Evolution towards the steady state in a hopf bifurcation: A scaling investigation

Abstract

Some scaling properties describing the convergence for the steady state in a Hopf bifurcation are discussed. Two different procedures are considered in the investigation: (i) a phenomenological description obtained from time series coming from the numerical integration of the system, leading to a set of critical exponents and hence to scaling laws; (ii) a direct solution of the differential equations, which is possible only in the normal form. At the bifurcation, the convergence to the stationary state obeys a generalized and homogeneous function. For short time, the dynamics giving by the distance from the fixed point is mostly constant when a critical time is reached hence changing the dynamics to a convergence for the steady state given by a power law. Both the size of the constant plateau and the characteristic crossover time depend on the initial distance from the fixed point. Near the bifurcation, the convergence is described by an exponential decay with a relaxation time given by a power law.