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      Volume 84, Issue 3

      March 2015,   pages  327-490

    • Dynamics of solitons in multicomponent long wave–short wave resonance interaction system

      T Kanna K Sakkaravarthi M Vijayajayanthi M Lakshmanan

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      In this paper, we study the formation of solitons, their propagation and collision behaviour in an integrable multicomponent (2+1)-dimensional long wave–short wave resonance interaction (𝑀-LSRI) system. First, we briefly revisit the earlier results on the dynamics of bright solitons and demonstrate the fascinating energy exchange collision of bright solitons appearing in the short-wave components of the 𝑀-LSRI system. Then, we explicitly construct the exact one-and two-multicomponent dark soliton solutions of the 𝑀-LSRI system by using the Hirota’s direct method and explore its propagation dynamics. Also, we study the features of dark soliton collisions.

    • Breathers and rogue waves: Demonstration with coupled nonlinear Schrödinger family of equations

      N Vishnu Priya M Senthilvelan M Lakshmanan

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      Different types of breathers and rogue waves (RWs) are some of the important coherent structures which have been recently realized in several physical phenomena in hydrodynamics, nonlinear optics, Bose–Einstein condensates, etc. Mathematically, they have been deduced in non-linear Schrödinger (NLS) equations. Here we show the existence of general breathers, Akhmediev breathers, Ma soliton and rogue wave solutions in coupled Manakov-type NLS equations and coupled generalized NLS equations representing four-wave mixing. We deduce their explicit forms using Hirota bilinearization procedure and bring out their exact structures and important properties. We also show the method to deduce the various breather solutions from rogue wave solutions using factorization form and the so-called imbricate series.

    • Empirical questions for collective-behaviour modelling

      Nicholas T Ouellette

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      The collective behaviour of groups of social animals has been an active topic of study across many disciplines, and has a long history of modelling. Classical models have been successful in capturing the large-scale patterns formed by animal aggregations, but fare less well in accounting for details, particularly for groups that do not display net motion. Inspired by recent measurements of swarming insects, which are not well described by the classical modelling paradigm, I pose a set of questions that must be answered by any collective-behaviour model. By explicitly stating the choices made in response to each of these questions, models can be more easily categorized and compared, and their expected range of validity can be clarified.

    • Classification of periodic, chaotic and random sequences using approximate entropy and Lempel–Ziv complexity measures

      Karthi Balasubramanian Silpa S Nair Nithin Nagaraj

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      ‘Complexity’ has several definitions in diverse fields. These measures are indicators of some aspects of the nature of the signal. Such measures are used to analyse and classify signals and as a signal diagnostics tool to distinguish between periodic, quasiperiodic, chaotic and random signals. Lempel–Ziv (LZ) complexity and approximate entropy (ApEn) are such popular complexity measures that are widely used for characterizing biological signals also. In this paper, we compare the utility of ApEn, LZ complexities and Shannon’s entropy in characterizing data from a nonlinear chaotic map (logistic map). In this work, we show that LZ and ApEn complexity measures can characterize the data complexities correctly for data sequences as short as 20 in length while Shannon’s entropy fails for length less than 50. In the case of noisy sequences with 10% uniform noise, Shannon’s entropy works only for lengths greater than 200 while LZ and ApEn are successful with sequences of lengths greater than 30 and 20, respectively.

    • Generalized virial theorem for the Liénard-type systems

      José Cariñena Anindya Ghose Choudhury Partha Guha

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      A geometrical description of the virial theorem (VT) of statistical mechanics is presented using the symplectic formalism. The character of the Clausius virial function is determined for second-order differential equations of the Liénard type. The explicit dependence of the virial function on the Jacobi last multiplier is illustrated. The latter displays a dual role, namely, as a position-dependent mass term and as an appropriate measure in the geometrical context.

    • Effect of heat source on the growth of dendritic drying patterns

      Kiran M Kolwankar Pulkit Prakash Shruthi Radhakrishnan Swadhini Sahu Aditya K Dharmadhikari Jayashree A Dharmadhikari Deepak Mathur

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      Shining a tightly-focussed but low-powered laser beam on an absorber dispersed in a biological fluid gives rise to spectacular growth of dendritic patterns. These result from localized drying of the fluid because of efficient absorption and conduction of optical energy by the absorber. We have carried out experiments in several biologically relevant fluids and have analysed patterns generated by different types of absorbers. We observe that the growth velocity of branches in the dendritic patterns can decrease below the value expected for natural drying.

    • Thermalized solutions, statistical mechanics and turbulence: An overview of some recent results

      Samriddhi Sankar Ray

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      In this study, we examine the intriguing connection between turbulence and equilibrium statistical mechanics. There are several recent works which emphasize this connection. Thus in the last few years, the first manifestations of the thermalization, predicted by T D Lee in 1952, was seen and a theoretical understanding of this was developed through detailed studies of finite-dimensional, Galerkin-truncated equations of hydrodynamics. Furthermore, the idea of the Galerkin truncation can be generalized for studying turbulence in non-integer (fractal) dimensions to yield a new, critical dimension with an equilibrium Gibbs state coinciding with a Kolmogorov spectrum. In this paper, we discuss these very exciting and recent developments in turbulence as well as open problems for the future.

    • Rolling motion in moving droplets

      Sumesh P Thampi Rama Govindarajan

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      Drops moving on a substrate under the action of gravity display both rolling and sliding motions. The two limits of a thin sheet-like drop in sliding motion on a surface, and a spherical drop in roll, have been extensively studied. We are interested in intermediate shapes. We quantify the contribution of rolling motion for any intermediate shape, and recently obtained a universal curve for the amount of roll as a function of a shape parameter using hybrid lattice Boltzmann simulations. In this paper, we discuss the linear relationship which is expected between the Capillary and Bond numbers, and provide detailed confirmation by simulations. We also show that the viscosity of the surrounding medium can qualitatively affect dynamics. Our results provide an answer to a natural question of whether drops roll or slide on a surface and carry implications for various applications where rolling motion may or may not be preferred.

    • State variable participation in the limit cycle of induction motor

      Krishnendu Chakrabarty Urmila Kar

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      The paper presents bifurcation behaviour of a single-phase induction motor. Study of bifurcation of a system gives the complete picture of its dynamical behaviour with the change in system’s parameters. The system is mathematically described by a set of differential equations in the state space. Induction motors are very widely used in domestic and commercial applications. Single-phase capacitor-run induction motors are commonly used as prime movers for fans, pumps and compressors. This paper provides a numerical approach to understand the dynamics of an induction motor in the light of bifurcation and chaos. It is seen that the dynamics of a capacitor-run single-phase induction motor cannot be ascertained by the profile of a single state variable. This paper also attempts to discuss the bifurcation behaviour of the system based on the evolution of different state variables. The bifurcation diagrams drawn looking at different state variables are different in terms of periodicity and route to chaos. The knowledge of the dynamics of the system obtained from bifurcation diagrams give useful guidelines to control the operation of the induction motor depending on the need of an application for better performance.

    • Non-linear dynamics in pulse combustor: A review

      Sirshendu Mondal Achintya Kukhopadhyay Swarnendu Sen

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      The state of the art of non-linear dynamics applied to pulse combustor theoretically and experimentally is reviewed. Pulse combustors are a class of air-breathing engines in which pulsations in combustion are utilized to improve the performance. As no analytical solution can be obtained for most of the nonlinear systems, the whole set of solutions can be investigated with the help of dynamical system theory. Many studies have been carried out on pulse combustors whose dynamics include limit cycle behaviour, Hopf bifurcation and period-doubling bifurcation. The dynamic signature has also been used for early prediction of extinction.

    • Building better oscillators using nonlinear dynamics and pattern formation

      M C Cross Eyal Kenig John-Mark A Allen

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      Frequency and time references play an essential role in modern technology and in living systems. The precision of self-sustained oscillations is limited by the effects of noise, which becomes evermore important as the sizes of the devices become smaller. In this paper, we review our recent theoretical results on using nonlinear dynamics and pattern formation to reduce the effects of noise and improve the frequency precision of oscillators, with particular reference to ongoing experiments on oscillators based on nanomechanical resonators. We discuss using resonator nonlinearity, novel oscillator architectures and the synchronization of arrays of oscillators, to improve the frequency precision.

    • Nonlinear dynamics of spin transfer nano-oscillators

      B Subash V K Chandrasekar M Lakshmanan

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      The evolution equation of a ferromagnetic spin system described by Heisenberg nearest-neighbour interaction is given by Landau–Lifshitz–Gilbert (LLG) equation, which is a fascinating nonlinear dynamical system. For a nanomagnetic trilayer structure (spin valve or pillar) an additional torque term due to spin-polarized current has been suggested by Slonczewski, which gives rise to a rich variety of dynamics in the free layer. Under appropriate conditions the spin-polarized current gives a time-varying resistance to the magnetic structure thereby inducing magnetization oscillations of frequency which lies in the microwave region. Such a device is called a spin transfer nanooscillator (STNO). However, this interesting nanoscale level source of microwaves lacks efficiency due to its low emitting power typically of the order of nWs. To over-come this difficulty, one has to consider the collective dynamics of synchronized arrays/networks of STNOs as suggested by Fert and coworkers so that the power can be enhanced 𝑁2 times that of a single STNO. We show that this goal can be achieved by applying a common microwave magnetic field to an array of STNOs. In order to make the system technically more feasible to practical level integration with CMOS circuits, we establish suitable electrical connections between the oscillators. Although the electrical connection makes the system more complex, the applied microwave magnetic field drives the system to synchronization in large regions of parameter space.

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