E. Padmanaban

TARGETING AND CONTROL OF SYNCHRONIZATION IN CHAOTIC OSCILLATORS

A method of targeting synchronization and its control is reported in chaotic oscillators. A design of appropriate coupling is proposed using an open-plus-closed-loop (OPCL) scheme based on Hurwitz stability to realize a desired state of synchrony between the oscillators. A general theory of the coupling definition is described for unidirectional as well as bidirectional mode. In a synchronization state, a chaotic attractor can be scaled up or down in size relative to another attractor. Additionally, a technique of controlling synchronization is introduced that allows a smooth transition from complete synchronization to antisynchronization or vice versa, by varying a parameter inserted in the coupling definition without loss of stability during the transition. A smooth scaling of the size of the attractor is also implemented. Numerical examples are given using a Sprott system. Physical realization of the OPCL coupling under bidirectional mode and control of synchronization under unidirectional mode is demonstrated in an electronic circuit.

Engineering synchronization of chaotic oscillators using controller based coupling design

We propose a general formulation of coupling for engineering synchronization in chaotic oscillators for unidirectional as well as bidirectional mode. In the synchronization regimes, it is possible to amplify or to attenuate a chaotic attractor with respect to other chaotic attractors. Numerical examples are presented for a Lorenz system, Rössler oscillator, and a Sprott system. We physically realized the controller based coupling design in electronic circuits to verify the theory. We extended the theory to a network of coupled oscillators and provided a numerical example with four Sprott oscillators.

Amplified response in coupled chaotic oscillators by induced heterogeneity.

The phenomenon of emergent amplified response is reported in two unidirectionally coupled identical chaotic systems when heterogeneity as a parameter mismatch is introduced in a state of complete synchrony. The amplified response emerges from the interplay of heterogeneity and a type of cross-feedback coupling. It is reflected as an expansion of the response attractor in some directions in the state space of the coupled system. The synchronization manifold is simply rotated by the parameter detuning while its stability in the transverse direction is still maintained. The amplification factor is linearly related to the amount of parameter detuning. The phenomenon is elaborated with examples of the paradigmatic Lorenz system, the Shimizu-Morioka single-mode laser model, the Rössler system, and a Sprott system. Experimental evidence of the phenomenon is obtained in an electronic circuit. The method may provide an engineering tool for distortion-free amplification of chaotic signals.

Anomalous behaviour of the bis(benzimidazolium)butane [2]pseudorotaxanes of dibenzo-24-crown-8: a comparative study of methane, ethane, propane and butane bis(benzimidazolium)alkane]dibenzo-24-crown-8 systems with variable spacers

Bis(benzimidazolium)butane (BBIM-butane) system acts as a new guest molecule for dibenzo-24-crown-8 (DB24C8). This is the only [BBIM]DB24C8 pseudorotaxanes where two conformers coexist (detected by 1H NMR and structure established by DFT studies); major conformer and minor conformer. Low temperature 1H NMR (240, 253xa0K) was done for a representative [BBIM-butane]DB24C8 pseudorotaxane that could not be detected at room-temperature. DFT calculations for this [BBIM-butane]DB24C8 pseudorotaxane were done both with B3LYP and MPW1PW91 functionals. The MPW1PW91 optimized structures identified possible stable major and minor conformer with the strongest H-bonding. We have compared the characteristics of [BBIM]DB24C8 [2]pseudorotaxanes comprising the [BBIM-methane]DB24C8, [BBIM-ethane]DB24C8, [BBIM-propane]DB24C8 and [BBIM-butane]DB24C8 systems. The feasibility of formation of these pseudorotaxanes decreases considerably from [BBIM-methane]DB24C8 to the [BBIM-butane]DB24C8 system. The presence of these threaded complexes was initially screened based on 1H NMR and substantiated further by 2D NMR spectroscopy (NOESY and COSY), X-ray crystallography, HR-ESI-MS and DFT calculations.Graphical Abstract

Design of coupling for mixed synchronization in chaotic oscillators

We report a design of coupling in chaotic oscillators for realizing a desired response: complete synchronization, antisynchronization and amplitude death simultaneously in different state variables of a system and thereby targeting a control of synchronization. This is robust to parameter mismatch and the route of transition to synchrony obeys a scaling law. Experimental evidence of the coupling is presented using an electronic circuit.

Coupling conditions for globally stable and robust synchrony of chaotic systems

We propose a set of general coupling conditions to select a coupling profile (a set of coupling matrices) from the linear flow matrix of dynamical systems for realizing global stability of complete synchronization (CS) in identical systems and robustness to parameter perturbation. The coupling matrices define the coupling links between any two oscillators in a network that consists of a conventional diffusive coupling link (self-coupling link) as well as a cross-coupling link. The addition of a selective cross-coupling link in particular plays constructive roles that ensure the global stability of synchrony and furthermore enables robustness of synchrony against small to nonsmall parameter perturbation. We elaborate the general conditions for the selection of coupling profiles for two coupled systems, three- and four-node network motifs analytically as well as numerically using benchmark models, the Lorenz system, the Hindmarsh-Rose neuron model, the Shimizu-Morioka laser model, the Rössler system, and a Sprott system. The role of the cross-coupling link is, particularly, exemplified with an example of a larger network, where it saves the network from a breakdown of synchrony against large parameter perturbation in any node. The perturbed node in the network transits from CS to generalized synchronization (GS) when all the other nodes remain in CS. The GS is manifested by an amplified response of the perturbed node in a coherent state.

Engineering synchronization of chaotic oscillators

We propose a controller based coupling design for engineering synchronization in chaotic oscillators for unidirectional as well as bi‐directional mode. In the synchronization regimes, it is possible to amplify/ attenuate a chaotic attractor with respect to other chaotic attractors. Numerical examples are presented for a Lorenz system, a Rossler oscillator, and a Sprott system. Physical implementation of the scheme is done in electronic circuit to design the controller for verification of the theory.