equilibrium point. In this communication, we initially consider small values of µ, for which homoclinic connections to the equilibrium point L3 are horseshoe-shaped, and then, other values of µ are considered. We compute the objects in the center manifold of L3, including the invariant manifolds associated with them. They are computed by purely numerical procedures, in order to avoid the convergence restrictions of the semi-analytical ones (typically used around L1 or L2). We deal with homoclinic connections of periodic orbits and develop some numerical tools in order to compute them. These tools can be extended to compute also homoclinic connections to invariant tori.

In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (which surround three equilibrium points called L3, L4 and L5), when the mass parameter µ is positive and small; we describe the structure of such families from the two-body problem (µ = 0). On the other hand, the region of existence of horseshoe periodic orbits for any value of µ ∈ (0, 1/2] implies the understanding of the behaviour of the invariant manifolds of L3. So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L3, and on the simple infinite and double infinite period homoclinic phenomena are also analysed. Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe periodic orbits are considered in detail.

We study homoclinic transport to Lyapunov orbits around a collinear libration point in the planar restricted three body problem. A method to compute homoclinic orbits is first described. Then we introduce the scattering map for this problem (defined on a suitable normally hyperbolic invariant manifold) and we show how to compute it using the information already obtained for the homoclinic orbits. An example application to Astrodynamics is also proposed.

In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (which surround three equilibrium points called L3, L4 and L5), when the mass parameter µ is positive and small; we describe the structure of such families from the two-body problem (µ = 0). On the other hand, the region of existence of horseshoe periodic orbits for any value of µ ∈ (0, 1/2] implies the understanding of the behaviour of the invariant manifolds of L3. So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L3, and on the simple infinite and double infinite period homoclinic phenomena are also analysed. Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe periodic orbits are considered in detail.

We consider a perturbation of an integrable Hamiltonian vector field with three degrees of freedom with a center–center–saddle equilibrium having a homoclinic orbit or loop. With the help of a Poincaré map (chosen based on the unperturbed homoclinic loop), we study the homoclinic intersections between the stable and unstable manifolds associated to persistent hyperbolic KAM tori, on the center manifold near the equilibrium. If the perturbation is such that the homoclinic loop is preserved (i.e. the perturbation also has a homoclinic loop inherited from the unperturbed one), we establish that, in general, the manifolds intersect along 8, 12 or 16 transverse homoclinic orbits. On the other hand, in a more generic situation (the loop is not preserved; a condition for this fact is obtained by means of a Melnikov-like method), the manifolds intersect along four transverse homoclinic orbits, though a small neighborhood of the loop has to be excluded. In a first approximation, those homoclinic orbits can be detected as nondegenerate critical points of a Melnikov potential defined on the 2-torus. The number of homoclinic orbits is given by Morse theory applied to the Melnikov potential.

We consider the billiard motion inside a C2-small perturbation of a ndimensional ellipsoid Q with a unique major axis. The diameter of the ellipsoid Q is a hyperbolic two-periodic trajectory whose stable and unstable invariant manifolds are doubled, so that there is a n-dimensional invariant set W of homoclinic orbits for the unperturbed billiard map. The set W is a stratified set with a complicated structure. For the perturbed billiard map the set W generically breaks down into isolated homoclinic orbits. We provide lower bounds for the number of primary homoclinic orbits of the perturbed billiard which are close to unperturbed homoclinic orbits in certain strata of W. The lower bound for the number of persisting primary homoclinic billiard orbits is deduced from a more general lower bound for exact perturbations of twist maps possessing a manifold of homoclinic orbits.

The splitting of separatrices for hyperbolic fixed points of twist maps with $d$ degrees of freedom is studied through a real-valued function, called the Melnikov potential. Its non-degenerate critical points are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, Morse theory can be applied to give lower bounds on the number of transverse homoclinic orbits. This theory is applied first to elliptic billiards, where non-integrability holds for any non-trivial entire symmetric perturbation. Next, symmetrically perturbed prolate billiards with $d>1$ degrees of freedom are considered. Several topics are studied about these billiards: existence of splitting, explicit computations of Melnikov potentials, existence of $8$ or $8d$ transverse homoclinic orbits, exponentially small splitting, etc

Quasiperiodic perturbations with two frequencies $(1/\varepsilon ,\gamma /\varepsilon )$ of a pendulum are considered, where $\gamma $ is the golden mean number. We study the splitting of the three-dimensional invariant manifolds associated to a two-dimensional invariant torus in a neighbourhood of the saddle point of the pendulum. Provided that some of the Fourier coefficients of the perturbation (the ones associated to Fibonacci numbers) are separated from zero, it is proved that the invariant manifolds split for $\varepsilon $ small enough. The value of the splitting, that turns out to be ${\rm O} (\exp (-{\rm const} /\sqrt{\varepsilon }) )$, is correctly predicted by the Melnikov function.

The principal investigator, together with two post-doctoral fellows, several graduate students, and colleagues, has applied the modern mathematical theory of nonlinear waves to problems in nonlinear optics. Projects included (1) the interaction of laser light with nematic liquid crystals, (2) propagation through random nonlinear media, (3) cross polarization instabilities and optical shocks for propagation along nonlinear optical fibers, and (4) the dynamics of bistable optical switches coupled through both diffusion and diffraction. In project (1) the extremely strong nonlinear response of a cw laser beam in a nematic liquid crystal medium produced striking undulation and filamentation of the cw beam which was observed experimentally and explained theoretically. In project (2) the interaction of randomness with nonlinearity was investigated, as well as an effective randomness due to the simultaneous presence of many nonlinear instabilities. In the polarization problems of project (3) theoretical hyperbolic structure (instabilities and homoclinic orbits) in the coupled pdes was identified and used to explain cross polarization instabilities in both the focusing and defocusing cases, as well as to describe optical shocking phenomena. For the coupled bistable optical switches of project (4), a numerical code was carefully developed in two spatial and one temporal dimensions. The code was used to study the decay of temporal transients to 'on-off' steady states in a geometry which includes forward and backward longitudinal propagation, together with one dimensional transverse coupling of both electromagnetic diffraction and carrier diffusion.

1. Nonlinear Vibrations. 2. Viscosity Solutions and Applications. 3. Wavelets and Applications. 4. Variational Methods for Image Segmentation.

This paper presents a geometric analysis of bifurcations leading to chaos for Hamiltonian systems with two degrees of freedom of the form x-dot = y, y-dot = -gradient V(x). Two bifurcation parameters are considered. One is the energy level and the other is an angle, Psi, between two homoclinic orbits. Though global non-linearities are necessary, the results are obtained by local analysis of the flow near the origin where it is assumed that (D-sq)V(0) = I, the 2 x 2 identity matrix. For a fixed energy level it is shown that as Psi decreases through 90 deg the two homoclinic orbits bifurcate into two homoclinic orbits, a periodic orbit, and connecting orbits. These orbits can then be used to define a compact region in R supercript 4. Now treating the energy as a parameter value the trajectory of orbits passing through this compact region can be described using symbolic dynamics. In this case it is shown that a single periodic orbit bifurcates into three periodic orbits whose stable and unstable manifold intersect transversely.

This research involves extensive studying of the problem of transverse homoclinic orbits of periodic orbits of functional differential equations (FDE's), and also systems of reaction diffusion equations and attempting to understand the effects of boundary conditions on the flow when the diffusion coefficients and the boundary conditions are varied.

This document establishes the existence of a homoclinic solution of a Hamiltonian system assuming that the potential V is T periodic in t, grows more rapidly than quadratically as the value of 9 approaches limit of infinity and satisfies some other technical conditions. The homoclinic solution is obtained as the limit of subharmonic solutions of (*). The subharmonic solutions are found using a minimax argument., Sponsored in part by Grant NSF-MCS81-10556.

This paper concerns Hamiltonian and non-Hamiltonian perturbations of integrable two degree of freedom Hamiltonian systems which contain homoclinic and periodic orbits. Our main example concerns perturbations of the uncoupled system consisting of the simple pendulum and the harmonic oscillator. We show that small coupling perturbations with, possibly, the addition of positive and negative dampling breaks the integrability by introducing horseshoes into the dynamics. (Author), Sponsored in part by Grant NSF-MCS78-06718.

We start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates. We use KAM theory to show that some of the resulting tori persist under small perturbations and use a vector of Melkinov integrals to show that, under suitable hypotheses, their stable and unstable manifolds intersect transversally. This transverse intersection is ultimately responsible for Arnold diffusion on each energy surface. The method is applied to a pendulum-oscillator systems. (Author), Sponsored in part by Grant NSF-MCS78-06718.

This conference will focus on nonlinear partial and integrodifferential equations by considering them as infinite-dimensional dynamical systems. One day will be devoted to new classes of equations that arise via mathematical modelling of large flexible space structures. Some of the topics which will be represented are: Nonlinear Semigroups; Dynamical Systems; Attractors; Reduction to Finite Dimensional Systems; Inertial Manifolds; Bifurcation Theory; Control Theory; Compensated Compactness; Nonlinear Evolution Equations; Reaction-Diffusion Equationsl and Stability Analysis of PDE's.

This paper studies the dynamics of coupled planar rigid bodies, concentrating on the case of two or three bodies coupled with a hinge joint. The Hamiltonian structure is non-canonical and is obtained using the methods of reduction, starting from canonical brackets on the cotangent bundle of the configuration space in material representation. The dynamics on the reduced space for two bodies occurs on cylinders in IR(3); stability of the equilibria is studied using the Energy-Casimir method and is confirmed numerically. The phase space of the two bodies contains a homoclinic orbit which produces chaotic solutions when the system is perturbed by a third body. This and a study of periodic orbits are discussed in part II. The number and stability of equilibria and their bifurcations for three bodies as system parameters are varied are studied here; in particular, it is found that there are always 4 or 6 equilibria.

info:eu-repo/semantics/published

info:eu-repo/semantics/published