Global generic dynamics close to symmetry
Copyright policy )Global generic dynamics close to symmetry
The paper studies the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in \(S^3\). The cycle is assumed to involve two saddle-foci of different type and is structurally stable within the class of \((\mathbb Z_2\oplus\mathbb Z_2)\)-symmetric vector fields. It is also assumed that the cycle contains a two-dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry-breaking perturbations. It is shown that breaking the symmetry in a two-parameter family leads to a wide range of dynamical behavior, i.e., an attracting periodic trajectory; other heteroclinic trajectories; homoclinic orbits; \(n\)-pulses; suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of Shilnikov's type. It is also shown that, generically, the coexistence of linked homoclinic orbits at the two saddle-foci has codimension two and occurs arbitrarily close to the symmetric cycle.
- Universidade Lusófona do Porto Portugal
Averaging method for ordinary differential equations, Bifurcation theory for ordinary differential equations, Dynamical aspects of attractors and their bifurcations, Perturbations of ordinary differential equations, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Symmetry breaking, Homoclinic and heteroclinic solutions to ordinary differential equations, Shilnikov's homoclinic cycles, symmetry breaking, Complex behavior and chaotic systems of ordinary differential equations, Dynamical systems with hyperbolic orbits and sets, Bykov cycles, heteroclinic networks, Symmetries, invariants of ordinary differential equations, Heteroclinic networks, Shilnikovʼs homoclinic cycles, Analysis
Averaging method for ordinary differential equations, Bifurcation theory for ordinary differential equations, Dynamical aspects of attractors and their bifurcations, Perturbations of ordinary differential equations, Topological structure of integral curves, singular points, limit cycles of ordinary differential equations, Symmetry breaking, Homoclinic and heteroclinic solutions to ordinary differential equations, Shilnikov's homoclinic cycles, symmetry breaking, Complex behavior and chaotic systems of ordinary differential equations, Dynamical systems with hyperbolic orbits and sets, Bykov cycles, heteroclinic networks, Symmetries, invariants of ordinary differential equations, Heteroclinic networks, Shilnikovʼs homoclinic cycles, Analysis
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