Intermittent large deviation of chaotic trajectory in Ikeda map: Signature of extreme events

Abstract

We notice signatures of extreme events-like behavior in a laser based Ikeda map. The trajectory of the system occasionally travels a large distance away from the bounded chaotic region, which appears as intermittent spiking events in the temporal dynamics. The large spiking events satisfy the conditions of extreme events as usually observed in dynamical systems. The probability density function of the large spiking events shows a long-tail distribution consistent with the characteristics of rare events. The inter-event intervals obey a Poisson-like distribution. We locate the parameter regions of extreme events in phase diagrams. Furthermore, we study two Ikeda maps to explore how and when extreme events terminate via mutual interaction. A pure diffusion of information exchange is unable to terminate extreme events where synchronous occurrence of extreme events is only possible even for large interaction. On the other hand, a threshold-activated coupling can terminate extreme events above a critical value of mutual interaction.PACS numbers: 89.75.Fb, 05.45.aRare and recurrent large amplitude deviations of normally bounded dynamics are seen in many systems. Such occasional large amplitude spiking events are larger than a nominal value and their statistical distribution of occurrence shows qualitative similarities, in dynamical sense, with data records of natural disasters, rogue waves, tsunami, flood and share market crashes. These observations draw attention of researchers to investigate similar sudden large intermittent events in dynamical systems for developing an understanding of the origin of extreme events and exploring the possibilities of prediction. A laser based Ikeda map was studied earlier to profess the origin of a new dynamical phenomenon, namely, interior crisis, that leads to a sudden expansion of a chaotic attractor. This sudden expansion of attractor is not always a permanent property of the system, and it could be intermittent, which shows similarities with extreme events and this signature was overlooked earlier. Here we explore this extreme value dynamical features of the Ikeda map to confirm the phenomenon and the statistical properties of events. An investigation with two coupled maps has also been made in search of an appropriate coupling scheme that is able to terminate these undesirable extreme events.

Emergence and mitigation of extreme events in a parametrically driven system with velocity-dependent potential

“…We observe that extreme events starts to originate at = 0.079. In particular, when = 0.081, from This is similar to what was observed in [24]. For all the values of , although the nature ofẋ is qualitatively similar to x in producing additional spikes after every increase in , we finḋ…”

“…[12][13][14][15]. As far as the dynamical systems are concerned, these events occur in FitzHugh-Nagumo oscillators, Hindmarsh-Rose model, climatic models, Linéard system, memristor-based Linénard system, micromechanical system, electronic circuits, coupled Ikeda map, network of moving agents, network of Josephson junctions, dispersive wave models, Ginzburg-Landau model and nonlinear Schrödinger equation [4,5,8,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Further, in real-time engineering applications, these events are found in laser systems, plasma, optical fibers, and superfluid helium [6,7,28,29].…”

“…The mechanism can be found by analysing the bifurcation scenario of the considered system particularly at the point of emergence of extreme events. A majority of the extreme events occur when a chaotic attractor bifurcates into another chaotic attractor or when a periodic orbit bifurcates into a chaotic attractor [5,12,24].…”

Synchronization to extreme events in moving agents

“…In most of the previous works [50][51][52][53][54][55], the studies on extreme events are done either by considering only single isolated dynamical system or by considering static dynamical network formalisms and the possible temporal evolution of the network itself are rather ignored. But, dynamical systems in nature are rarely isolated and more importantly, in most of the cases interacting dynamical units in physical, biological and social networks undergo through time-varying connections instead of being coupled via static regular topologies.…”

“…Evidence of extreme events in coupled Hindmarsh-Rosebursting neurons interacting through chemical synaptic and gap junctional diffusive coupling is presented in the [54]. Intermittent large deviations of chaotic trajectory indicating the emergence of such rare events in a laser based Ikeda map, has been reported very recently [55].…”