André Vanderbauwhede

Centre Manifolds, Normal Forms and Elementary Bifurcations

These notes originated from a seminar on dynamical systems held at the University of Louvain-la-Neuve (Belgium) in the spring of 1985. Our guide for that seminar was the book Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by Guckenheimer and Holmes [9]. This otherwise excellent book has one disadvantage for mathematicians: it contains very few proofs, and hence one is forced to go searching in the literature if one wants to fill in the details. When I tried to do this for chapter 3 of the book (on centre manifolds, normal form theory, and codimension one bifurcations), I rapidly got frustrated by the rather sketchy way in which most texts deal with the more technical parts of this theory. Around the same time I found in the PhD thesis of S. Van Gils [36] the idea of using spaces of exponentially growing functions in order to formulate and prove the centre manifold theorem. This seemed (at least for me) to be a more natural approach, and later we found out that also others had already used this idea to some extent. Stimulated by a few colleagues I then started on the project of writing some notes which would contain a reasonably complete account of the theory, which would use the new approach as its main guiding principle, and which would be suitable for use in seminars and graduate courses. Of course I had seriously underestimated the efforts needed to finish such a project, and it was only with a considerable delay that I was able to produce a first version during the summer of 1986. Since then this version has circulated among some colleagues and was tried out at a few seminars. The many remarks and suggestions which I received were taken into consideration when I wrote the version presented here; the main difference with the earlier version is the use of the fibre contraction theorem to prove the differentiability of the centre manifold. I leave it to the reader to judge to what extent the text still reflects its original goals.

Center Manifold Theory in Infinite Dimensions

Center manifold theory forms one of the cornerstones of the theory of dynamical systems. This is already true for finite-dimensional systems, but it holds a fortiori in the infinite-dimensional case. In its simplest form center manifold theory reduces the study of a system near a (non-hyperbolic) equilibrium point to that of an ordinary differential equation on a low-dimensional invariant center manifold. For finite-dimensional systems this means a (sometimes considerable) reduction of the dimension, leading to simpler calculations and a better geometric insight. When the starting point is an infinite-dimensional problem, such as a partial, a functional or an integro differential equation, then the reduction forms also a qualitative simplification. Indeed, most infinite-dimensional systems lack some of the nice properties which we use almost automatically in the case of finite-dimensional flows. For example, the initial value problem may not be well posed, or backward solutions may not exist; and one has to worry about the domains of operators or the regularity of solutions. Therefore the reduction to a finite-dimensional center manifold, when it is possible, forms a most welcome tool, since it allows us to recover the familiar and easy setting of an ordinary differential equation.

Center manifolds and contractions on a scale of Banach spaces

We give a new proof for the existence of a Ck-center manifold at a nonhyperbolic equilibrium point of a finite-dimensional vector field of class Ck. The problem is reduced to a fixed point problem on a scale of Banach spaces; these Banach spaces consist of mappings with a certain maximal exponential growth at infinity. We give conditions under which there is a unique fixed point depending differentiably on the parameters; the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of Banach spaces.

Homoclinic period blow-up in reversible and conservative systems

We show that in conservative systems each non-degenerate homoclinic orbit asymptotic to a hyperbolic equilibrium possesses an associated family of periodic orbits. The family is parametrized by the period, and the periodic orbits accumulate on the homoclinic orbit as the period tends to infinity. A similar result holds for symmetric homoclinic orbits in reversible systems. Our results extend earlier work by Devaney and Henrard, and provide a positive answer to a conjecture of Strömgren. We present a unified approach to both the conservative and the reversible case, based on a technique introduced recently by X.-B. Lin.

Continuation of periodic orbits in conservative and Hamiltonian systems

We introduce and justify a computational scheme for the continuation of periodic orbits in systems with one or more first integrals, and in particular in Hamiltonian systems having several independent symmetries. Our method is based on a generalization of the concept of a normal periodic orbit as introduced by Sepulchre and MacKay [Nonlinearity 10 (1997) 679]. We illustrate the continuation method on some integrable Hamiltonian systems with two degrees of freedom and briefly discuss some further applications.

ELEMENTAL PERIODIC ORBITS ASSOCIATED WITH THE LIBRATION POINTS IN THE CIRCULAR RESTRICTED 3-BODY PROBLEM

We present an overview of detailed computational results for families of periodic orbits that emanate from the five libration points in the Circular Restricted 3-Body Problem, as well as for various secondary bifurcating families. Our extensive overview covers all values of the mass-ratio parameter, and includes many known families that have been studied in the past. The numerical continuation and bifurcation algorithms employed in our study are based on boundary value techniques, as implemented in the numerical continuation and bifurcation software AUTO.

Computation of Periodic Solutions of Conservative Systems with Application to the 3-Body Problem

We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Simo, as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.

Bifurcation of degenerate homoclinics

We analyze the continuation and bifurcation of homoclinic orbits near a given degenerate homoclinic orbit. We show that the existence of such degenerate homoclinic orbit is a codimension three phenomenon, and that generically the set of parametervalues at which a nearby homoclinic exists forms a codimension one surface which shows a singularity of Whitney umbrella type at the critical parametervalue. The line of self-intersecting points of such surface corresponds to systems which have two nearby homoclinics.

Hopf bifurcation for equivariant conservative and time-reversible systems

We study the bifurcation of small periodic solutions at a non-semi-simple 1:1-resonance in equivariant conservative or equivariant time-reversible systems. By using an equivariant Liapunov-Schmidt method and restricting to solutions with an appropriate isotropy, we reduce the problem to a scalar bifurcation equation. The analysis of this equation shows a bifurcation behaviour similar to that found for the Hamiltonian Hopf bifurcation.

Subharmonic branching in reversible systems

The branching of subharmonic solutions at a symmetric periodic solution of an autonomous reversible system is studied. Here “symmetric” means “invariant under time reversal.” It is shown that generically each such symmetric periodic solution belongs to a one-parameter family of similar periodic solutions. Along such a family solutions having multipliers that are roots of unity can be met generically. It is shown that at such solutions further branching of subharmonic solutions will generically occur.