Critical and scaling behavior of delayed bifurcations in nonlinear systems with dynamic disorder

Abstract

We explore the critical and scaling behavior of delayed bifurcations in systems with dynamic disorder well-known in kinetics when the control parameter in the rate is made to sweep stochastically across the critical point. The analysis is based on an extended Hamiltonian system that includes noise as an additional degree of freedom conjugate to the relevant dynamical variable. Appropriate quantifiers for measuring the time delay in reaching the bifurcation point have been introduced. We show that the time delay (i.e., the time difference between the static and dynamic bifurcation times) exhibits a power law decay with the deviation of the control parameter from its critical value, the decay constant being close to unity and is independent of the nature of bifurcations. The characteristic time to reach the zero solution state decreases algebraically with the deviation of the dynamical variable from its critical value and the decay exponent scales as the highest power of nonlinearity characterizing the nature of bifurcations.

Graphical abstract

Bifurcations are essential features of various chemical kinetics; biological and ecological evolution. Critical and scaling behavior of delayed bifurcations in such systems with dynamic disorder has been explored when the control parameter in the rate is varied stochastically across the critical point.