Bifurcations, crisis, unstable dimension variability and the spreading transition in the coupled sine circle map system

The dynamical behavior of spatially extended dynamical systems can have interesting consequences for their statistics. We demonstrate this in a specific context, a system of coupled sine circle maps, and discuss the interconnection between the statistical and dynamical behaviors of the system. The system has an interesting phase diagram in parameter space wherein a spreading transition is seen across an infection line, with spatio-temporal and spatial intermittency of distinct universality classes (directed percolation and non-directed percolation) seen in the spreading/non-spreading regimes. The dynamical origins of the spreading transition, lie in a crisis arising from a tangent bifurcation in the system. In addition to changing the statistics, and therefore the universality class of the system, the crisis also has dynamical consequences. Unstable dimension variability is seen in the neighbourhood of this crisis, and multiple routes to crisis are seen due to the presence of multi-attractor solutions. We examine the system using a variety of characterizers such as finite time Lyapunov exponents and their distributions. We discuss the signatures of the phenomena seen in the quantifiers, and also whether similar techniques can be extended to other situations. Finally, we demonstrate the success of the quantifiers in another regime, spatio-temporal intermittency with travelling wave laminar solutions, and a solitonic regime.