Learning unidirectional coupling using echo-state network

Reservoir Computing has found many potential applications in the field of complex dynamics. In this article, we exploit the exceptional capability of the echo-state network (ESN) model to make it learn a unidirectional coupling scheme from only a few time series data of the system. We show that, once trained with a few example dynamics of a drive-response system, the machine is able to predict the response system's dynamics for any driver signal with the same coupling. Only a few time series data of an $A-B$ type drive-response system in training is sufficient for the ESN to learn the coupling scheme. After training even if we replace drive system $A$ with a different system $C$, the ESN can reproduce the dynamics of response system $B$ using the dynamics of new drive system $C$ only

Under what kind of parametric fluctuations is spatiotemporal regularity the most robust?

It was observed that the spatiotemporal chaos in lattices of coupled chaotic maps was suppressed to a spatiotemporal fixed point when some fraction of the regular coupling connections were replaced by random links. Here we investigate the effects of different kinds of parametric fluctuations on the robustness of this spatiotemporal fixed point regime. In particular we study the spatiotemporal dynamics of the network with noisy interaction parameters, namely fluctuating fraction of random links and fluctuating coupling strengths. We consider three types of fluctuations: (i) noisy in time, but homogeneous in space; (ii) noisy in space, but fixed in time; (iii) noisy in both space and time. We find that the effect of different kinds of parameteric noise on the dynamics is quite distinct: quenched spatial fluctuations are the most detrimental to spatiotemporal regularity; spatiotemporal fluctuations yield phenomena similar to that observed when parameters are held constant at the mean-value; and interestingly, spatiotemporal regularity is most robust under spatially uniform temporal fluctuations, which in fact yields a larger fixed point range than that obtained under constant mean-value parameters

Enhancement of spatiotemporal regularity in an optimal window of random coupling

We investigate the spatiotemporal dynamics of a lattice of coupled chaotic maps whose coupling connections are dynamically rewired to random sites with probability p, namely at any instance of time, with probability p a regular link is switched to a random one. In a range of weak coupling, where spatiotemporal chaos exists for regular lattices (i.e. for p = 0), we find that p > 0 yields synchronized periodic orbits. Further we observe that this regularity occurs over a window of p values, beyond which the basin of attraction of the synchronized cycle shrinks to zero. Thus we have evidence of an optimal range of randomness in coupling connections, where spatiotemporal regularity is efficiently obtained. This is in contrast to the commonly observed monotonic increase of synchronization with increasing p, as seen for instance, in the strong coupling regime of the very same system.Comment: 10 pages, 6 figure

Explosive death induced by mean–field diffusion in identical oscillators

We report the occurrence of an explosive death transition for the first time in an ensemble of identical limit cycle and chaotic oscillators coupled via mean–field diffusion. In both systems, the variation of the normalized amplitude with the coupling strength exhibits an abrupt and irreversible transition to death state from an oscillatory state and this first order phase transition to death state is independent of the size of the system. This transition is quite general and has been found in all the coupled systems where in–phase oscillations co–exist with a coupling dependent homogeneous steady state. The backward transition point for this phase transition has been calculated using linear stability analysis which is in complete agreement with the numerics

Phase ordering at crises

Phase order, namely the average direction of sequential iterations, is studied in the family of unimodal maps x→1≄μ|x|z on the interval [≄1,1]. The average phase order or "magnetization" M is sensitive to local changes in the dynamics. At merging crises, this quantity increases from zero with the scaling behaviour M ∼(μ≄μc)1/z, while at exterior crises, M decreases, also having the same scaling exponent. We find that the exponent z is governed by the singularities of the invariant density ρ(x) at the edges of the interval: as x→±1, ρ(x)∼(1≄|x|z)-βwith β=1≄1/z