Experimental realization of chaos control by thresholding

Emergent order in adaptively rewired networks

We explore adaptive link change strategies that can lead a system to network configurations that yield ordered dynamical states. We propose two adaptive strategies based on feedback from the global synchronization error. In the first strategy, the connectivity matrix changes if the instantaneous synchronization error is larger than a prescribed threshold. In the second strategy, the probability of a link changing at any instant of time is proportional to the magnitude of the instantaneous synchronization error. We demonstrate that both these strategies are capable of guiding networks to chaos suppression within a prescribed tolerance, in two prototypical systems of coupled chaotic maps. So, the adaptation works effectively as an efficient search in the vast space of connectivities for a configuration that serves to yield a targeted pattern. The mean synchronization error shows the presence of a sharply defined transition to very low values after a critical coupling strength, in all cases. For the first strategy, the total time during which a network undergoes link adaptation also exhibits a distinct transition to a small value under increasing coupling strength. Analogously, for the second strategy, the mean fraction of links that change in the network over time, after transience, drops to nearly zero, after a critical coupling strength, implying that the network reaches a static link configuration that yields the desired dynamics. These ideas can then potentially help us to devise control methods for extended interactive systems, as well as suggest natural mechanisms capable of regularizing complex networks.

Regulating dynamics through intermittent interactions

In this article we experimentally demonstrate an efficient scheme to regulate the behavior of coupled nonlinear oscillators through dynamic control of their interaction. It is observed that introducing intermittency in the interaction term as a function of time or the system state predictably alters the dynamics of the constituent oscillators. Choosing the nature of the interaction, attractive or repulsive, allows for either suppression of oscillations or stimulation of activity. Two parameters Δ and τ, that reign the extent of interaction among subsystems, are introduced. They serve as a harness to access the entire range of possible behaviors from fixed points to chaos. For fixed values of system parameters and coupling strength, changing Δ and τ offers fine control over the dynamics of coupled subsystems. We show this experimentally using coupled Chua's circuits and elucidate their behavior for a range of coupling parameters through detailed numerical simulations.

Asymmetry induced suppression of chaos

We explore the dynamics of a group of unconnected chaotic relaxation oscillators realized by mercury beating heart systems, coupled to a markedly different common external chaotic system realized by an electronic circuit. Counter-intuitively, we find that this single dissimilar chaotic oscillator manages to effectively steer the group of oscillators on to steady states, when the coupling is sufficiently strong. We further verify this unusual observation in numerical simulations of model relaxation oscillator systems mimicking this interaction through coupled differential equations. Interestingly, the ensemble of oscillators is suppressed most efficiently when coupled to a completely dissimilar chaotic external system, rather than to a regular external system or an external system identical to those of the group. So this experimentally demonstrable controllability of groups of oscillators via a distinct external system indicates a potent control strategy. It also illustrates the general principle that symmetry in the emergent dynamics may arise from asymmetry in the constituent systems, suggesting that diversity or heterogeneity may have a crucial role in aiding regularity in interactive systems.

Qualitative models and experimental investigation of chaotic NOR gates and set/reset flip-flops

It has been observed through experiments and SPICE simulations that logical circuits based upon Chua's circuit exhibit complex dynamical behaviour. This behaviour can be used to design analogues of more complex logic families and some properties can be exploited for electronics applications. Some of these circuits have been modelled as systems of ordinary differential equations. However, as the number of components in newer circuits increases so does the complexity. This renders continuous dynamical systems models impractical and necessitates new modelling techniques. In recent years, some discrete dynamical models have been developed using various simplifying assumptions. To create a robust modelling framework for chaotic logical circuits, we developed both deterministic and stochastic discrete dynamical models, which exploit the natural recurrence behaviour, for two chaotic NOR gates and a chaotic set/reset flip-flop. This work presents a complete applied mathematical investigation of logical circuits. Experiments on our own designs of the above circuits are modelled and the models are rigorously analysed and simulated showing surprisingly close qualitative agreement with the experiments. Furthermore, the models are designed to accommodate dynamics of similarly designed circuits. This will allow researchers to develop ever more complex chaotic logical circuits with a simple modelling framework.

Threshold-activated transport stabilizes chaotic populations to steady states

We explore Random Scale-Free networks of populations, modelled by chaotic Ricker maps, connected by transport that is triggered when population density in a patch is in excess of a critical threshold level. Our central result is that threshold-activated dispersal leads to stable fixed populations, for a wide range of threshold levels. Further, suppression of chaos is facilitated when the threshold-activated migration is more rapid than the intrinsic population dynamics of a patch. Additionally, networks with large number of nodes open to the environment, readily yield stable steady states. Lastly we demonstrate that in networks with very few open nodes, the degree and betweeness centrality of the node open to the environment has a pronounced influence on control. All qualitative trends are corroborated by quantitative measures, reflecting the efficiency of control, and the width of the steady state window.

Exploiting chaos for applications

We discuss how understanding the nature of chaotic dynamics allows us to control these systems. A controlled chaotic system can then serve as a versatile pattern generator that can be used for a range of application. Specifically, we will discuss the application of controlled chaos to the design of novel computational paradigms. Thus, we present an illustrative research arc, starting with ideas of control, based on the general understanding of chaos, moving over to applications that influence the course of building better devices.

On a bifurcation structure mimicking period adding

In this work, we investigate a piecewise-linear discontinuous scalar map defined on three partitions. This map is specifically constructed in such a way that it shows a recently discovered bifurcation scenario in its pure form. Owing to its structure on the one hand and the similarities to the nested period-adding scenario on the other hand, we denoted the new bifurcation scenario as nested period-incrementing bifurcation scenario. The new bifurcation scenario occurs in several physical and electronical systems but usually not isolated, which makes the description complicated. By isolating the scenario and using a suitable symbolic description for the asymptotically stable periodic orbits, we derive a set of rules in the space of symbolic sequences that explain the structure of the stable periodic domain in the parameter space entirely. Hence, the presented work is a necessary step for the understanding of the more complicated bifurcation scenarios mentioned above.

Chaogates: morphing logic gates that exploit dynamical patterns

Chaotic systems can yield a wide variety of patterns. Here we use this feature to generate all possible fundamental logic gate functions. This forms the basis of the design of a dynamical computing device, a chaogate, that can be rapidly morphed to become any desired logic gate. Here we review the basic concepts underlying this and present an extension of the formalism to include asymmetric logic functions.

Adaptive steady-state stabilization for nonlinear dynamical systems

By means of LaSalle's invariance principle, we propose an adaptive controller with the aim of stabilizing an unstable steady state for a wide class of nonlinear dynamical systems. The control technique does not require analytical knowledge of the system dynamics and operates without any explicit knowledge of the desired steady-state position. The control input is achieved using only system states with no computer analysis of the dynamics. The proposed strategy is tested on Lorentz, van der Pol, and pendulum equations.

Asynchronous updating induces order in threshold coupled systems

We study a class of models incorporating threshold-activated coupling on a lattice of chaotic elements, evolving under updating rules incorporating varying degrees of synchronicity. Interestingly, we observe that asynchronous updating, both random and sequential, yields more spatiotemporal order than parallel (synchronous) updating. Further, the order induced by random asynchronous updating is very robust and occurs even for small asynchronicities in the temporal evolution of the local dynamics. So this case study suggests a very different mechanism for inducing regularity in extended systems.

Multifolded torus chaotic attractors: design and implementation

This paper proposes a systematic methodology for creating multifolded torus chaotic attractors from a simple three-dimensional piecewise-linear system. Theoretical analysis shows that the multifolded torus chaotic attractors can be generated via alternative switchings between two basic linear systems. The theoretical design principle and the underlying dynamic mechanism are then further investigated by analyzing the emerging bifurcation and the stable and unstable subspaces of the two basic linear systems. A novel block circuit diagram is also designed for hardware implementation of 3-, 5-, 7-, 9-folded torus chaotic attractors via switching the corresponding switches. This is the first time a 9-folded torus chaotic attractor generated by an analog circuit has been verified experimentally. Furthermore, some recursive formulas of system parameters are rigorously derived, which is useful for improving hardware implementation.

Exploiting the controlled responses of chaotic elements to design configurable hardware

We discuss how threshold mechanisms can be effectively employed to control chaotic systems onto stable fixed points and limit cycles of widely varying periodicities. Then, we outline the theory and experimental realization of fundamental logic-gates from a chaotic system, using thresholding to effect control. A key feature of this implementation is that a single chaotic 'processor' can be flexibly configured (and re-configured) to emulate different fixed or dynamic logic gates through the simple manipulation of a threshold level.

A family of driving forces to suppress chaos in jerk equations: Laplace domain design

A family of driving forces is discussed in the context of chaos suppression in the Laplace domain. This idea can be attained by increasing the order of the polynomial in the expressions of the driving force to account for the robustness and/or the performance of the closed loop. The motivation arises from the fact that chaotic systems can be controlled by increasing the order of the Laplace controllers even to track arbitrary orbits. However, a larger order in the driving forces can induce an undesirable frequency response, and the control efforts can result in either peaking or large energy accumulation. We overcame these problems by showing that considering the frequency response (interpreted by norms), the closed-loop execution can be improved by designing the feedback suppressor in the Laplace domain. In this manner, the stabilization of the chaotic behavior in jerk-like systems is achieved experimentally. Jerk systems are particularly sensitive to control performance (and robustness issues) because the acceleration time-derivative is involved in their models. Thus, jerky systems are especially helped by a robust control design.