Dependence of Topological Conjugacies on Parameters

For impulsive differential equations, we establish the existence of invariant stable manifolds under sufficiently small perturbations of a linear equation. We consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy. One of the main advantages of our work is that our results are optimal, in the sense that for vector fields of class C ¹ outside the jumping times, we show that the invariant manifolds are also of class C ¹ outside these times. The novelty of our proof is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, using the same approach we can also consider linear perturbations.     

How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

     

Nonuniform Exponential Dichotomies and Lyapunov Regularity

The notion of exponential dichotomy plays a central role in the Hadamard–Perron theory of invariant manifolds for dynamical systems. The more general notion of nonuniform exponential dichotomy plays a similar role under much weaker assumptions. On the other hand, for nonautonomous linear equations v′ = A(t)v with global solutions, we show here that this more general notion is in fact as weak as possible: namely, any such equation possesses a nonuniform exponential dichotomy. It turns out that the construction of invariant manifolds under the existence of a nonuniform exponential dichotomy requires the nonuniformity to be sufficiently small when compared to the Lyapunov exponents. Thus, it is crucial to estimate the deviation from the uniform exponential behavior. This deviation can be measured by the so-called regularity coefficient, in the context of the classical Lyapunov–Perron regularity theory. We obtain here lower and upper sharp estimates for the regularity coefficient, expressed solely in terms of the matrices A(t).     

Characterization of transversal homothetic solutions in the n-body problem

Homothetic solutions of the n-body problem can be seen as heteroclinic orbits when the dynamical variables are changed via the McGehee blow-up and the time is suitably scaled. Transversality of the invariant asymptotic manifolds which contain the heteroclinic orbits is related to some structural stability. We fully characterize the cases in which such transversality is obtained for the n-body problem in any dimension.     

Parameter Dependence of Stable Manifolds for Delay Equations with Polynomial Dichotomies

We establish the existence of Lipschitz stable invariant manifolds for semiflows generated by a delay equation x′ = L(t)x _t + f (t, x _t , λ), assuming that the linear equation x′ = L(t)x _t admits a polynomial dichotomy and that f is a sufficiently small Lipschitz perturbation. Moreover, we show that the stable invariant manifolds are Lipschitz in the parameter λ. We also consider the general case of nonuniform polynomial dichotomies.     

Stable Manifolds for Impulsive Equations Under Nonuniform Hyperbolicity

For impulsive differential equations, we establish the existence of invariant stable manifolds under sufficiently small perturbations of a linear equation. We consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy. One of the main advantages of our work is that our results are optimal, in the sense that for vector fields of class C ¹ outside the jumping times, we show that the invariant manifolds are also of class C ¹ outside these times. The novelty of our proof is the use of the fiber contraction principle to establish the smoothness of the invariant manifolds. In addition, using the same approach we can also consider linear perturbations.     

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.     

<Emphasis Type="Italic">J</Emphasis><Subscript>2</Subscript> invariant relative orbits via differential correction algorithm

This paper describes a practical method for finding the invariant orbits in J ₂ relative dynamics. Working with the Hamiltonian model of the relative motion including the J ₂ perturbation, the effective differential correction algorithm for finding periodic orbits in three-body problem is extended to formation flying of Earth’s orbiters. Rather than using orbital elements, the analysis is done directly in physical space, which makes a direct connection with physical requirements. The asymptotic behavior of the invariant orbit is indicated by its stable and unstable manifolds. The period of the relative orbits is proved numerically to be slightly different from the ascending node period of the leader satellite, and a preliminary explanation for this phenomenon is presented. Then the compatibility between J ₂ invariant orbit and desired relative geometry is considered, and the design procedure for the initial values of the compatible configuration is proposed. The influences of measure errors on the invariant orbit are also investigated by the Monte–Carlo simulation. The project supported by the Innovation Foundation of Beihang University for Ph.D. Graduates, and the National Natural Science Foundation of China (60535010).     

Invariant manifolds of one class of systems of impulsive differential equations

We consider general problems related to the existence of invariant toroidal sets for linear and weakly nonlinear systems of impulsive differential equations defined in the direct product of an m-dimensional torus and an n-dimensional Euclidean space. We investigate classes of problems for which the conditions for the existence of invariant toroidal manifolds are satisfied.     

On the Global Geometric Structure of the Dynamics of the Elastic Pendulum

We approach the planar elastic pendulum as a singular perturbation of the pendulum to show that its dynamics are governed by global two-dimensional invariant manifolds of motion. One of the manifolds is nonlinear and carries purely slow periodic oscillations. The other one, on the other hand, is linear and carries purely fast radial oscillations. For sufficiently small coupling between the angular and radial degrees of freedom, both manifolds are global and orbitally stable up to energy levels exceeding that of the unstable equilibrium of the system. For fixed value of coupling, the fast invariant manifold bifurcates transversely to create unstable radial oscillations exhibiting energy transfer. Poincaré sections of iso-energetic manifolds reveal that only motions on and near a separatrix emanating from the unstable region of the fast invariant manifold exhibit energy transfer.     

Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations

Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perrons method. Then, we prove the smoothness of these invariant manifolds.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Nonexistence of invariant manifolds in fractional-order dynamical systems

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional-order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional-order systems.

     

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.     

An invariant manifold approach for studying waves in a one-dimensional array of non-linear oscillators

We consider a one-dimensional linear spring-mass array coupled to a one-dimensional array of uncoupled pendula. The principal aim of this study is to investigate the non-linear dynamics of this large-scale system in the limit of weak non-linearities, i.e. when the (fast) non-linear pendulum effects are small compared to the underlying (slow) linear dynamics of the linear spring-mass chain. We approach the dynamics in the context of invariant manifolds of motion. In particular, we prove the existence of an invariant manifold containing the (predominantly) slow dynamics of the system, with the fast pendulum dynamics providing small perturbations to the motions on the invariant manifold. By restricting the motion on the slow invariant manifold and performing asymptotic analysis we prove that the non-linear large-scale system possesses propagation and attenuation zones (PZs and AZs) in the frequency domain, similarly to the corresponding zones of the linearized system. Inside PZs non-linear travelling wave solutions exist, whereas in AZs only attenuating waves are permissible.     

Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R2

We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R ² and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows us to extend those results in a number of ways.     

Invariant Manifolds for Steady Boltzmann Flows and Applications

We develop a theory of invariant manifolds for the steady Boltzmann equation and apply it to the study of boundary layers and nonlinear waves. The steady Boltzmann equation is an infinite dimensional differential equation, so the standard center manifold theory for differential equations based on spectral information does not apply here. Instead, we employ a time-asymptotic approach using the pointwise information of Green’s function for the construction of the linear invariant manifolds. At the resonance cases when the Mach number at the far field is around one of the critical values of ?1, 0 or 1, the truly nonlinear theory arises. In such a case, there are wave patterns combining the fast decaying Knudsen-type and slow varying fluid-like waves. The key Knudsen manifolds consisting of only Knudsentype layers are constructed through delicate analysis of identifying the singular behavior around the critical Mach numbers. Around Mach number ± 1, the fluidlike waves are compressive and expansive waves; and around the Mach number 0, they are linear thermal layers. The quantitative analysis of the fluid-like waves is done using the reduction of dimensions to the center manifolds.Two-scale nonlinear dynamics based on those on the Knudsen and center manifolds are formulated for the study of the global dynamics of the combined wave patterns. There are striking bifurcations in the transition of evaporation to condensation and in the transition of the Milne’s problem with a subsonic far field to one with a supersonic far field. The analysis of these wave patterns allows us to understand the Sone Diagram for the study of the complete condensation boundary value problem. The monotonicity of the Boltzmann shock profiles, a problem that initially motivated the present study, is shown as a consequence of the quantitative analysis of the nonlinear fluid-like waves.     

Inviscid Dynamical Structures Near Couette Flow

Consider inviscid fluids in a channel {-1\leqq y\leqq1}{\{-1\leqq y\leqq1\}} . For the Couette flow u ₀ = (y, 0), the vertical velocity of solutions to the linearized Euler equation at u ₀ decays in time. Whether the same happens at the non-linear level is an open question. Here we study issues related to this problem. First, we show that in any (vorticity) H^s(s < \frac32){H^{s}\left(s<\frac{3}{2}\right)} neighborhood of Couette flow, there exist non-parallel steady flows with arbitrary minimal horizontal periods. This implies that nonlinear inviscid damping is not true in any (vorticity) H^s(s < \frac32){H^{s}\left(s<\frac{3}{2}\right)} neighborhood of Couette flow for any horizontal period. Indeed, the long time behaviors in such neighborhoods are very rich, including nontrivial steady flows and stable and unstable manifolds of nearby unstable shears. Second, in the (vorticity) ${H^{s}\left(s>\frac{3}{2}\right)}${H^{s}\left(s>\frac{3}{2}\right)} neighborhoods of Couette flow, we show that there exist no non-parallel steadily travelling flows \varvecv(x-ct,y){\varvec{v}\left(x-ct,y\right)} , and no unstable shears. This suggests that the long time dynamics in ${H^{s}\left(s>\frac{3}{2}\right)}${H^{s}\left(s>\frac{3}{2}\right)} neighborhoods of Couette flow might be much simpler. Such contrasting dynamics in H ^s spaces with the critical power s=\frac32{s=\frac{3}{2}} is a truly nonlinear phenomena, since the linear inviscid damping near Couette flow is true for any initial vorticity in L ².     

Quasi-Periodic Stability of Subfamilies of an Unfolded Skew Hopf Bifurcation

In the skew Hopf bifurcation a quasi-periodic attractor with nontrivial normal linear dynamics loses hyperbolicity. The simplest setting concerns rotationally symmetric diffeomorphisms of S ¹×R ². Their dynamics involve periodicity, quasi-periodicity and chaos, including mixed spectrum. The present paper deals with the persistence under symmetry-breaking of quasi-periodic invariant circles in this bifurcation. It turns out that, when adding sufficiently many unfolding parameters, the invariant circle persists for a large Hausdorff measure subset of a submanifold in parameter space. Accepted: September 3, 1999     

Unfolding a Bykov Attractor: From an Attracting Torus to Strange Attractors

In this paper we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a three-dimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different Morse indices. After slightly increasing both parameters, while keeping the one-dimensional connections unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the equilibria do not intersect. Under some conditions on the parameters and on the eigenvalues of the linearisation of the vector field at the saddle-foci, we prove the existence of many complicated dynamical objects, ranging from an attracting quasi-periodic torus to Hénon-like strange attractors, as a consequence of the Torus-Breakdown Theory. The mechanism for the creation of horseshoes and strange attractors is also discussed. Theoretical results are applied to show the occurrence of strange attractors in some analytic unfoldings of a Hopf-zero singularity.

     

The invariant manifold approach applied to nonlinear dynamics of a rotor-bearing system

The invariant manifold approach is used to explore the dynamics of a nonlinear rotor, by determining the nonlinear normal modes, constructing a reduced order model and evaluating its performance in the case of response to an initial condition. The procedure to determine the approximation of the invariant manifolds is discussed and a strategy to retain the speed dependent effects on the manifolds without solving the eigenvalue problem for each spin speed is presented. The performance of the reduced system is analysed in function of the spin speed.