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.
scientific article; zbMATH DE number 6081512
scientific article