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Quiescent phases and stability in discrete time dynamical systems

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  • 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 phases that have been thoroughly studied. In the continuous time case quiescent phases with equal rates for all components stabilize against the onset of Hopf bifurcations (but not against eigenvalues passing through zero) while unequal rates may induce Hopf bifurcations unless the Jacobian matrix has a `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 stability boundary 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, quiescent phases may destabilize a stationary point. In this case we find (for two components) a notion of `strong stability' for the Jacobian matrix such that the stationary point cannot be excited. Discrete time predator prey models serve as examples for the damping and excitation phenomena.
    Mathematics Subject Classification: Primary: 37C75, 37G10; Secondary: 37G35, 92D40.

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