arXiv subjects

V. K. Chandrasekar

Publications and source records attributed to V. K. Chandrasekar.

At least 109 records · Page 6Linked to original sources

arXiv2013

Adaptive coupling induced multi-stable states in complex networks

Adaptive coupling, where the coupling is dynamical and depends on the behaviour of the oscillators in a complex system, is one of the most crucial factors to control the dynamics and streamline various processes in complex networks. In this paper, we have demonstrated the occurrence of multi-stable states in a system of identical phase oscillators that are dynamically coupled. We find that the multi-stable state is comprised of a two cluster synchronization state where the clusters are in anti-phase relationship with each other and a desynchronization state. We also find that the phase relationship between the oscillators is asymptotically stable irrespective of whether there is synchronization or desynchronization in the system. The time scale of the coupling affects the size of the clusters in the two cluster state. We also investigate the effect of both the coupling asymmetry and plasticity asymmetry on the multi-stable states. In the absence of coupling asymmetry, increasing the plasticity asymmetry causes the system to go from a two clustered state to a desynchronization state and then to a two clustered state. Further, the coupling asymmetry, if present, also affects this transition. We also analytically find the occurrence of the above mentioned multi-stable-desynchronization-multi-stable state transition. A brief discussion on the phase evolution of nonidentical oscillators is also provided. Our analytical results are in good agreement with our numerical observations.

nlin.AO↗

arXiv2013

A nonlocal connection between certain linear and nonlinear ordinary differential equations - Part II: Complex nonlinear oscillators

In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear complex ordinary differential equations (ODEs) with complex linear ODEs, thereby proving the integrability of the former. We also show how to construct the solutions using the two types of nonlocal transformations with several physically interesting cases as examples.

nlin.SI↗

arXiv2013

Generating Finite Dimensional Integrable Nonlinear Dynamical Systems

In this article, we present a brief overview of some of the recent progress made in identifying and generating finite dimensional integrable nonlinear dynamical systems, exhibiting interesting oscillatory and other solution properties, including quantum aspects. Particularly we concentrate on Lienard type nonlinear oscillators and their generalizations and coupled versions. Specific systems include Mathews-Lakshmanan oscillators, modified Emden equations, isochronous oscillators and generalizations. Nonstandard Lagrangian and Hamiltonian formulations of some of these systems are also briefly touched upon. Nonlocal transformations and linearization aspects are also discussed.

nlin.SI↗

arXiv2013

Synchronization of an array of spin torque nano oscillators in periodic applied external magnetic field

Considering an array of spin torque transfer nano oscillators (STNOs), we have investigated the synchronization property of the system under the action of a common periodically driven applied external magnetic field by numerically analyzing the underlying system of Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equations for the macro-spin variables. We find the novel result that the applied external magnetic field can act as a medium to induce synchronization of periodic oscillations, both in-phase and anti-phase, even without coupling through spin current, thereby leading to the exciting possibility of enhancement of microwave power in a straightforward way.

nlin.CD↗

arXiv2012

Method of generating $N$-dimensional isochronous nonsingular Hamiltonian systems

In this paper we develop a straightforward procedure to construct higher dimensional isochronous Hamiltonian systems. We first show that a class of singular Hamiltonian systems obtained through the $Ω$-modified procedure is equivalent to constrained Newtonian systems. Even though such systems admit isochronous oscillations, they are effectively one degree of freedom systems due to the constraints. Then we generalize the procedure in terms of $Ω_i$-modified Hamiltonians and identify suitable canonically conjugate coordinates such that the constructed $Ω_i$-modified Hamiltonian is \emph{nonsingular} and the corresponding Newton's equation of motion is constraint free. The procedure is first illustrated for two dimensional systems and subsequently extended to $N$-dimensional systems. The general solution of these systems are obtained by integrating the underlying equations and is shown to admit isochronous as well as amplitude independent quasiperiodic solutions depending on the choice of parameters.

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arXiv2012

On the complete integrability of a nonlinear oscillator from group theoretical perspective

In this paper, we investigate the integrability aspects of a physically important nonlinear oscillator which lacks sufficient number of Lie point symmetries but can be integrated by quadrature. We explore the hidden symmetry, construct a second integral and derive the general solution of this oscillator by employing the recently introduced $λ$-symmetry approach and thereby establish the complete integrability of this nonlinear oscillator equation from a group theoretical perspective.

nlin.SI↗

arXiv2012

A class of solvable coupled nonlinear oscillators with amplitude independent frequencies

Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Liénard type oscillators exhibit this interesting property. We show that a specific subset can be explicitly solved from which we demonstrate the existence of periodic and quasiperiodic solutions. Another set of $N$-coupled nonlinear oscillators, possessing the amplitude independent nature of frequencies, is almost integrable in the sense that the system can be reduced to a single nonautonomous first order scalar differential equation which can be easily integrated numerically.

nlin.SI↗

arXiv2012

Exact solutions of coupled Liénard-type nonlinear systems using factorization technique

General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find although powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases particular solutions may be found with little effort if it is possible to factorize the equation in terms of first order differential operators. In our present study we use this factorization technique to address the problem of finding solutions of a system of general two-coupled Liénard type nonlinear differential equations. We describe a generic algorithm to identify specific classes of Liénard type systems for which solutions may be found. We demonstrate this method by identifying a class of two-coupled equations for which the particular solution can be found by solving a Bernoulli equation. This class of equations include coupled generalization of the modified Emden equation. We further deduce the general solution of a class of coupled ordinary differential equations using the factorization procedure discussed in this manuscript.

nlin.SI↗

arXiv2012

A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: II. Extension to coupled ODEs

In this second paper on the method of deriving linearizing transformations for nonlinear ODEs, we extend the method to a set of two coupled second order nonlinear ODEs. We show that besides the conventional point, Sundman and generalized linearizing transformations one can also find a large class of mixed or hybrid type linearizing transformations like point-Sundman, point-generalized linearizing transformation and Sundman-generalized linearizing transformation in coupled second order ODEs using the integrals of motion. We propose suitable algorithms to identify all these transformations (with maximal in number) in a straightforward manner. We illustrate the method of deriving each one of the linearizing transformations with a suitable example.

nlin.SI↗

arXiv2012

A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: I. Scalar case

In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focuss our attention on a system of two coupled second order ODEs. In the case of scalar ODEs, we consider second and third order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.

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arXiv2011

Nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order

In this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations (ODEs) of arbitrary order related to linear ODEs through nonlocal relations. The procedure makes use of the Lie point symmetries of the linear ODEs and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear ODEs. Using these nonlocal symmetries we obtain reduction transformations and reduced equations to specific examples. We find the reduced equations can be explicitly integrated to deduce the general solutions for these cases. We also extend this procedure to coupled higher order nonlinear ODEs with specific reference to second order nonlinear ODEs.

nlin.SI↗

arXiv2011

A general coupled nonlinear oscillator model for event-related (de)synchronization

Changes in the level of synchronization and desynchronization in coupled oscillator systems due to an external stimulus is called event related synchronization or desynchronization (ERS/ERD). Such changes occur in real life systems where the collective activity of the entities of a coupled system is affected by some external influence. In order to understand the role played by the external influence in the occurrence of ERD and ERS, we study a system of coupled nonlinear oscillators in the presence of an external stimulus signal. We find that the phenomena of ERS and ERD are generic and occur in all types of coupled oscillator systems. We also find that the same external stimulus signal can cause ERS and ERD depending upon the strength of the signal. We identify the stability of the ERS and ERD states and also find analytical and numerical boundaries between the different synchronization regimes involved in the occurrence of ERD and ERS.

nlin.AO↗

arXiv2011

Mass synchronization: Occurrence and its control with possible applications to brain dynamics

Occurrence of strong or mass synchronization of a large number of neuronal populations in the brain characterizes its pathological states. In order to establish an understanding of the mechanism underlying such pathological synchronization we present a model of coupled populations of phase oscillators representing the interacting neuronal populations. Through numerical analysis, we discuss the occurrence of mass synchronization in the model, where a source population which gets strongly synchronized drives the target populations onto mass synchronization. We hypothesize and identify a possible cause for the occurrence of such a synchronization, which is so far unknown: Pathological synchronization is caused not just because of the increase in the strength of coupling between the populations but also because of the strength of the strong synchronization of the drive population. We propose a demand-controlled method to control this pathological synchronization by providing a delayed feedback where the strength and frequency of the synchronization determines the strength and the time delay of the feedback. We provide an analytical explanation for the occurrence of pathological synchronization and its control in the thermodynamic limit.

q-bio.NC↗

arXiv2010

A nonlocal connection between certain linear and nonlinear ordinary differential equations : Extension to coupled equations

Identifying integrable coupled nonlinear ordinary differential equations (ODEs) of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics. In this paper we undertake these problems and unearth two classes of integrable coupled nonlinear ODEs of arbitrary order. To achieve these goals we introduce suitable nonlocal transformations in certain linear ODEs and generate the coupled nonlinear ODEs. In particular, we show that the problem of solving these classes of coupled nonlinear ODEs of any order, effectively reduces to solving a single first order nonlinear ODE. We then describe a procedure to derive explicit general solutions for the identified integrable coupled ODEs, when the above mentioned first order nonlinear ODE reduces to a Bernoulli equation. The equations which we generate and solve include the two coupled versions of modified Emden equations (in second order), coupled versions of Chazy equations (in third order) and their variants, higher dimensional coupled Ricatti and Abel chains as well as a new integrable chain and higher order equations.

nlin.SI↗

arXiv2010

Chimera and globally clustered chimera: Impact of time delay

Following a short report of our preliminary results [Phys. Rev. E 79, 055203(R) (2009)], we present a more detailed study of the effects of coupling delay in diffusively coupled phase oscillator populations. We find that coupling delay induces chimera and globally clustered chimera (GCC) states in delay coupled populations. We show the existence of multi-clustered states that act as link between the chimera and the GCC states. A stable GCC state goes through a variety of GCC states, namely periodic, aperiodic, long-- and short--period breathers and becomes unstable GCC leading to global synchronization in the system, on increasing time delay. We provide numerical evidence and theoretical explanations for the above results and discuss possible applications of the observed phenomena.

nlin.AO↗

arXiv2010

Recursive Generation of Isochronous Hamiltonian Systems

We propose a simple procedure to identify the collective coordinate $Q$ which is used to generate the isochronous Hamiltonian. The new isochronous Hamiltonian generates more and more isochronous oscillators, recursively.

nlin.SI↗

arXiv2010

On certain new integrable second order nonlinear differential equations and their connection with two dimensional Lotka-Volterra system

In this paper, we consider a second order nonlinear ordinary differential equation of the form $\ddot{x}+k_1\frac{\dot{x}^2}{x}+(k_2+k_3x)\dot{x}+k_4x^3+k_5x^2+k_6x=0$, where $k_i$'s, $i=1,2,...,6,$ are arbitrary parameters. By using the modified Prelle-Singer procedure, we identify five new integrable cases in this equation besides two known integrable cases, namely (i) $k_2=0, k_3=0$ and (ii) $k_1=0, k_2=0, k_5=0$. Among these five, four equations admit time dependent first integrals and the remaining one admits time independent first integral. From the time independent first integral, nonstandard Hamiltonian structure is deduced thereby proving the Liouville sense of integrability. In the case of time dependent integrals, we either explicitly integrate the system or transform to a time-independent case and deduce the underlying Hamiltonian structure. We also demonstrate that the above second order ordinary differential equation is intimately related to the two-dimensional Lotka-Volterra (LV) system. From the integrable parameters of above nonlinear equation and all the known integrable cases of the latter can be deduced thereby.

nlin.SI↗

arXiv2009

Event--related desynchronization in diffusively coupled oscillator models

We seek explanation for the neurophysiological phenomenon of event related desynchronization (ERD) by using models of diffusively coupled nonlinear oscillators. We demonstrate that when the strength of the event is sufficient, ERD is found to emerge and the accomplishment of a behavioral/functional task is determined by the nature of the desynchronized state. We illustrate the phenomenon for the case of limit cycle and chaotic systems. We numerically demonstrate the occurrence of ERD and provide analytical explanation. We also discuss possible applications of the observed phenomenon in real physical systems other than the brain.

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