Narrow Results

The spatiotemporal symmetry properties of a periodically forced network of three Continuous Stirred Tank Reactors are assessed. The symmetry is induced to the system by a cyclic switching of the feed and discharge positions. The symmetry properties imply that the Poincaré map is the third iterate of another nonstroboscopic map. This feature is here used to characterize the symmetry of the regime solutions, and to carry out bifurcation analysis. Possible bifurcation scenarios and, in particular, symmetry-breaking bifurcations are discussed. As the switch time is varied, different transitions have been identified: among the others, an important role in the birth of asymmetric regimes is played by frequency-locking phenomena. In addition, different routes to chaotic regimes (both symmetric and asymmetric) are reported.

Symmetry often plays an important role in the formation of complicated structures in the dynamics of vector fields. Here, we study a specific family of systems defined on ℝ³, which are invariant under the D₂ symmetry group. Under the assumption that they are polynomial of degree at most two, they belong to a two-parameter family of vector fields, called the D₂ model. We describe the global behavior of the system, for most parameter values, and locate a region of parameter space where complicated structures occur. The existence of heteroclinic and homoclinic orbits is shown, as well as of heteroclinic cycles (for other parameter values), implying the presence of (different types of) Shil'nikov type of chaos in the D₂ systems. We then employ Poincaré maps to illustrate the bifurcations leading to this behavior. The global bifurcations exhibited by its strange attractors are explained as an effect of symmetry. We conclude by describing the behavior of the system at infinity.