FONDECYT Postdoctoral No. 3130497 - Global Dynamics and Bifurcations: Insight into Theory and Applications

AM2V Members: Juan Peypouquet, Pablo Aguirre
Start Date: 10-01-2012 - End Date: 09-30-2015
Status: finished

We focus on the consequences of homoclinic and heteroclinic bifurcations in some concrete model vector fields in order to describe and characterize the topological changes of the dynamics given by these global bifurcations. Indeed, homoclinic and heteroclinic orbits are two frequent examples of global topological transitions in many applications. How such bifurcations (and their corresponding unfoldings) arise is well understood, and software packages exist to detect and follow them in parameters.

In this project, however, we address an issue that it is far less well understood, namely, these bifurcations are characterized by interactions of the (un)stable manifolds of equilibria and periodic orbits involved, leading to a reorganization of the overall dynamics in phase space. Moreover, these global objects act as separatrices between the basins of attraction of the stable invariant objects in phase space. Hence, the following questions arise: how do the associated higher-dimensional stable manifolds change both topologically and geometrically during the given bifurcation? and how does this affect the organization of the phase space?

We focus the investigation on three main problems that cover both theoretical and applied aspects of global bifurcations:

  • The role of global (non)orientable invariant manifolds near codimension-two resonant bifurcations and their change of orientability at flip bifurcations (of type B) and the topological description of basin boundaries of attracting objects in a model by Sandstede.
  • Global bifurcations of invariant manifolds near a codimension-two non-central saddle-node homoclinic bifurcation and their role in the organization of excitable dynamics in a model of a laser with optical injection.
  • Bifurcation analysis of limit cycles and connecting orbits in a predator-prey model with Allee effect and the configurations of separatrices leading to extinction/survival scenarios.

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. In this way, we are able to explain how the global bifurcations of interest may lead to quite dramatic changes of the overall dynamics.

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