arXiv subjects

M. Lakshmanan

Publications and source records attributed to M. Lakshmanan.

At least 235 records · Page 13Linked to original sources

arXiv2005

On the complete integrability and linearization of nonlinear ordinary differential equations - Part II: Third order equations

We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle-Singer method. We describe a procedure to deduce all the integrals of motion associated with the given equation so that the general solution follows straightforwardly from these integrals. The method is illustrated with several examples. Further, we propose a powerful method of identifying linearizing transformations. The proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation (GLT). In addition to the above, we provide an algorithm to invert the nonlocal linearizing transformation. Through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation.

nlin.SI↗

arXiv2005

Existence of anticipatory, complete and lag synchronizations in time-delay systems

Existence of different kinds of synchronizations, namely anticipatory, complete and lag type synchronizations (both exact and approximate), are shown to be possible in time-delay coupled piecewise linear systems. We deduce stability condition for synchronization of such unidirectionally coupled systems following Krasovskii-Lyapunov theory. Transition from anticipatory to lag synchronization via complete synchronization as a function of coupling delay is discussed. The existence of exact synchronization is preceded by a region of approximate synchronization from desynchronized state as a function of a system parameter, whose value determines the stability condition for synchronization. The results are corroborated by the nature of similarity functions. A new type of oscillating synchronization that oscillates between anticipatory, complete and lag synchronization, is identified as a consequence of delay time modulation with suitable stability condition.

nlin.CD↗

arXiv2005

Transition from anticipatory to lag synchronization via complete synchronization in time-delay systems

The existence of anticipatory, complete and lag synchronization in a single system having two different time-delays, that is feedback delay $τ_1$ and coupling delay $τ_2$, is identified. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay $τ_2$ with suitable stability condition is discussed. The existence of anticipatory and lag synchronization is characterized both by the minimum of similarity function and the transition from on-off intermittency to periodic structure in laminar phase distribution.

nlin.CD↗

arXiv2005

Desynchronized wave patterns in synchronized chaotic regions of coupled map lattices

We analyze the size limits of coupled map lattices with diffusive coupling at the crossover of low-dimensional to high-dimensional chaos. We investigate the existence of standing-wave-type periodic patterns, within the low-dimensional limit, in addition to the stable synchronous chaotic states depending upon the initial conditions. Further, we bring out a controlling mechanism to explain the emergence of standing-wave patterns in the coupled map lattices. Finally, we give an analytic expression in terms of the unstable periodic orbits of the isolated map to represent the standing-wave patterns.

nlin.CD↗

arXiv2005

Quantum dynamical echo in two-level systems

Evolution of quantum fidelity for two-level systems is studied in the context of periodic echo. From a general treatment for time independent case, we obtain a simple condition on the governing Hamiltonians under which the systems display periodic quantum echo. For a specific time dependent problem the quantum fidelity is shown to exhibit Rabi oscillation. This may be considered as a simple mechanism to generate periodic echo, except for a specific initial superpositional state in which case the fidelity remains invariant.

nlin.SI↗

arXiv2005

On the complete integrability and linearization of certain second order nonlinear ordinary differential equations

A method of finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle-Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the PS method. In addition to the above we introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second order nonlinear ordinary differential equations to free particle equation. We illustrate the theory with several potentially important examples and show that our procedure is widely applicable.

nlin.SI↗

arXiv2005

An unusual Liénard type nonlinear oscillator with properties of a linear harmonic oscillator

A Liénard type nonlinear oscillator of the form $\ddot{x}+kx\dot{x}+\frac{k^2}{9}x^3+λ_1 x=0$, which may also be considered as a generalized Emden type equation, is shown to possess unusual nonlinear dynamical properties. It is shown to admit explicit nonisolated periodic orbits of conservative Hamiltonian type for $λ_1>0$. These periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator. This is completely contrary to the standard characteristic property of nonlinear oscillators. Interestingly, the system though appears deceptively a dissipative type for $λ_1\leq0$ does admit a conserved Hamiltonian description, where the characteristic decay time is also independent of the amplitude. The results also show that the criterion for conservative Hamiltonian system in terms of divergence of flow function needs to be generalized.

nlin.SI↗

arXiv2005

Equatorial and related non-equilibrium states in magnetization dynamics of ferromagnets: Generalization of Suhl's spin-wave instabilities

We investigate the nonlinear dynamics underlying the evolution of a 2-D nanoscale ferromagnetic film with uniaxial anisotropy in the presence of perpendicular pumping. Considering the associated Landau-Lifshitz spin evolution equation with Gilbert damping together with Maxwell equation for the demagnetization field, we study the dynamics in terms of the stereographic variable. We identify several new fixed points for suitable choice of external field in a rotating frame of reference. In particular, we identify explicit equatorial and related fixed points of the spin vector in the plane transverse to the anisotropy axis when the pumping frequency coincides with the amplitude of the static parallel field. We then study the linear stability of these novel fixed points under homogeneous and spin wave perturbations and obtain a generalized Suhl's instability criterion, giving the condition for exponential growth of P-modes under spin wave perturbations. Two parameter phase diagrams (in terms of amplitudes of static parallel and oscillatory perpendicular magnetic fields) for stability are obtained, which differ qualitatively from those for the conventional ferromagnetic resonance near thermal equilibrium and are amenable to experimental tests.

nlin.CD↗

arXiv2004

Localized Coherent Structures of Ishimori Equation I through Hirota's Bilinearization method:Time dependent/Stationary boundaries

Ishimori equation is a $(2+1)$ dimensional generalization of the $(1+1)$ dimensional integrable classical continuous Heisenberg ferromagnetic spin equation. The richness of the coherent structures admitted by Ishimori equation I such as dromion, lump and rationally- exponentially localized solutions, have been demonstrated in the literature through $\bar \partial$ technique and binary Darboux transformation method. To our knowledge Hirota's method had been adopted to construct only the vortex solutions of Ishimori equation II. For the first time, the various types of localized coherent structures mentioned above have been constructed in this paper for the Ishimori equation I using the Hirota's direct method. In particular we have brought out the significance of boundaries and arbitrary functions in generating all these types of localized structures and proved that the absence of such boundaries leads only to line soliton solutions.

nlin.SI↗

arXiv2004

Bifurcations and Chaos in Time Delayed Piecewise Linear Dynamical Systems

We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of bifurcations and chaos associated with it as a function of the delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. Significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for low values of time delay which is evidenced by multiple positive Lyapunov exponents are brought out. The study is extended to the case of two coupled systems, one with delay and the other one without delay.

nlin.CD↗

arXiv2004

Estimation of System Parameters and Predicting the Flow Function from Time Series of Continuous Dynamical Systems

We introduce a simple method to estimate the system parameters in continuous dynamical systems from the time series. In this method, we construct a modified system by introducing some constants (controlling constants) into the given (original) system. Then the system parameters and the controlling constants are determined by solving a set of nonlinear simultaneous algebraic equations obtained from the relation connecting original and modified systems. Finally, the method is extended to find the form of the evolution equation of the system itself. The major advantage of the method is that it needs only a minimal number of time series data and is applicable to dynamical systems of any dimension. The method also works extremely well even in the presence of noise in the time series. This method is illustrated for the case of Lorenz system.

nlin.CD↗

arXiv2004

On the Integrability, BÄcklund Transformation and Symmetry Aspects of a Generalized Fisher Type Nonlinear Reaction-Diffusion Equation

The dynamics of nonlinear reaction-diffusion systems is dominated by the onset of patterns and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized Fisher equation in both (1+1) and (2+1) dimensions. A Painlevé singularity structure analysis singles out a special case ($m=2$) as integrable. More interestingly, a Bäcklund transformation is shown to give rise to a linearizing transformation for the integrable case. A Lie symmetry analysis again separates out the same $m=2$ case as the integrable one and hence we report several physically interesting solutions via similarity reductions. Thus we give a group theoretical interpretation for the system under study. Explicit and numerical solutions for specific cases of nonintegrable systems are also given. In particular, the system is found to exhibit different types of travelling wave solutions and patterns, static structures and localized structures. Besides the Lie symmetry analysis, nonclassical and generalized conditional symmetry analysis are also carried out.

nlin.SI↗

arXiv2004

Symmetries and Integrability Properties of Generalized Fisher Type Nonlinear Diffusion Equation

Nonlinear reaction-diffusion systems are known to exhibit very many novel spatiotemporal patterns. Fisher equation is a prototype of diffusive equations. In this contribution we investigate the integrability properties of the generalized Fisher type equation to obtain physically interesting solutions using Lie symmetry analysis. In particular, we report several travelling wave patterns, static patterns and localized structures depending upon the choice of the parameters involved.

nlin.SI↗

arXiv2004

Spatio-temporal dynamics of coupled array of Murali-Lakshmanan-Chua circuits

The circuit recently proposed by Murali, Lakshmanan and Chua (MLC) is one of the simplest non-autonomous nonlinear electronic circuits which shows a variety of dynamical phenomena including various bifurcations, chaos and so on. In this paper we study the spatio-temporal dynamics in one and two dimensional arrays of coupled MLC circuits both in the absence as well as in the presence of external periodic force. In the absence of any external force, the propagation phenomena of travelling wave front and its failure have been observed from numerical simulations....

patt-sol↗

arXiv2004

New aspects of integrability of force-free Duffing-van der Pol oscillator and related nonlinear systems

In this letter we show that the force-free Duffing-van der Pol oscillator is completely integrable for a specific parametric choice. We derive a general solution for this parametric choice. Further, we describe a procedure to construct the transformation which removes the time dependent part from the first integral and provide the general solution by quadrature. The procedure is shown to have a wider applicability through additional examples. We also show that through our method one can deduce linearizing transformations in a simple and straightforward way and illustrate it with a specific example.

nlin.SI↗

arXiv2004

Nonlinear Dynamics of Moving Curves and Surfaces: Applications to Physical Systems

The subject of moving curves (and surfaces) in three dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a variety of physical problems in different disciplines. Making use of the underlying geometry, one can very often relate the associated evolution equations to many interesting nonlinear evolution equations, including soliton possessing nonlinear dynamical systems. Typical examples include dynamics of filament vortices in ordinary and superfluids, spin systems, phases in classical optics, various systems encountered in physics of soft matter, etc. Such interrelations between geometric evolution and physical systems have yielded considerable insight into the underlying dynamics. We present a succinct tutorial analysis of these developments in this article, and indicate further directions. We also point out how evolution equations for moving surfaces are often intimately related to soliton equations in higher dimensions.

nlin.PS↗

arXiv2004

Estimation of System Parameters in Discrete Dynamical Systems from Time Series

We propose a simple method to estimate the parameters involved in discrete dynamical systems from time series. The method is based on the concept of controlling chaos by constant feedback. The major advantages of the method are that it needs a minimal number of time series data and is applicable to dynamical systems of any dimension. The method also works extremely well even in the presence of noise in the time series. The method is specifically illustrated by means of logistic and Henon maps.

nlin.CD↗

arXiv2004

Exponentially Localized Solutions of Mel'nikov Equation

The Mel'nikov equation is a (2+1) dimensional nonlinear evolution equation admitting boomeron type solutions. In this paper, after showing that it satisfies the Painlevé property, we obtain exponentially localized dromion type solutions from the bilinearized version which have not been reported so far. We also obtain more general dromion type solutions with spatially varying amplitude as well as induced multi-dromion solutions.

nlin.SI↗