Engineering synchronization of chaotic oscillators using controller based coupling design (original) (raw)
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Design of coupling for synchronization of chaotic oscillators
Physical Review E, 2009
A general procedure is discussed to formulate a coupling function capable of targeting desired responses such as synchronization, antisynchronization, and amplitude death in identical as well as mismatched chaotic oscillators. The coupling function is derived for unidirectional, mutual, and matrix type coupling. The matrix coupling, particularly, is able to induce synchronization, antisynchronization, and amplitude death simultaneously in different state variables of a response system. The applicability of the coupling is demonstrated in spiking-bursting Hindmarsh-Rose neuron model, Rössler oscillator, Lorenz system, Sprott system, and a double scroll system. We also report a scaling law that defines a process of transition to synchronization.
Targeting and Control of Synchronization in Chaotic Oscillators
International Journal of Bifurcation and Chaos, 2012
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 unid...
Engineering generalized synchronization in chaotic oscillators
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2011
Synchrony is an amazing natural phenomenon ubiquitous in the living world and also verified in laboratory scale systems: chemical reaction, laser, and electronic circuits. Over the last two decades, the theoretical framework of synchronization of nonlinear oscillators, limit cycle and chaotic, is mainly established under unidirectional or mutual interactions and with complex topology. However, most of the theories start with an approximation of identical oscillators to reduce the burden of mathematical complexity to develop an understanding ...
Control of partial synchronization in chaotic oscillators
Pramana, 2015
A design of coupling is proposed to control partial synchronization in two chaotic oscillators in a driver-response mode. A control of synchrony between one response variables is made possible (a transition from a complete synchronization to antisynchronization via amplitude death and vice versa without loss of synchrony) keeping the other pairs of variables undisturbed in their pre-desired states of coherence. Further, one of the response variables can be controlled so as to follow the dynamics of an external signal (periodic or chaotic) while keeping the coherent status of other variables unchanged. The stability of synchronization is established using the Hurwitz matrix criterion. Numerical example of an ecological foodweb model is presented. The control scheme is demonstrated in an electronic circuit of the Sprott system.
Coupling conditions for globally stable and robust synchrony of chaotic systems
Physical review. E, 2017
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...
SN Applied Sciences
A model for synchronization of coupled Nakano's chaotic circuits is studied. The Nakano circuit consists of a simple RLC circuit with a switch voltage-depending reset rule which generates a discontinuous dynamics. Thus, the model that we study is a network of identical spiking oscillators with integrate-and-fire dynamics. The coupling between oscillators is linear, but the network is subject to a common regime of reset depending on the global state of the oscillator population. This constitutes the simplest way of build pulse-coupled networks with arbitrary topology for this type of oscillators, and it allows the emergence of synchronous states and different reset regimes. The main result is that under certain hypothesis over the weight matrix (that represents the network topology) the different reset regimes match and the formalism of the master stability function can be generalized in order to study the stability of the synchronous state and the discontinuous dynamic of the network. Also, the low dimensionality of the Nakano's circuit allows to implement the saltation-matrix method and numerical simulations can be performed in order to analyze the role of the coupling mode in the synchronization regime of the network and the influence of the voltage-type variables.
Chaotic synchronization through coupling strategies
Chaos, 2006
Usually, complete synchronization (CS) is regarded as the form of synchronization proper of identical chaotic systems, while generalized synchronization (GS) extends CS in nonidentical systems. However, this generally accepted view ignores the role that the coupling plays in determining the type of synchronization. In this work, we show that by choosing appropriate coupling strategies, CS can be observed in coupled
Synchronization transition in neuronal networks composed of chaotic or non-chaotic oscillators
Scientific Reports
Chaotic dynamics has been shown in the dynamics of neurons and neural networks, in experimental data and numerical simulations. Theoretical studies have proposed an underlying role of chaos in neural systems. Nevertheless, whether chaotic neural oscillators make a significant contribution to network behaviour and whether the dynamical richness of neural networks is sensitive to the dynamics of isolated neurons, still remain open questions. We investigated synchronization transitions in heterogeneous neural networks of neurons connected by electrical coupling in a small world topology. The nodes in our model are oscillatory neurons that-when isolated-can exhibit either chaotic or nonchaotic behaviour, depending on conductance parameters. We found that the heterogeneity of firing rates and firing patterns make a greater contribution than chaos to the steepness of the synchronization transition curve. We also show that chaotic dynamics of the isolated neurons do not always make a visible difference in the transition to full synchrony. Moreover, macroscopic chaos is observed regardless of the dynamics nature of the neurons. However, performing a Functional Connectivity Dynamics analysis, we show that chaotic nodes can promote what is known as multi-stable behaviour, where the network dynamically switches between a number of different semi-synchronized, metastable states. Over the past decades, a number of observations of chaos have been reported in the analysis of time series from a variety of neural systems, ranging from the simplest to the more complex 1,2. It is generally accepted that the inherent instability of chaos in nonlinear systems dynamics, facilitates the extraordinary ability of neural systems to respond quickly to changes in their external inputs 3 , to make transitions from one pattern of behaviour to another when the environment is altered 4 , and to create a rich variety of patterns endowing neuronal circuits with remarkable computational capabilities 5. These features are all suggestive of an underlying role of chaos in neural systems (For reviews, see 5-7), however these ideas may have not been put to test thoroughly. Chaotic dynamics in neural networks can emerge in a variety of ways, including intrinsic mechanisms within individual neurons 8-12 or by interactions between neurons 3,13-21. The first type of chaotic dynamics in neural systems is typically accompanied by microscopic chaotic dynamics at the level of individual oscillators. The presence of this chaos has been observed in networks of Hindmarsh-Rose neurons 8 and biophysical conductance-based neurons 9-12. The second type of chaotic firing pattern is the synchronous chaos. Synchronous chaos has been demonstrated in networks of both biophysical and non-biophysical neurons 3,13,15,17,22-24 , where neurons display synchronous chaotic firing-rate fluctuations. In the latter cases, the chaotic behaviour is a result of network connectivity, since isolated neurons do not display chaotic dynamics or burst firing. More recently, it has been shown that asynchronous chaos, where neurons exhibit asynchronous chaotic firing-rate fluctuations, emerge generically from balanced networks with multiple time scales in their synaptic dynamics 20. Different modelling approaches have been used to uncover important conditions for observing these types of chaotic behaviour (in particular, synchronous and asynchronous chaos) in neural networks, such as the synaptic strength 25-27 , heterogeneity of the numbers of synapses and their synaptic strengths 28,29 , and lately the balance of excitation and inhibition 21. The results obtained by Sompolinsky et al. 25 showed that, when the synaptic strength is increased, neural networks display a highly heterogeneous chaotic state via a transition from an inactive state. Other studies demonstrated that chaotic behaviour emerges in the presence of weak and strong heterogeneities, for example a coupled heterogeneous population of neural oscillators with different synaptic strengths 28-30. Recently, Kadmon et al. 21 highlighted the importance of the balance between excitation and inhibition on a
Enhancing synchronization in chaotic oscillators by induced heterogeneity
The European Physical Journal Special Topics
We report enhancing of complete synchronization in identical chaotic oscillators when their interaction is mediated by a mismatched oscillator. The identical oscillators now interact indirectly through the intermediate relay oscillator. The induced heterogeneity in the intermediate oscillator plays a constructive role in reducing the critical coupling for a transition to complete synchronization. A common lag synchronization emerges between the mismatched relay oscillator and its neighboring identical oscillators that leads to this enhancing effect. We present examples of one-dimensional open array, a ring, a star network and a two-dimensional lattice of dynamical systems to demonstrate how this enhancing effect occurs. The paradigmatic Rössler oscillator is used as a dynamical unit, in our numerical experiment, for different networks to reveal the enhancing phenomenon.