Convergence towards asymptotic state in 1-D mappings: a scaling investigation

TL;DR

This paper investigates how one-dimensional logistic-like mappings approach their steady states, revealing universal scaling laws and critical exponents near bifurcations through numerical and theoretical analysis.

Contribution

It introduces a phenomenological and theoretical framework describing decay to steady states in 1-D mappings, highlighting universal critical exponents near bifurcations.

Findings

Decay characterized by a homogeneous function with three critical exponents

Exponential decay near bifurcation points with power-law relaxation time

Formalism applicable to other dissipative mappings

Abstract

Decay to asymptotic steady state in one-dimensional logistic-like mappings is characterized by considering a phenomenological description supported by numerical simulations and confirmed by a theoretical description. As the control parameter is varied bifurcations in the fixed points appear. We verified at the bifurcation point in both; the transcritical, pitchfork and period-doubling bifurcations, that the decay for the stationary point is characterized via a homogeneous function with three critical exponents depending on the nonlinearity of the mapping. Near the bifurcation the decay to the fixed point is exponential with a relaxation time given by a power law whose slope is independent of the nonlinearity. The formalism is general and can be extended to other dissipative mappings.