Energetic and entropic cost due to overlapping of Turing-Hopf instabilities in the presence of cross diffusion

A systematic introduction to nonequilibrium thermodynamics of dynamical instabilities are considered for an open nonlinear system beyond conventional Turing pattern in presence of cross diffusion. An altered condition of Turing instability in presence of cross diffusion is best reflected through a critical control parameter and wave number containing both the self- and cross-diffusion coefficients. Our main focus is on entropic and energetic cost of Turing-Hopf interplay in stationary pattern formation. Depending on the relative dispositions of Turing-Hopf codimensional instabilities from the reaction-diffusion equation it clarifies two aspects: energy cost of pattern formation, especially how Hopf instability can be utilized to dictate a stationary concentration profile, and the possibility of revealing nonequilibrium phase transition. In the Brusselator model, to understand these phenomena, we have analyzed through the relevant complex Ginzberg-Landau equation using multiscale Krylov-Bogolyubov averaging method. Due to Hopf instability it is observed that the cross-diffusion parameters can be a source of huge change in free-energy and concentration profiles.