Bałtycki Istytut Matematyki,

Gdynia, Poland

Mini-course:

The role of global invariant manifolds in the

organisation of chaotic dynamics

This course is concerned with the global organization of phase space near global bifurcations

of three-dimensional vector fields. Typically, these phenomena cause global re-arrangements

of the dynamics, including the (dis)appearance of periodic orbits, changes to basins of attractions

and the generation of chaotic dynamics. This is of special interest to understand the nature of systems

near global bifurcations such as in laser dynamics, nerve impulses in neurons, electrochemical reactions,

communication systems based on chaos, food chains in predation models, etc.

Our approach is to study how the relevant two-dimensional (un)stable manifold rearrange themselves globally

near homoclinic bifurcations to reconfigure the overall organization of the dynamics in phase space.

In order to deal with these questions we make extensive use of analytical tools from dynamical systems theory

in combination with advanced numerical methods for the accurate computation of higher-dimensional global

(un)stable manifolds. More specifically, we compute these manifolds via the numerical continuation of orbit

segments as solutions of a boundary value problem.

This course will focus in three main areas:

1) An introduction on the basic mathematical background: Bifurcations of vector fields, invariant manifolds of

equilibria and periodic orbits, routes to chaos, applications.

2) Computational methods: Numerical bifurcation analysis, detection and continuation of homoclinic bifurcations,

computation and visualisation of global invariant manifolds.

3) An overview of recent research topics: invariant manifolds in the unfolding of codimension-one and codimension-two

homoclinic bifurcations, Shilnikov chaos, the (non)orientability of a homoclinic bifurcation, the Lorenz manifold, etc.

All from both theoretical and applied contexts. And future challenges: Bifurcations to non-hyperbolic equilibria, higher

codimension phenomena, etc.

In this way, we are able to explain how the global

bifurcations of interest may lead to quite dramatic changes of the overall dynamics.

References:

P. Aguirre, E. Doedel, B. Krauskopf and H.~M.~Osinga,

Investigating the consequences of global bifurcations for two-dimensional manifolds of vector fields,

Discrete and Continuous Dynamical Systems - Series A, 29, pp. 1309-1344, 2011.

P. Aguirre, B. Krauskopf and H. M. Osinga,

Global invariant manifolds near a Shilnikov homoclinic bifurcation,

to appear in Journal of Computational Dynamics.

P. Aguirre, B. Krauskopf and H. M. Osinga,

Global invariant manifolds near homoclinic orbits to a real saddle: (non)orientability and flip bifurcation,

to appear in SIAM Journal on Applied Dynamical Systems.

J. Guckenheimer and P. Holmes,

Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,

Second ed., Springer-Verlag, New York/Berlin, 1986.

Yu. Ilyashenko and W. Li,

Nonlocal Bifurcations,

Mathematical Surveys and Monographs, vol. 66, American Mathematical Society, Providence, RI, 1999.

B. Krauskopf and H. M. Osinga,

Computing invariant manifolds via the continuation of orbit segments,

in Numerical Continuation Methods for Dynamical Systems, Underst. Complex Syst.,

Springer-Verlag, New York, 2007, pp. 117-154.

Yu. A. Kuznetsov,

Elements of Applied Bifurcation Theory,

Third ed., Springer-Verlag, New York/Berlin, 2004.

Universidad Técnica Federico Santa María / Departamento de Matemática / P Aguirre