Quiescent phases and stability in discrete time dynamical systems

Abstract

We study coupled maps where a map representing an `active phase' is coupled to the identity which represents a `quiescent phase'.The resulting system in double dimension is a natural analogue of differential equations with quiescent phasesthat have been thoroughly studied.In the continuous time case quiescent phases with equal rates for all componentsstabilize against the onset of Hopf bifurcations (but not againsteigenvalues passing through zero) while unequal rates may induce Hopf bifurcations unless the Jacobian matrix hasa `strong stability' property.Here we show that similar effects occur in the discrete time case. In the case of equal rates we determine the exact stabilityboundary as an algebraic curve of fourth order.It is shown that large quiescence rates may completely inhibit period doubling bifurcations. If the rates are unequal, quiescentphases may destabilize a stationary point. In this casewe find (for two components) a notion of `strong stability' for the Jacobian matrix such that the stationary pointcannot be excited. Discrete time predator prey models serve as examples for the damping and excitation phenomena.