We describe the rich spectrum of spatio-temporal phenomena emerging from a class of models incorporating adaptive dynamics on a lattice of nonlinear (typically chaotic) elements. The investigation is based on extensive numerical simulations which reveal many novel dynamical phases, ranging from spatio-temporal fixed points and cycles of all orders, to parameter regimes displaying marked scaling properties (as manifest in distinct 1/f spectral characteristics and power law distributions of spatial quantities).

Motivated by studies onq-deformed physical systems related to quantum group structures, and by the elements of Tsallis statistical mechanics, the concept ofq-deformed nonlinear maps is introduced. As a specific example, aq-deformation procedure is applied to the logistic map. Compared to the canonical logistic map, the resulting family ofq-logistic maps is shown to have a wider spectrum of interesting behaviours, including the co-existence of attractors — a phenomenon rare in one-dimensional maps.

We report the first experimental realization of all the fundamental logic gates, flexibly, using a chaotic circuit. In our scheme a simple threshold mechanism allows the chaotic unit to switch easily between behaviours emulating the different gates. We also demonstrate the combination of gates through a half-adder implementation.

We study a network of chaotic model neurons incorporating threshold activated coupling. We obtain a wide range of spatiotemporal patterns under varying degrees of asynchronicity in the evolution of the neuronal components. For instance, we find that sequential updating of threshold-coupled chaotic neurons can yield dynamical switching of the individual neurons between two states. So varying the asynchronicity in the updating scheme can serve as a control mechanism to extract different responses, and this can have possible applications in computation and information processing.

Here we introduce a model of parametrically coupled chaotic maps on a one-dimensional lattice. In this model, each element has its internal self-regulatory dynamics, whereby at fixed intervals of time the nonlinearity parameter at each site is adjusted by feedback from its past evolution. Additionally, the maps are coupled sequentially and unidirectionally, to their nearest neighbor, through the difference of their parametric variations. Interestingly we find that this model asymptotically yields clusters of superstable oscillators with different periods. We observe that the sizes of these oscillator clusters have a power-law distribution. Moreover, we find that the transient dynamics gives rise to a $1/f$ power spectrum. All these characteristics indicate self-organization and emergent scaling behavior in this system. We also interpret the power-law characteristics of the proposed system from an ecological point of view.

It was observed that the spatiotemporal chaos in lattices of coupled chaotic maps was suppressed to a spatiotemporal fixed point when some fractions 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 parametric noise on the dynamics is quite distinct: quenched spatial fluctuations are the most detrimental to spatiotemporal regularity; patiotemporal 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.

In this paper, we review and extend the results from our recently published work [Scientific Reports (Nature) 4, 4308] on taming explosive growth in spatially extended systems. Specifically, we consider collections of relaxation oscillators, which are relevant to modelling phenomena ranging from engineering to biology, under varying coupling topologies. We find that the system witnesses unbounded growth under regular connections on a ring, for sufficiently strong coupling strengths. However, when a fraction of the regular connections are dynamically rewired to random links, this blow-up is suppressed. We present the critical value of random links necessary for successful prevention of explosive growth in the oscillators for varying network rewiring time-scales. Further, we outline our analysis on the possible mechanisms behind the occurrence of catastrophes and how the switching of links helps to suppress them.

We investigate the influence of diversity on the temporal regularity of spiking in a ring of coupled model neurons. We find diversity-induced coherence in the spike events, with an optimal amount of parametric heterogeneity at the nodal level yielding the greatest regularity in the spike train. Further, we investigate the system under random spatial connections, where the links are both dynamic and quenched, and in all the cases we observe marked diversity-induced coherence. We quantitatively find the effect of coupling strength and random rewiring probability, on the optimal coherence that can be achieved under diversity. Our results indicate that the largest coherence in the spike events emerge when the coupling strength is high, and when the underlying connections are mostly random and dynamically changing.