Rate-induced transitions for parameter shift systems

Rate-induced transitions have recently emerged as an identifiable type of instability of attractors in nonautonomous dynamical systems. In most studies so far, these attractors can be associated with equilibria of an autonomous limiting system, but this is not necessarily the case. For a specific class of systems with a parameter shift between two autonomous systems, we consider how the breakdown of the quasistatic approximation for attractors can lead to rate-induced transitions, where nonautonomous instability can be characterised in terms of a critical rate of the parameter shift. We find a number of new phenomena for non-equilibrium attractors: weak tracking where the pullback attractor of the system limits to a proper subset of the attractor of the future limit system, partial tipping where certain phases of the pullback attractor tip and others track the quasistatic attractor, em invisible tipping where the critical rate of partial tipping is isolated and separates two parameter regions where the system exhibits end-point tracking. For a model parameter shift system with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system. Considering weak tracking for a nonautonomous Rossler system, we show that there are infinitely many critical rates at which a pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor

Weak tracking in nonautonomous chaotic systems

This is the final version. Available from the American Physical Society via the DOI in this recordPrevious studies have shown that rate-induced transitions can occur in pullback attractors of systems subject to "parameter shifts" between two asymptotically steady values of a system parameter. For cases where the attractors limit to equilibrium or periodic orbit in past and future limits of such an nonautonomous systems, these can occur as the parameter change passes through a critical rate. Such rate-induced transitions for attractors that limit to chaotic attractors in past or future limits has been less examined. In this paper, we identify a new phenomenon is associated with more complex attractors in the future limit: weak tracking, where a pullback attractor of the system limits to a proper subset of an attractor of the future limit system. We demonstrate weak tracking in a nonautonomous Rössler system, and argue there are infinitely many critical rates at each of which the pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor. We also state some necessary conditions that are needed for weak tracking.European CommissionEnterprise IrelandLaya HealthcareEuropean Union Horizon 202