Bifurcations phenomena and route to chaos via the cosymmetry breaking in the Darcy convection problem

Abstract

The purpose of this paper is to examine and exhibit bifurcations arising from cosymmetry breaking in dynamical systems. Curves of equilibria are typical of cosymmetric dynamical systems. Perturbations violating cosymmetry destroy these curves, leading to different attractors. An example of a cosymmetric problem is the Darcy convection in a rectangular vessel with a linear temperature profile on its boundary. Low-intensity internal heat sources or weak through-flow violate the cosymmetry. The article presented numerical results of bifurcations in the convection problem caused by these perturbations. It achieved the discretization of partial differential equations using the Bubnov–Galerkin method. It has developed an algorithm based on cosymmetry theory to calculate equilibrium curves and analyse their destruction. A long-time integration using high-order methods applies for the searching of attractors. The study reveals that both perturbations breaking cosymmetry have similar effects on system dynamics. When all equilibria on a curve are stable, the curve can break into finite sets of equilibria or exhibit slow periodic motions. If arcs of unstable equilibria exist, then new quasi-periodic and chaotic attractors may arise. The dynamics on these attractors include slow movements close to destroyed arcs of equilibria and fast transitions between arcs. We may interpret this scenario as the route to chaos via the cosymmetry breaking. The results also show the typicality of multistability in cosymmetric and near-cosymmetric problems.