Dynamic boundary crisis in the Lorenz-type map

Abstract

Effects of the slowly varying control parameters on bifurcations of the equilibria and limit cycles have been previously studied in detail. In this paper, the concept of dynamic bifurcations is extended to chaotic phenomena. We consider this problem for a Lorenz-type map. As the control parameter passes through a critical value, the dynamic boundary crisis of a chaotic attractor takes place. We discover and analyze the effects of delayed exit from the chaotic region and non-exponential decay of the number of surviving trajectory points. The property of the delay increase with increasing rate of the control parameter change has also been demonstrated and explained.

Transient sequences in a hypernetwork generated by an adaptive network of spiking neurons

“…The map-based approach in modelling neuronal networks is described in more detail in [47][48][49][50]. The system (4.1) was used in a number of research papers for modelling the collective activity of different neuronal systems [51][52][53][54][55][56] and studying rather theoretical issues on chaotic dynamics in maps [57,58]. Note that the chaos itself is not necessarily needed to generate a hypernetwork and the general results in §2 do not depend on whether chaotic or regular dynamics is exhibited by the network.…”

Transient dynamics of the Lorenz system with a parameter drift

“…The area enclosed in the hysteresis diagram depends on the adiabatic parameter, ε, and this relation is defined by its corresponding scaling law [Berglund & Kunz, 1999]. As far as we know, the crossing of a bifurcation due to a slow parameter drift when chaotic attractors are implied has been only studied so far for maps, for instance, in the Lorenz map in [Maslennikov & Nekorkin, 2013] and [Maslennikov et al, 2018]. Here, we study the dynamic heteroclinic bifurcation at r = 24.06 for the Lorenz system.…”

“…To characterize a nonattracting chaotic set, as the chaotic saddle responsible for the transient chaos, we may analyze the decay in the number of trajectories that still present a chaotic behavior [Maslennikov & Nekorkin, 2013]. In Fig.…”

“…However, when chaotic attractors are involved, dynamic bifurcations have been only studied so far for maps. In particular, the case of a Lorenztype map when the control parameter varies monotonically with time and when it induces a periodic forcing, has been analyzed in [Maslennikov & Nekorkin, 2013;Maslennikov et al, 2018] respectively. Here, we aim to broaden the current knowledge on dynamic bifurcations to flows presenting strange attractors.…”

Evolving dynamical networks with transient cluster activity

“…Note that for the parameter values chosen, the first equation in ( 1) is a Lorenztype map and has a chaotic attractor for fixed values y i from some range. The variation of y i in time forms the relaxation dynamics of (1) with both regular and chaotic features (for more details, see [23,25]). Thus, the nodal response of the nodes generally displays intrinsic chaotic dynamics.…”