On local attraction properties and a stability index for heteroclinic connections

Some invariant sets may attract a nearby set of initial conditions but nonetheless repel a complementary nearby set of initial conditions. For a given invariant set X [subset] [real number[supn]] with a basin of attraction N, we define a stability index [sigma](x) of a point x [in] X that characteri...

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Bibliographic Details
Published in:Nonlinearity 2011-03, Vol.24 (3), p.887-929
Main Authors: Podvigina, Olga, Ashwin, Peter
Format: Article
Language:English
Subjects:
Attraction
Basins
Eigenvalues
Fields (mathematics)
Initial conditions
Invariants
Stability
Steady state
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Summary:Some invariant sets may attract a nearby set of initial conditions but nonetheless repel a complementary nearby set of initial conditions. For a given invariant set X [subset] [real number[supn]] with a basin of attraction N, we define a stability index [sigma](x) of a point x [in] X that characterizes the local extent of the basin. We adapt the definition to local basins of attraction (i e where N is defined as the set of initial conditions that are in the basin and whose trajectories remain local to X) This stability index is particularly useful for discussing the stability of robust heteroclinic cycles, where several authors have studied the appearance of cusps of instability near cycles that are Milnor attractors. We study simple (robust heteroclinic) cycles in [real number[sup 4]] and show that the local stability indices can be calculated in terms of the eigenvalues of the linearization of the vector field at steady states on the cycle.
ISSN:0951-7715
1361-6544
DOI:10.1088/0951-7715/24/3/009