Numerical calculation of invariant manifolds for maps

Many processes in the sciences and in engineering are modelled by dynamical systems and—in discretized version—by nonlinear maps. To understand the often complicated dynamical behaviour it is a well established tool to use the concept of invariant manifolds of the system. In this way it is often possible to reduce the dimension of the system considerably. In this paper we propose a new method to calculate numerically invariant manifolds near fixed points of maps. We prove convergence of our procedure and provide an error estimation. Finally, the application of the method is illustrated by examples.     

Stable invariant manifolds for parabolic dynamics

We consider nonautonomous equations v^′=A(t)v in a Banach space that exhibit stable and unstable behaviors with respect to arbitrary growth rates ecρ(t) for some function ρ(t). This corresponds to the existence of a “generalized” exponential dichotomy, which is known to be robust. When ρ(t)≠t this behavior can be described as a type of parabolic dynamics. We consider the general case of nonuniform exponential dichotomies, for which the Lyapunov stability is not uniform. We show that for any sufficiently small perturbation f of a “generalized” exponential dichotomy there is a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v). We also consider the case of exponential contractions, which allow a simpler treatment, and we show that they persist under sufficiently small nonlinear perturbations.     

The parameterization method for invariant manifolds III: overview and applications

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.     

The evolution of invariant manifolds in Hamiltonian-Hopf bifurcations

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter, ν. The eigenvalues of the linearized system are complex for ν<0 and pure imaginary for ν>0. Thus, for ν<0 the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for ν>0 these stable and unstable manifolds are gone. If the sign of a certain term in the normal form is positive then for small negative ν the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set.     

Bifurcation and invariant manifolds of the logistic competition model

In this paper, we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important centre manifolds, and study their bifurcation. Saddle-node and period-doubling bifurcation route to chaos are exhibited via numerical simulations.     

Two-dimensional invariant manifolds and global bifurcations: some approximation and visualization studies

We illustrate and discuss the computer-assisted study (approximation and visualization) of two-dimensional (un)stable manifolds of steady states and saddle-type limit cycles for flows in R ⁿ . Our investigation highlights a number of computational issues arising in this task, along with our solutions and “quick-fixes” for some of these problems. Two examples illustrative of both successes and shortcomings of our current approach are presented. Representative “snapshots” demonstrate the dependence of two-dimensional invariant manifolds on a bifurcation parameter as well as their interactions. Such approximation and visualization studies are a necessary component of the computer-assisted study and understanding of global bifurcations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Leaving the Moon by means of invariant manifolds of libration point orbits

The aim of this work is to look for rescue trajectories that leave the surface of the Moon, belonging to the hyperbolic manifolds associated with the central manifold of the Lagrangian points L₁ and L₂ of the Earth–Moon system. The model used for the Earth–Moon system is the Circular Restricted Three-Body Problem. We consider as nominal arrival orbits halo orbits and square Lissajous orbits around L₁ and L₂ and we show, for a given Δv, the regions of the Moon’s surface from which we can reach them. The key point of this work is the geometry of the hyperbolic manifolds associated with libration point orbits. Both periodic/quasi-periodic orbits and their corresponding stable invariant manifold are approximated by means of the Lindstedt–Poincaré semi-analytical approach.     

Construction of circle bifurcations of a two-dimensional spatially periodic flow

The study by Yudovich [V.I. Yudovich, Example of the generation of a secondary stationary or periodic flow when there is loss of stability of the laminar flow of a viscous incompressible fluid, J. Math. Mech. 29 (1965) 587-603] on spatially periodic flows forced by a single Fourier mode proved the existence of two-dimensional spectral spaces and each space gives rise to a bifurcating steady-state solution. The investigation discussed herein provides a structure of secondary steady-state flows. It is constructed explicitly by an expansion that when the Reynolds number increases across each of its critical values, a unique steady-state solution bifurcates from the basic flow along each normal vector of the two-dimensional spectral space. Thus, at a single Reynolds number supercritical value, the bifurcating steady-state solutions arising from the basic solution form a circle.     

Imperfect transcritical and pitchfork bifurcations

Imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on Banach spaces. In particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established. Applications include semilinear elliptic equations, imperfect Euler buckling beam problem and perturbed diffusive logistic equation.     

Minimal dynamics on Menger manifolds

The Menger universal spaces are realized as invariant sets of noninvertible, expanding maps. Minimal actions on these spaces of free groups with two or three generators are exhibited.     

Stable manifolds associated to fixed points with linear part equal to identity

We consider maps defined on an open set of having a fixed point whose linear part is the identity. We provide sufficient conditions for the existence of a stable manifold in terms of the nonlinear part of the map.These maps arise naturally in some problems of Celestial Mechanics. We apply the results to prove the existence of parabolic orbits of the spatial elliptic three-body problem.     

Orbits heteroclinic to invariant manifolds

Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3–5].Supported by the National Natural Science Foundation of China and Shanghai Natural Science Foundation.     

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results

In this paper we prove rigorous results on persistence of invariant tori and their whiskers. The proofs are based on the parameterization method of [X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J. 52 (2) (2003) 283-328; X. Cabré, E. Fontich, R. de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters, Indiana Univ. Math. J. 52 (2) (2003) 329-360]. The invariant manifolds results proved here include as particular cases of the usual (strong) stable and (strong) unstable manifolds, but also include other non-resonant manifolds. The method lends itself to numerical implementations whose analysis and implementation is studied in [A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms, preprint, 2005; A. Haro, R. de la Llave, A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical implementation and examples, preprint, 2005]. The results are stated as a posteriori results. Namely, that if one has an approximate solution which is not degenerate, then, one has a true solution not too far from the approximate one. This can be used to validate the results of numerical computations.     

Stable manifolds for nonuniform polynomial dichotomies

We establish the existence of smooth stable manifolds in Banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy. This new dichotomy is more restrictive in the “nonuniform part” but allow the “uniform part” to obey a polynomial law instead of an exponential (more restrictive) law. We consider two families of perturbations. For one of the families we obtain local Lipschitz stable manifolds and for the other family, assuming more restrictive conditions on the perturbations and its derivatives, we obtain C¹ global stable manifolds. Finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family.     

Intersecting invariant manifolds in spatial restricted three-body problems: Design and optimization of Earth-to-halo transfers in the Sun-Earth-Moon scenario

This work deals with the design of transfers connecting LEOs with halo orbits around libration points of the Earth-Moon CRTBP using impulsive maneuvers. Exploiting the coupled circular restricted three-body problem approximation, suitable first guess trajectories are derived detecting intersections between stable manifolds related to halo orbits of EM spatial CRTBP and Earth-escaping trajectories integrated in planar Sun-Earth CRTBP. The accuracy of the intersections in configuration space and the discontinuities in terms of Δv are controlled through the box covering structure implemented in the software GAIO. Finally first guess solutions are optimized in the bicircular four-body problem and single-impulse and two-impulse transfers are presented.     

Stable invariant manifolds for delay equations with piecewise constant argument

We construct smooth stable invariant manifolds for a class of delay equations with piecewise constant delay, for any sufficiently small perturbation of a nonuniform exponential dichotomy. We build on former work for perturbations of a uniform exponential dichotomy, also for delay equations with piecewise constant delay. These equations can be described as delay equations with an impulsive behavior of the derivative, such that at certain times the derivative changes abruptly.     

Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy

We establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that the linear equation v^′=A(t)v admits a nonuniform exponential dichotomy. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of Ck functions with α-Hölder continuous kth derivative. This is a closed subset of the space of continuous functions with the supremum norm, by an apparently not so well-known lemma of Henry (see Proposition 3). The estimates showing that the functions maintain the original bounds when transformed under the fixed-point operator are obtained through a careful application of the Faà di Bruno formula for the higher derivatives of the compositions (see (31) and (35)). As a consequence, we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class Ck.     

Invariant manifolds of maps close to a product of rotations: close to the rotation axis

We consider families of maps depending on a parameter ε such that for ε=0 the map becomes a product of linear rotations in and for ε≠0 the map is weakly attracting in the product of the rotation planes and weakly repelling in some complementary subspace. We prove that the unstable manifold converges to the complementary subspace in the C^r topology, the case r=∞ included. We consider both the local and the global manifolds. For that we prove some results on families of maps near a norm one linear map, which are of independent interest.     

A K-theoretical invariant and bifurcation for a parameterized family of functionals

Let F:={fx:x∈X} be a family of functionals defined on a Hilbert manifold and smoothly parameterized by a compact connected orientable n-dimensional manifold X, and let be a smooth section of critical points of F. The aim of this paper is to give a sufficient topological condition on the parameter space X which detects bifurcation of critical points for F from the trivial branch. Finally we are able to give some quantitative properties of the bifurcation set for perturbed geodesics on semi-Riemannian manifolds.