- Homoclinic explosions and implosions
- Philosophical Transactions of the Royal Society A - Mathematical, Physical and Engineering Sciences
- Volume | Issue number
- 354 | 1709
- Pages (from-to)
- Document type
- Faculty of Science (FNWI)
- Korteweg-de Vries Institute for Mathematics (KdVI)
In this paper we study the interaction of local saddle node and Hopf bifurcations with global recurrent orbits. We consider a four-parameter family of three-dimensional vector fields which are small perturbations of an integrable system possessing a line of degenerate saddle points connected by a manifold of homoclinic loops. Our most striking finding is that homoclinic bifurcations occur in which a unique connecting orbit is replaced by a countable infinity of such orbits (an `explosion') and in which a countable infinity collapses to a unique orbit (an `implosion'). We also discover a rich variety of heteroclinic connections among fixed points and periodic orbits. The family we study was motivated by the search for travelling `structures' such as fronts and domain walls in the complex Ginzburg-Landau partial differential equation which models weakly nonlinear wave interactions near the onset of instability, and which leads to a special case of our more general unfolding.
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