Using Floquet theory to unravel far-from equilibrium dynamics in reaction-diffusion systems

The interplay between reaction kinetics and diffusion leads to a wide range of spatiotemporal behaviors in reaction-diffusion (RD) systems. This article presents a theoretical and computational study of a RD system inspired by experimental observations of the Rho-GEF-Myosin signaling network controlling cell contraction dynamics. The temporal reaction system dynamics range from periodicity to bistability. By employing a dual hybrid dynamical systems approach of numerical bifurcation analysis and Floquet theory, we characterize the spatiotemporal dynamics when stable limit-cycle oscillators or homogeneous time-periodic solutions undergo the process of Floquet-Turing-diffusion-driven-instability (FTDDI). FTDDI refers to the emergence of spatially nonuniform patterns when diffusion destabilizes an otherwise temporally stable limit cycle of the underlying reaction kinetics. For the temporal reaction system, numerical bifurcation allows us to identify regions in a two-parameter space defining the stability of the uniform steady states and regions where the system exhibits limit cycles, which are characterized by employing the Floquet theory. In the presence of spatial variations, Floquet theory classifies regions where diffusion destabilizes the limit cycle or maintains their homogeneous stability and the emerging spatiotemporal dynamics of the full system, far from equilibrium. In the bistable regime, diffusion differentiates regions into those that exhibit classical Turing diffusion-driven instability, leading to pattern formation; and those that exhibit FTDDI, leading to space-time periodic patterns, spatially inhomogeneous patterns, oscillatory pulses, and wave propagation, and those that remain unaffected by diffusion. These findings provide theoretical insights into understanding complex spatiotemporal dynamics relevant to biological, chemical, and ecological spatiotemporal systems.