Manish Dev Shrimali

Controlling Dynamics of Hidden Attractors

Amplitude death (AD) in hidden attractors is attained with a scheme of linear augmentation. This linear control scheme is capable of stabilizing the system to a fixed point state even when the original system does not have any fixed point. Depending on the control parameter, different routes to AD such as boundary crises and Hopf bifurcation are observed. Lyapunov exponent and amplitude index are used to study the dynamical properties of the system.

2008 Special Issue: Threshold control of chaotic neural network

The chaotic neural network constructed with chaotic neurons exhibits rich dynamic behaviour with a nonperiodic associative memory. In the chaotic neural network, however, it is difficult to distinguish the stored patterns in the output patterns because of the chaotic state of the network. In order to apply the nonperiodic associative memory into information search, pattern recognition etc. it is necessary to control chaos in the chaotic neural network. We have studied the chaotic neural network with threshold activated coupling, which provides a controlled network with associative memory dynamics. The network converges to one of its stored patterns or/and reverse patterns which has the smallest Hamming distance from the initial state of the network. The range of the threshold applied to control the neurons in the network depends on the noise level in the initial pattern and decreases with the increase of noise. The chaos control in the chaotic neural network by threshold activated coupling at varying time interval provides controlled output patterns with different temporal periods which depend upon the control parameters.

THE NATURE OF ATTRACTOR BASINS IN MULTISTABLE SYSTEMS

In systems that exhibit multistability, namely those that have more than one coexisting attractor, the basins of attraction evolve in specific ways with the creation of each new attractor. These multiple attractors can be created via different mechanisms. When an attractor is formed via a saddle-node bifurcation, the size of its basin increases as a power-law in the bifurcation parameter. In systems with weak dissipation, the basins of low-order periodic attractors increase linearly, while those of high-order periodic attractors decay exponentially as the dissipation is increased. These general features are illustrated for autonomous as well as driven mappings. In addition, the boundaries of the basins can also change from being smooth to fractal when a new attractor appears. Transitions in the basin boundary morphology are reflected in abrupt changes in the dependence of the uncertainty exponent on the bifurcation parameter.

Phase-flip transition in nonlinear oscillators coupled by dynamic environment.

We study the dynamics of nonlinear oscillators indirectly coupled through a dynamical environment or a common medium. We observed that this form of indirect coupling leads to synchronization and phase-flip transition in periodic as well as chaotic regime of oscillators. The phase-flip transition from in- to anti-phase synchronization or vise-versa is analyzed in the parameter plane with examples of Landau-Stuart and Rössler oscillators. The dynamical transitions are characterized using various indices such as average phase difference, frequency, and Lyapunov exponents. Experimental evidence of the phase-flip transition is shown using an electronic version of the van der Pol oscillators.

Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a hidden attractor in the case of multistability as well as a classical self-excited attractor. The hidden attractor in this system can be localized by analytical/numerical methods based on the continuation and perpetual points. The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of attractors. A conjecture on the Lyapunov dimension of self-excited attractors and the notion of exact Lyapunov dimension are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents and dimension by different algorithms is presented. An adaptive algorithm for studying the dynamics of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden attractor and hidden transient chaotic set in the case of multistability are given.

Pinning control of threshold coupled chaotic neuronal maps.

Chaotic neuronal maps are studied with threshold activated coupling at selected pinning sites with increasing pinning density. A transition from spatiotemporal chaos to a fixed spatial profile with synchronized temporal cycles is observed. There is an optimal fraction of sites where it is necessary to apply the control algorithm in order to effectively suppress chaotic dynamics.

Control and synchronization of chaotic neurons under threshold activated coupling

We have studied the spatiotemporal behaviour of threshold coupled chaotic neurons. We observe that the chaos is controlled by threshold activated coupling, and the system yields synchronized temporally periodic states under the threshold response. Varying the frequency of thresholding provides different higher order periodic behaviors, and can serve as a simple mechanism for stabilising a large range of regular temporal patterns in chaotic systems. Further, we have obtained a transition from spatiotemporal chaos to fixed spatiotemporal profiles, by lengthening the relaxation time scale.

Phase-flip and oscillation-quenching-state transitions through environmental diffusive coupling

We study the dynamics of nonlinear oscillators coupled through environmental diffusive coupling. The interaction between the dynamical systems is maintained through its agents which, in turn, interact globally with each other in the common dynamical environment. We show that this form of coupling scheme can induce an important transition like phase-flip transition as well transitions among oscillation quenching states in identical limit-cycle oscillators. This behavior is analyzed in the parameter plane by analytical and numerical studies of specific cases of the Stuart-Landau oscillator and van der Pol oscillator. Experimental evidences of the phase-flip transition and quenching states are shown using an electronic version of the van der Pol oscillators.

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.

Chaos control in a neural network with threshold activated coupling

The chaotic neural network is studied under the threshold activated coupling, which provides a controlled output patterns. In general the chaotic neural network constructed with chaotic neurons exhibits very rich dynamic behaviors with a nonperiodic associative memory. In the chaotic neural network, it is difficult to distinguish the stored patterns from others, because of the chaotic states of output of the network. In order to apply the nonperiodic associative memory into information search and pattern recognition, etc, it is necessary to control chaos in this chaotic neural network. Chaos in the chaotic neural network is controlled with threshold activated coupling method and the network converges on one of its stored patterns or their reverses which has the smallest Hamming distance with the initial state of the network or provides a controlled neural network with spatio-temporal patterns depending upon the nature of control.