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In this paper, we study the existence of transversal homoclinic orbits in a planar circular restricted four-body problem, based on the perturbation theory of integrable Hamiltonian systems. We start from a planar circular restricted four-body model and regard it as a perturbation of the two-body model. Then, in order to conveniently study unbounded orbits, we transform the infinite points to finite points by a non-canonical transformation, arriving at a non-Hamiltonian system with degenerate fixed points. According to the extended Melnikov method, we finally prove that there exist transversal homoclinic orbits in this four-body model.

We give new sufficient conditions for the integrability and unique integrability of continuous tangent sub-bundles on manifolds of arbitrary dimension, generalizing Frobenius' classical Theorem for C^1 sub-bundles. Using these conditions we derive new criteria for uniqueness of solutions to ODE's and PDE's and for the integrability of invariant bundles in dynamical systems. In particular we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings., Comment: 40 pages, 5 figures. To appear in Crelle's Journal

We address the following question raised by M. Entov and L. Polterovich: given a continuous map $f:X\to X$ of a metric space $X$, closed subsets $A,B\subset X$, and an integer $n\geq 1$, when is it possible to find a continuous function $\theta:X\to\mathbb{R}$ such that \[ \theta f-\theta\leq 1, \quad \theta|A\leq 0, \quad\text{and}\quad \theta|B> n\,? \] To keep things as simple as possible, we solve the problem when $A$ is compact. The non-compact case will be treated in a later work.

We provide numerical evidence that a finite-dimensional inertial manifold on which the dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for Kuramoto-Sivashinsky system, and find it to be equal to the `physical dimension' computed previously via the hyperbolicity properties of covariant Lyapunov vectors., Comment: 6 pages, 3 pdf figures, uses revtex4

In this paper, the homotopy perturbation method is applied to approximate the stable manifold of a nonlinear autonomous system of ordinary differential equations. The results reveal that the proposed method is very effective and simple.

A new algorithm is presented to compute both one dimensional stable and unstable manifolds of planar maps. The global manifold is grown from a local manifold and one point is added each step. It is proved that the gradient of the global manifold can be predicted by the known points on the manifold with a gradient prediction scheme and it can be used to locate the image or preimage of the new point quickly. A new accuracy criterion is derived from the gradient prediction scheme. The performance of the algorithm is tested with several well-known examples.

The new algorithm uses the idea of growing the manifold. The preimage of the new point is found quickly with a method called gradient prediction scheme and a new accuracy criterion is proposed. Furthermore, our algorithm is capable of computing both stable and unstable one dimensional manifold.

A new algorithm is presented for computing one dimensional unstable manifold of a map and Hénon map is taken as an example to test the performance of the algorithm. The unstable manifold is grown with new point added at each step and the distance between consecutive points is adjusted according to the local curvature. It is proved that the gradient of the manifold at the new point can be predicted by the known points on the manifold and in this way the preimage of the new point could be located immediately. During the simulation, it is found that the unstable manifold of Hénon map coincides with its direct iteration when canonical parameters are chosen which means order is obtained out of chaos. In the other several groups of parameters the two branches of the unstable manifolds are nearly symmetric, and they serve as the borderline of the Hénon map iteration sequence. We hope that this would contribute to the further exploration of Hénon map.

An analytical method is proposed to find geometric structures of stable, unstable and center manifolds of the collinear Lagrange points. In a transformed space, where the linearized equations are in Jordan canonical form, these invariant manifolds can be approximated arbitrarily closely as Taylor series around Lagrange points. These invariant manifolds are represented by algebraic equations containing the state variables only without the help of time. Thus the so-called geometric structure of these invariant manifolds is obtained. The stable, unstable and center manifolds are tangent to the stable, unstable and center eigenspaces, respectively. As an example of applicability, the invariant manifolds of L1 point of the Sun-Earth system are considered. The stable and unstable manifolds are symmetric about the line from the Sun to the Earth, and they both reach near the Earth, so that the low energy transfer trajectory can be found based on the stable and unstable manifolds. The periodic or quasi-periodic orbits, which are chosen as nominal arrival orbits, can be obtained based on the center manifold.

We study the dynamics of the Burgers equation on the unit interval driven by affine linear noise. Mild solutions of the Burgers stochastic partial differential equation generate a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. We establish a local stable manifold theorem near a hyperbolic stationary point, as well as the existence of local smooth invariant manifolds with finite codimension and a countable global invariant foliation of the energy space relative to an ergodic stationary point.

We show the existence of local Lipschitzian stable and unstable manifolds for the ill-posed problem of perturbations of hyperbolic bi-semigroups. We do not assume backward nor forward uniqueness of solutions. We do not use cut-off functions because we do not assume global smallness conditions on the nonlinearities. We introduce what we call dichotomous flows which recovers the symmetry between the past and the future. Thus, we need to prove only a stable manifold theorem. We modify the Conley–McGehee–Moeckel approach to avoid appealing to Wazewski principle and Brouwer degree theory. Hence we allow both the stable and unstable directions to be infinite dimensional. We apply our theorem to the elliptic system uξξ+Δu=g(u,uξ) in an infinite cylinder R×Ω.

This study proposes an analytical method that determines the center manifold and identifies the reduced system on the center manifold. The proposed method expresses the center manifold through general equations containing only state variables, and not functions with respect to time. This is the so-called geometric structure of the center manifold. The location of periodic or quasi-periodic orbits is identified after the geometric structure of the center manifold is determined. The reduced system on the center manifold is described using ordinary differential equations, so that periodic or quasi-periodic orbits can be computed by numerically integrating the reduced system. The results indicate that the analytical method proposed in this study has higher precision compared with the Lindstedt-Poincare method of the same order.

In this paper we propose a numerical method for approximating connecting orbits on a manifold and its bifurcation parameters. First we extend the standard nondegeneracy condition to the connecting orbits on a manifold. Then we construct a well-posed system such that the nondegenerate connecting orbit pair on a manifold is its regular solution. We use a difference method to discretize the ODE part in this well-posed system and we find that the numerical solutions still remain on the same manifold. We also set up a modified projection boundary condition to truncate connecting orbits on a manifold onto a finite interval. Then we prove the existence of truncated approximate connecting orbit pairs and derive error estimates. Finally, we carry out some numerical experiments to illustrate the theoretical estimates.

Holmgren's theorem guarantees unique continuation across non-characteristic surfaces of class C 1 for solutions to homogeneous linear partial differential equations with analytic coefficients. Based on this result a global uniqueness theorem for solutions to hyperbolic differential equations with analytic coefficients in a space-time cylinder Q = (0, T) × Ω is established. Zero Cauchy data on a part of the lateral C 1-boundary will force every solution to a homogeneous hyperbolic PDE to vanish at half time T/2 provided that T > T 0. The minimal time T 0 depends on the geometry of the slowness surface of the hyperbolic operator, and can be determined explicitly as demonstrated by several examples including the system of transversely isotropic elasticity and Maxwell's equations. Furthermore, it is shown that this global uniqueness result holds for hyperbolic operators with C 1-coefficients as long as they satisfy the conclusion of Holmgren's Theorem.

We obtain a local stable manifold theorem for perturbations of nonautonomous linear difference equations possessing a very general type of nonuniform dichotomy, possibly with different growth rates in the uniform and nonuniform parts. Note that we consider situations where the classical Lyapunov exponents can be zero. Additionally, we study how the manifolds decay along the orbit of a point as well as the behavior under perturbations and give examples of nonautonomous linear difference equations that admit the dichotomies considered.

We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Frechet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods. We already derived such a theorem in [11], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations.

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf. [20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]).

Using a fixed point theorem of Krasnosel’skii type, the paper proves the existence of asymptotically stable solutions for a Volterra–Hammerstein integral equation.

We study the ergodic theory of non-conservative C 1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C 1-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C 1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ. In addition, confirming a claim made by R. Mane in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin’s Stable Manifold Theorem, even if the diffeomorphism is only C 1.

We consider weakly coupled map lattices with a decaying interaction. That is, we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites. We develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds. This framework is very well suited to study systematically invariant objects. One obtains that the invariant objects are essentially local. We use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties (i.e. the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small). Other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets, in the companion paper by Fontich et al. (2011) [10].

In this paper, we study the stability properties of solutions of a class of functional differential equations with variable delay. By using the fixed point theory under an exponentially weighted metric, we obtain some interesting sufficient conditions ensuring that the zero solution of the equations is stable and asymptotically stable.

In this paper we establish three different existence results for periodic solutions for a class of first-order neutral differential equations. The first one is based on a generalized version of the Poincar?-Birkhoff fixed point theorem where we establish conditions on f which guarantee that a first-order neutral differential equations has infinitely many periodic solutions. The second one is based on Mawhin?s continuation theorem and the third one is based on Krasnoselskii fixed point theorem.

By using the method of the invariant subspaces for unbounded linear operators and Schauder’s fixed point theorem, we give an existence theorem of mild pseudo-almost periodic solutions for some semilinear differential equations with a Stepanov-like pseudo-almost periodic term under some suitable assumptions. For this purpose, we show a new composition theorem of Stepanov-like pseudo-almost periodic functions. As applications, we examine the existence of mild pseudo-almost periodic solutions to some second-order hyperbolic equations. Our work is done under a “uniform continuity” condition instead of the “Lipschitz” condition assumed in the literature.

Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits, are used to connect the initial and final orbits. The transfer asymptotically departs the initial orbit on a trajectory contained within the initial orbit’s unstable manifold and later asymptotically approaches the final orbit on a trajectory contained within the stable manifold of the final orbit. The manifold trajectories are connected by the execution of impulsive maneuvers. Two-body parameters dictate the selection of the individual manifold trajectories used to construct efficient transfers. A bounding sphere centered on the secondary, with a radius less than the sphere of influence of the secondary, is used to study the manifold trajectories. A two-body parameter, κ, is computed within the bounding sphere, where the gravitational effects of the secondary dominate. The parameter κ is defined as the sum of two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on the unstable manifold and a point on the stable manifold. It is numerically demonstrated that as the κ parameter decreases, the total cost to complete the transfer decreases. Preliminary results indicate that this method of constructing transfers produces a significant cost savings over methods that do not employ the use of invariant manifolds.

For delay difference equations with infinite delay we consider the notion of nonuniform exponential dichotomy. This includes the notion of uniform exponential dichotomy as a very special case. Our main aim is to establish a stable manifold theorem under sufficiently small nonlinear perturbations. We also establish the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. Finally, we characterize the nonuniform exponential dichotomies in terms of strict Lyapunov sequences. In particular, we construct explicitly a strict Lyapunov sequence for each exponential dichotomy.

We consider positive, radial and exponentially decaying steady state solutions of the general reaction–diffusion and Klein–Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction–diffusion equation and a precise center-stable manifold theorem for the Klein–Gordon equation, which include the co-dimension of the manifolds and the decay rates for even perturbations.

We show the persistence of hyperbolic bounded solutions to nonautonomous difference and retarded functional differential equations under parameter perturbation, where hyperbolicity is given in terms of an exponential dichotomy in variation. Our functional-analytical approach is based on a formulation of dynamical systems as operator equations in ambient sequence or function spaces. This yields short proofs, in particular of the stable manifold theorem. As an ad hoc application, the behavior of hyperbolic solutions and stable manifolds for ODEs under numerical discretization with varying step-sizes is studied.

We extend a result of Minsky to show that, for a map of a surface to a hyperbolic [math] –manifold, which is [math] –incompressible rel a geodesic link with a definite tube radius, the set of noncontractible simple loops with bounded length representatives is quasi-convex in the complex of curves of the surface. We also show how wide product regions can be used to find a geodesic link with a definite tube radius with respect to which a map is [math] –incompressible.

A method of greatly decreasing the activation time of a control method based on stable manifold information is proposed.

We show how to effectively link covering relations with cone conditions. We give a new, ‘geometric,’ proof of the stable manifold theorem for hyperbolic fixed point of a map.

The Tychonoff-Schauder fixed point theorem is applied to prove the existence of bounded solutions of systems of ordinary differential equations that are C0-close to linear hyperbolic systems. Bibliography: 3 titles.

We consider a system of ordinary differential equations used to describe the dynamics of two coupled single-mode semiconductor lasers. In particular, we study solutions corresponding to the amplitude synchronization. It is shown that the set of these solutions forms a three-dimensional invariant manifold in the phase space. We study the stability of trajectories on this manifold both in the tangential direction and in the transverse direction. We establish conditions for the existence of globally asymptotically stable solutions of equations on the manifold synchronized with respect to the amplitude.

We consider a singularly perturbed system depending on two parameters with two (possibly the same) normally hyperbolic centre manifolds. We assume that the unperturbed system has an orbit connecting a hyperbolic fixed point on one centre manifold to a hyperbolic fixed point on the other. Then we prove some old and new results concerning the persistence of these connecting orbits and apply the results to find examples of systems in dimensions greater than three which possess Sil’nikov saddle-focus homoclinic orbits. Розглянуто сингулярно збурену систему, що залежить вiд двох параметрiв та має два (можливо, однаковi) нормально гiперболiчнi центрованi многовиди. При цьому припускається, що незбурена система має орбiту, яка поєднує гiперболiчну нерухому точку на одному центрованому многовидi з гiперболiчною нерухомою точкою на iншому. Доведено деякi вiдомi та новi результати щодо збереження цих орбiт та наведено приклади систем розмiрностi бiльше, нiж три, що мають сiдловi фокуснi гомоклiнiчнi орбiти Сiльнiкова.

This paper presents a new numerical method for computing global stable manifolds and global stable sets of nonlinear discrete dynamical systems. For a given map f:ℝ d →ℝ d , the proposed method is capable of yielding large parts of stable manifolds and sets within a certain compact region M⊂ℝ d . The algorithm divides the region M in sets and uses an adaptive subdivision technique to approximate an outer covering of the manifolds. In contrast to similar approaches, the method requires neither the system’s inverse nor its Jacobian. Hence, it can also be applied to noninvertible and piecewise-smooth maps. The successful application of the method is illustrated by computation of one- and two-dimensional stable manifolds and global stable sets.

We obtain a necessary as well as a sufficient condition for the existence of bifurcation points of a coincidence equation, and, in particular, of a parametrized fixed point problem. In both cases the trivial solutions are assumed to form a finite-dimensional submanifold of a Banach manifold. An application is given to a delay differential equation on a manifold: we detect periodic solutions that rotate close to an equilibrium point.

A classical result, studied, among others, by Caratheodory [C. Caratheodory, Calculus of Variations and Partial Differential Equations of the First Order, Chelsea, New York, 1989], states that, for second-order, scalar equations, nondegenerate periodic minimizers are hyperbolic. Consequently, the Stable/Unstable Manifold Theorem applies, and implies that, at least locally, the stable and unstable sets are regular curves intersecting transversally at the nondegenerate minimizer. For analytic equations, there is a version of this fact which holds for isolated, but possibly degenerate, minimizers.

We establish a Bowen type rigidity theorem for the fundamental group of a noncompact hyperbolic manifold of finite volume (with dimension at least 3).

In a broad sense, model reduction means producing a low-dimensional dynamical system that replicates either approximately, or more strictly, exactly and topologically, the output of a dynamical system. Model reduction has an important role in the study of dynamical systems and also with engineering problems. In many cases, there exists a good low-dimensional model for even very high-dimensional systems, even infinite dimensional systems in the case of a PDE with a low-dimensional attractor. The theory of global attractors approaches these issues analytically, and focuses on finding (depending on the question at hand), a slow-manifold, inertial manifold, or center manifold, on which a restricted dynamical system represents the interesting behavior of the dynamical system; the main issue depends on defining a stable invariant manifold in which the dynamical system is invariant. These approaches are analytical in nature, however, and are therefore not always appropriate for dynamical systems known only empirically through a dataset. Empirically, the collection of tools available are much more restricted, and are essentially linear in nature. Usually variants of Galerkin's method, project the dynamical system onto a function linear subspace spanned by modes of some chosen spanning set. Even the popular Karhunen–Loeve decomposition, or POD, method is exactly such a method. As such, it is forced to either make severe errors in the case that the invariant space is intrinsically a highly nonlinear manifold, or bypass low-dimensionality by retaining many modes in order to capture the manifold. In this work, we present a method of modeling a low-dimensional nonlinear manifold known only through the dataset. The manifold is modeled as a discrete graph structure. Intrinsic manifold coordinates will be found specifically through the ISOMAP algorithm recently developed in the Machine Learning community originally for purposes of image recognition.

An essentially nonlinear system of differential equations (i.e., a system without linear terms) is considered. It is proved that there exists a local-integral manifold of neutral type (central manifold) near an equilibrium point. Coefficient conditions for logarithmic norms are used.

A closed hyperbolic 3-manifold is exceptional if its shortest geodesic does not have an embedded tube of radius $\ln(3)/2$. D. Gabai, R. Meyerhoff and N. Thurston identified seven families of exceptional manifolds in their proof of the homotopy rigidity theorem. They identified the hyperbolic manifold known as Vol3 in the literature as the exceptional manifold associated to one of the families. It is conjectured that there are exactly 6 exceptional manifolds. We find hyperbolic 3-manifolds, some from the SnapPea's census of closed hyperbolic 3-manifolds, associated to 5 other families. Along with the hyperbolic 3-manifold found by Lipyanskiy associated to the seventh family we show that any exceptional manifold is covered by one of these manifolds. We also find their group coefficient fields and invariant trace fields., Comment: 19 pages, minor revisions, added appendix by Alan Reid, published by Experimental Mathematics

The idea of generating functions and maps is presented, first in global symplectic geometry and then in the theory of invariant manifolds, as introduced by McGehee and Sander in 1996. Their result on the stable manifold theorem is generalised and simplified; the proofs no longer use any functional analysis. Then comes an original “non-autonomous” version of the previous results, yielding–besides Pesin’s invariant laminations–seemingly unrelated results on invariant manifolds and conjugacies, presented in the end after a basic example.

We reprove the $\lambda$-Lemma for finite dimensional gradient flows by generalizing the well-known contraction method proof of the local (un)stable manifold theorem. This only relies on the forward Cauchy problem. We obtain a rather quantitative description of (un)stable foliations which allows to equip each leaf with a copy of the flow on the central leaf -- the local (un)stable manifold. These dynamical thickenings are key tools in our recent work [Web]. The present paper provides their construction., Comment: 35 pages, 9 figures

This paper considers the Field-Noyes model. There are two possibilities if we consider only the case that the positive equilibrium of the system p is hyperbolic. Either p is asymptotically stable or it is unstable with a one-dimensional stable manifold. Both cases can occur, depending on the values of the parameters. Condition for the existence of a nontrivial periodic orbit are given in the second case.

In these notes, we discuss obstructions to the existence of local invariant manifolds of some smoothness class, near hyperbolic fixed points of diffeomorphisms. We present an elementary construction for continuously differentiable invariant manifolds that are not necessarily normally hyperbolic, near attracting fixed points. The analogous theory for invariant manifold near hyperbolic equilibria of differential equations is included. For differential equations, we construct one dimensional invariant manifolds of higher smoothness class.Keywords:Hyperbolic fixed points, Smoothness, Diffeomorphisms, Differential equations

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.

A design of fuel-cell manifold to evenly distribute the reactants among the channels within bipolar plates is presented. Neglecting the friction term from the momentum equation, one-dimensional governing equations are manipulated to develop the manifold equation to achieve the required design. Some manifold shapes that satisfy the manifold equation, are suggested, together with the distribution of static pressure within the manifold itself.

We demonstrate that application of a generalized centre manifold is advantageous when low-dimensional reductions of continuous dynamical systems of hydrodynamic type are considered. The centre manifold theorem that we use differs from the standard one in the following way. We define a centre manifold as an invariant manifold, tangent to the eigenspace of the linearization of the mapping defining a dynamical system, where the eigenspace is associated with eigenvalues with small but not necessarily zero real parts (usually only eigenvalues with zero real parts are employed). As a result, several bifurcations following the onset of instability of the trivial steady state are reproduced in the eight-dimensional system that we obtain, whilst in the six-dimensional system constructed with application of the conventional centre manifold theorem only the first bifurcation of the steady state is reproduced correctly.

A dynamical system admitting an invariant manifold can be interpreted as a single element of an infinite class of dynamical systems that all exhibit the same behaviour on the invariant manifold. This observation is used in the context of autonomous ordinary differential equations to generalize a global stability result of Li and Muldowney. The new result is demonstrated on an epidemiological model.