M. Lakshmanan

• Bharathidasan University

The Chua's circuit is examined using a State Controlled-Cellular Neural Network (SC-CNN) framework with two logical square wave input signals. We illustrate, in particular, that this nonlinear circuit can generate all the basic logic operations, including OR/NOR, AND/NAND, and XOR/XNOR gates, by making use of the hopping of attractors which this ci...

In this paper, we demonstrate the emergence of non-degenerate bright solitons and summarize their several interesting features in a completely integrable two-component long-wave-short-wave resonance interaction model with a general form of nonlinearity coefficients. Through the classical Hirota's bilinear method, we obtain a fully non-degenerate $N...

In recent times, bound soliton states have often been referred to as soliton molecules in the nonlinear optics literature. The striking analogies between photonic bound states and matter molecular structures in chemistry and physics have intensified studies on optical soliton molecules in both conservative and dissipative systems. In this paper, we...

Mixed Mode Oscillations (MMOs) correspond to a kind of dynamic behaviour wherein the system alternates between large amplitude oscillations (LAOs) with short time scales and small amplitude oscillations (SAOs) with longer time scales as time progresses. In the present work, MMOs have been reported to occur in the memristive Murali-Lakshmanan-Chua (...

Many dynamical systems exhibit unexpected large amplitude excursions in the chronological progression of a state variable. In the present work, we consider the dynamics associated with the one-dimensional Higgs oscillator, which is realized through gnomonic projection of a harmonic oscillator defined on a spherical space of constant curvature onto...

Many dynamical systems exhibit unexpected large amplitude excursions in the chronological progression of a state variable. In the present work, we consider the dynamics associated with the one-dimensional Higgs oscillator which is realized through gnomic projection of a harmonic oscillator defined on a spherical space of constant curvature onto a E...

We investigate the dynamics of modulational instability (MI) in $\cal PT$-symmetric fiber Bragg gratings with a phenomenon of intermodulation known as four-wave mixing (FWM). Although the impact of FWM has already been analyzed in the conventional systems, the inclusion of gain and loss, which induces the notion of $\cal PT$- symmetry, gives rise t...

This paper discusses how to generalize the invariant subspace method for deriving the analytical solutions of the multi-component (N+1)-dimensional coupled nonlinear time-fractional PDEs (NTFPDEs) in the sense of Caputo fractional-order derivative for the first time. Specifically, we describe how to systematically find different invariant product l...

Recent studies on reservoir computing essentially involve a high-dimensional dynamical system as the reservoir, which transforms and stores the input as a higher-dimensional state for temporal and nontemporal data processing. We demonstrate here a method to predict temporal and nontemporal tasks by constructing virtual nodes as constituting a reser...

Implementation of logic gates has been investigated in nonlinear dynamical systems from various perspectives over the years. Specifically, logic gates have been implemented in both single nonlinear systems and coupled nonlinear oscillators. The majority of the works in the literature have been done on the evolution of single oscillators into OR/AND...

We report that conventional saturable periodic structures, in sharp contrast to the conventional systems with different nonlinearities which exhibit the typical S- shaped optical bi- and multi-stable states, reveal some unusual and unique nonlinear dynamics. These include the onset of ramp-like optical bistability (OB) and optical multistability (O...

In this article, we systematically explain how to apply the analytical technique called the invariant subspace method to find various types of analytical solutions for a coupled nonlinear time-fractional system of partial differential equations with time delays. Also, the present work explicitly studies a systematic way to obtain various kinds of f...

The influence of Gilbert damping on the propagation of electromagnetic waves (EMWs) in an anisotropic ferromagnetic medium is investigated theoretically. The interaction of the magnetic field component of the electromagnetic wave with the magnetization of a ferromagnetic medium has been studied by solving the associated Maxwell's equations coupled...

This article develops how to generalize the invariant subspace method for deriving the analytical solutions of the multi-component (N+1)-dimensional coupled nonlinear time-fractional PDEs (NTFPDEs) in the sense of Caputo fractional-order derivative for the first time. Specifically, we describe how to systematically find different invariant product...

In this article, we develop a systematic approach of the invariant subspace method combined with variable transformation to find the generalized separable exact solutions of the nonlinear two-component system of time-fractional PDEs (TFPDEs) in (2+1)-dimensions for the first time. Also, we explicitly explain how to construct various kinds of finite...

In this article, we systematically explain how to apply the analytical technique called the invariant subspace method to find various types of analytical solutions for a coupled nonlinear time-fractional system of partial differential equations with time delays. Also, the present work explicitly studies a systematic way to obtain various kinds of f...

Liénard-type nonlinear oscillators with linear and nonlinear damping terms exhibit diverse dynamical behavior in both the classical and quantum regimes. In this paper, we consider examples of various one-dimensional Liénard type-I and type-II oscillators. The associated Euler–Lagrange equations are divided into groups based on the characteristics o...

In this paper, we investigate the role of four-wave mixing effect on the structure of nondegenerate vector solitons and their collision dynamics. For this purpose, we consider the generalized coupled nonlinear Schrödinger (GCNLS) system which describes the evolution and nonlinear interaction of the two optical modes. The fundamental as well as high...

We consider a modified damped version of Hénon–Heiles system and investigate its integrability. By extending the Painlevé analysis of ordinary differential equations we find that the modified Hénon–Heiles system possesses the Painlevé property for three distinct parametric restrictions. For each of the identified cases, we construct two independent...

We report the occurrence of vibrational resonance and the underlying mechanism in a simple piecewise linear electronic circuit, namely, the Murali–Lakshmanan–Chua circuit, driven by an additional biharmonic signal with widely different frequencies. When the amplitude of the high-frequency force is tuned, the resultant vibrational resonance is used...

We report the occurrence of vibrational resonance (VR) and the underlying mechanism in a simple piecewise linear electronic circuit, namely the Murali-Lakshmanan-Chua (MLC) circuit, driven by an additional biharmonic signal with widely different frequency. When the amplitude of the high-frequency force is tuned, the resultant vibrational resonance...

We investigate the influence of field-like torque and the direction of the external magnetic field on a one-dimensional array of serially connected spin-torque nano oscillators (STNOs), having free layers with perpendicular anisotropy, to achieve complete synchronization between them by analyzing the associated Landau–Lifshitz–Gilbert–Slonczewski e...

We investigate the influence of field-like torque and the direction of the external magnetic field on a one-dimensional array of serially connected spin-torque nano oscillators (STNOs), having free layers with perpendicular anisotropy, to achieve complete synchronization between them by analyzing the associated Landau-Lifshitz-Gilbert-Slonczewski e...

The universal optical logic gates, namely, nand and nor gates, have been theoretically simulated by employing the energy sharing collision of bright optical solitons in the Manakov system, governing pulse propagation in a highly birefringent fiber. Further, we also realize the two-input optical logic gates, such as and, or, xor, xnor, for completen...

We revisit the Fermi-Pasta-Ulam-Tsingou lattice (FPUT) with quadratic and cubic nonlinear interactions in the continuous limit by deducing the Gardner equation. Through the Hirota bilinear method, multi-soliton solutions are obtained for the Gardner equation. Based on these solutions, we show the excitation of an interesting class of table-top soli...

We revisit the Fermi-Pasta-Ulam-Tsingou lattice (FPUT) with quadratic and cubic nonlinear interactions in the continuous limit by deducing the Gard-ner equation. Through the Hirota bilinear method, the fundamental as well as multi-soliton, particularly two-, three-and four-soliton, solutions are obtained for the Gardner equation. Based on these mul...

We examine how shear influences the emergence of symmetry-breaking dynamical states in a globally coupled Stuart–Landau (SL) oscillator system with combined attractive and repulsive interactions. In the absence of the shear parameter, the system exhibits synchronization, nontrivial oscillation death states and oscillation death states. However, wit...

We examine how shear influences the emergence of symmetry-breaking dynamical states in a globally coupled Stuart-Landau (SL) oscillator system with combined attractive and repulsive interactions. In the absence of the shear parameter, the system exhibits synchronization, nontrivial oscillation death states, and oscillation death states. However, wi...

Exchange coupling in an interfacial context is crucial for spin-torque nano-oscillator (STNO) that consists of a nonmagnetic spacer which is alloyed with a ferromagnetic material. Currently, investigations on the dynamics of the free-layer magnetization and frequency enhancement in the STNO with bilinear coupling are still being actively pursued. I...

Exchange coupling in an interfacial context is crucial for spin-torque nano oscillator (STNO) that consists of a non-magnetic spacer which is alloyed with a ferromagnetic material. Currently, investigations on the dynamics of the free layer magnetization and frequency enhancement in the STNO with bilinear coupling are still being actively pursued....

Exchange coupling in an interfacial context is crucial for spin-torque nano-oscillator (STNO) that consists of a nonmagnetic spacer which is alloyed with a ferromagnetic material. Currently, investigations on the dynamics of the free-layer magnetization and frequency enhancement in the STNO with bilinear coupling are still being actively pursued. I...

In the present work, we examine the role of nonlinearity in vibrational resonance of a forced and damped form of a velocity-dependent potential system. Many studies have focused on studying the vibrational resonance in different potentials, like bistable potential, asymmetrically deformed potential, and rough potential. In this connection, velocity...

In this paper, we investigate the role of four-wave mixing effect on the structure of nondegenerate vector solitons and their collision dynamics. For this purpose, we consider the generalized coupled nonlinear Schr\"odinger (GCNLS) system which describes the evolution and nonlinear interaction of the two optical modes. The fundamental as well as hi...

We investigate the nontrivial characteristics of modulational instability (MI) in a system of Bragg gratings with saturable nonlinearity. We also introduce an equal amount of gain and loss into the existing system, which gives rise to an additional degree of freedom, due to the concept of PT symmetry. We obtain the nonlinear dispersion relation of...

In the present work, we examine the role of nonlinearity in vibrational resonance (VR) of a forced and damped form of a velocity-dependent potential system. Many studies have focused on studying the vibrational resonance in different potentials, like bistable potential, asymmetrically deformed potential, and rough potential. In this connection, vel...

This paper systematically explains how to apply the invariant subspace method using variable transformation for finding the exact solutions of the (k+1)-dimensional nonlinear time-fractional PDEs in detail. More precisely, we have shown how to transform the given (k+1)-dimensional nonlinear time-fractional PDEs into (1+1)-dimensional nonlinear time...

In this paper, we generalize the theory of the invariant subspace method to (m + 1)-dimensional non-linear time-fractional partial differential equations for the first time. More specifically, the applicability and efficacy of the method have been systematically investigated through the (3 + 1)-dimensional generalized non-linear time-fractional dif...

We investigate the nontrivial characteristics of modulational instability (MI) in a system of Bragg gratings with saturable nonlinearity. We also introduce an equal amount of gain and loss into the existing system which gives rise to an additional degree of freedom, thanks to the concept of $\cal PT$- symmetry. We obtain the nonlinear dispersion re...

In the background of homogeneous Kantowski–Sachs (KS) space-time geometry, the anisotropic Skyrme fluid with zero heat flux has been considered in the framework of Einstein gravity. The homothetic group of the physical space is determined and the generation of the Noether point symmetry is discussed. Further, the Wheeler–DeWitt equation is construc...

This paper systematically explains how to apply the invariant subspace method using variable transformation for finding the exact solutions of the (k+1)-dimensional nonlinear time-fractional PDEs in detail. More precisely, we have shown how to transform the given (k+1)-dimensional nonlinear time-fractional PDEs into (1+1)-dimensional nonlinear time...

In this work, we present a new approach to find non-local symmetries and contact symmetries from the admitted Lie point symmetries of the considered system of nonlinear differential equations. By introducing a new function in both the numerator and denominator in the relation which relates the \(\lambda \)-symmetry function and the Lie point symmet...

Identifying higher dimensional nonlinear ordinary differential equations (ODEs) possessing a Lagrangian structure is a challenging problem. In this paper, we obtain a set of constraints in the form of Helmholtz conditions which are to be satisfied by a system of two coupled second-order ODEs in order to posses a Lagrangian structure. We propose a s...

SolitonA class of nonlinear dispersive wave equations in (1+1) dimensions having a delicate balance between dispersion and nonlinearity admit localized solitary waves which under interaction retain their shapes and speeds asymptotically. Such waves are called solitons because of their particle like elastic collision property. The systems include Ko...

The dynamics of the magnetization of the free layer in a spin-torque nano oscillator (STNO) influenced by a noncollinear alignment between the magnetizations of the free and pinned layers due to an interlayer exchange coupling has been investigated theoretically. The orientations of the magnetization of the free layer with that of the pinned layer...

COVID-19 will be a continuous threat to human population despite having a few vaccines at hand until we reach the endemic state through natural herd immunity and total immunization through universal vaccination. However, the vaccine acts as a practical tool for reducing the massive public health problem and the emerging economic consequences that t...

We analyze the dynamics of vortex solitons and the formation of soliton clusters with controllable attributes in a nonlinear metamaterial waveguide. We consider the nonlinear Schrödinger equation with cubic and quintic nonlinearities as the propagation model and identify stable as well as filamentation regions with different combinations of nonline...

Many natural and man-made systems require suitable feedback to function properly. In this study, we aim to investigate the impact of additional complex conjugate feedback on globally coupled Stuart-Landau oscillators. We find that this additional feedback results in the onset of symmetry breaking clusters and out-of-phase clusters. Interestingly, w...

Many natural and man-made systems require suitable feedback to function properly. In this study, we aim to investigate the impact of additional complex conjugate feedback on globally coupled Stuart-Landau oscillators. We find that this additional feedback results in the onset of symmetry breaking clusters and out-of-phase clusters. Interestingly, w...

COVID-19 will be a continuous threat to human population despite having a few vaccines at hand until we reach the endemic state through natural herd immunity and total immunization through universal vaccination. However, the vaccine acts as a practical tool for reducing the massive public health problem and the emerging economic consequences that t...

The dynamics of the magnetization of the free layer in a spin-torque nano oscillator (STNO) influenced by a noncollinear alignment between the magnetizations of the free and pinned layers due to an interlayer exchange coupling has been investigated theoretically. The orientations of the magnetization of the free layer with that of the pinned layer...

The dynamics of the magnetization of the free layer in a spin-torque nano oscillator (STNO) influenced by a noncollinear alignment between the magnetizations of the free and pinned layers due to an interlayer exchange coupling has been investigated theoretically. The orientations of the magnetization of the free layer with that of the pinned layer...

In this paper, we discuss the quantum dynamics of a nonlinear system that admits temporally localized solutions at the classical level. We consider a general ordered position-dependent mass Hamiltonian in which the ordering parameters of the mass term are treated as arbitrary. The mass function here is singular at the origin. We observe that the qu...

In this paper, we discuss the quantum dynamics of a nonlinear system that admits temporally localized solutions at the classical level. We consider a general ordered position-dependent mass Hamiltonian in which the ordering parameters of the mass term are treated as arbitrary. The mass function here is singular at the origin. We observe that the qu...

In the context of parity–time ( ${\cal P}{\cal T}$ )-symmetric fiber Bragg gratings, tailoring the nonlinear profile along the propagation coordinate serves as a new direction for realizing low-power all-optical switches. The scheme is fruitful only when the nonlinearity profile will be either linearly decreasing or increasing form. If the rate of...

In this paper, a generalized long-wave shortwave resonance interaction system, which describes the nonlinear interaction between a shortwave and a long-wave in fluid dynamics, plasma physics and nonlinear optics, is considered. Using the Hirota bilinear method, the general N-bright and N-dark soliton solutions are deduced and their Gram determinant...

In the context of $\mathcal{PT}$-symmetric fiber Bragg gratings, tailoring the nonlinear profile along the propagation coordinate serves to be a new direction for realizing low-power all-optical switches. The scheme is fruitful only when the nonlinearity profile will be either linearly decreasing or increasing form. If the rate of variation of the...

In this work, we present a method of generating a class of nonlinear ordinary differential equations (ODEs), representing the dynamics of appropriate nonlinear oscillators, that have the characteristics of either amplitude independent frequency of oscillations or amplitude dependent frequency of oscillations from the integrals of the simple harmoni...

In this paper, we study the dynamics of an interesting class of vector solitons in the long-wave-short-wave resonance interaction (LSRI) system. The model that we consider here describes the nonlinear interaction of long wave and two short waves and it generically appears in several physical settings. To derive this class of nondegenerate vector so...

In this paper, we consider a generalized long-wave short-wave resonance interaction system, which describes the nonlinear interaction between a short-wave (SW) and a long-wave (LW). The general N -bright and N -dark soliton solutions are derived using the Hirota bilinear method and they are written in a compact way using Gram determinants. Very int...

We explore the aging transition in a network of globally coupled Stuart-Landau oscillators under a discrete time-dependent coupling. In this coupling, the connections among the oscillators are turned ON and OFF in a systematic manner, having either a symmetric or an asymmetric time interval. We discover that depending upon the time period and duty...

Frequency plays a crucial role in exhibiting various collective dynamics in the coexisting corotating and counter-rotating systems. To illustrate the impact of counter-rotating frequencies, we consider a network of nonidentical and globally coupled Stuart-Landau oscillators with additional perturbation. Primarily, we investigate the dynamical trans...

We investigate the phase diagram of the Sakaguchi-Kuramoto model with a higher-order interaction along with the traditional pairwise interaction. We also introduce asymmetry parameters in both the interaction terms and investigate the collective dynamics and their transitions in the phase diagrams under both unimodal and bimodal frequency distribut...

In this paper we carry out a theoretical investigation on the propagation of spatiotemporal solitons (light bullets) in the nonlinear metamaterial waveguides. Our theoretical study is based on the formulation of Lagrangian variational analysis with a suitable ansatz, followed by a split-step Fourier method in confirming the previous outcomes numeri...

In this work, we explore the different measures of quantum correlations and quantum teleportation in the Heisenberg XY model for two different cases, namely without PT-symmetric operation and with PT-symmetric operation. Initially, we inspect the quantum correlation measures of thermally entangled states without PT-symmetric operation. Among the di...

We investigate the phase diagram of the Sakaguchi-Kuramoto model with a higher order interaction along with the traditional pairwise interaction. We also introduce asymmetry parameters in both the interaction terms and investigate the collective dynamics and their transitions in the phase diagrams under both unimodal and bimodal frequency distribut...

Frequency plays a crucial role in exhibiting various collective dynamics in the coexisting co- and counter-rotating (CR) systems. To illustrate the impact of CR frequencies, we consider a network of non-identical and globally coupled Stuart-Landau oscillators with additional perturbation. Primarily, we investigate the dynamical transitions in the a...

In this work, we present a method of generating a class of nonlinear ordinary differential equations (ODEs), representing the dynamics of appropriate nonlinear oscillators, that have the characteristics of either amplitude independent frequency of oscillations or amplitude-dependent frequency of oscillations from the integrals of the simple harmoni...

In this paper, we carry out a theoretical investigation on the propagation of spatio-temporal solitons (light bullets) in the nonlinear metamaterial waveguides. Our theoretical study is based on the formulation of Lagrangian variational analysis with a suitable ansatz followed by a split-step Fourier method in confirming the outcomes former numeric...

In this paper, we generalize the theory of the invariant subspace method to (m+1)-dimensional non-linear time-fractional partial differential equations for the first time. More specifically, the applicability and efficacy of the method have been systematically investigated through the (3+1)-dimensional generalized non-linear time-fractional diffusi...

We explore the aging transition in a network of globally coupled Stuart-Landau oscillators under a discrete time-dependent coupling. In this coupling, the connections among the oscillators are turned ON and OFF in a systematic manner, having either a symmetric or an asymmetric time interval. We discover that depending upon the time period and duty...

The central theme of this paper is the analysis of the usefulness of introducing four-wave mixing or modulation of Kerr nonlinearity in a nonuniform grating structure with gain and loss. To do so, we propose an inhomogeneous system in which the nonlinearity of the parity-time ( ${\cal P}{\cal T}$ )-symmetric grating is modulated. It is proven that...

India was under a grave threat from the second wave of the COVID-19 pandemic particularly in the beginning of May 2021. The situation appeared rather gloomy as the number of infected individuals/active cases had increased alarmingly during the months of May and June 2021 compared to the first wave peak. Indian government/state governments have been...

We study the dynamics of a spin torque nano oscillator that consists of parallelly magnetized free and pinned layers by numerically solving the associated Landau-Lifshitz-Gilbert-Slonczewski equation in the presence of a field-like torque. We observe that an in-plane magnetic field which is applied for a short interval of time ($<$1ns) triggers the...

We study the dynamics of a spin torque nano oscillator that consists of parallelly magnetized free and pinned layers by numerically solving the associated Landau–Lifshitz–Gilbert–Slonczewski equation in the presence of a field-like torque. We observe that an in-plane magnetic field which is applied for a short interval of time (<1 ns) triggers the...

The central theme of this article is the analysis of the usefulness of introducing four-wave mixing or modulation of Kerr nonlinearity in a nonuniform grating structure with gain and loss. To do so, we propose an inhomogeneous system in which the nonlinearity of the $\mathcal{PT}$-symmetric grating is modulated. It is proven that the proposed schem...

We study the dynamics of a spin torque nano oscillator that consists of parallelly magnetized free and pinned layers by numerically solving the associated Landau-Lifshitz-Gilbert-Slonczewski equation in the presence of a field-like torque. We observe that an in-plane magnetic field which is applied for a short interval of time ($<$1ns) triggers the...

We investigate the existence of collective dynamical states in nonlocally coupled Stuart–Landau oscillators with symmetry breaking included in the coupling term. We find that the radius of nonlocal interaction and nonisochronicity parameter play important roles in identifying the swing of synchronized states through amplitude chimera states. Collec...

We perform a detailed analysis of the behaviour of a non-autonomous prey–predator model where age-based growth with age discriminatory harvesting in prey and predator’s reliance upon alternative food in the absence of that particular prey are considered. We begin by deriving certain sufficient conditions for permanence and positive invariance and t...

It is well-known that third-order dispersion (TOD) plays no role in the onset of modulation instability in the nonlinear Schrödinger (NLS) family of systems. Here we demonstrate that the phenomenology and characteristics of modulational instability can be dramatically altered by the TOD in two different nonlinear systems, namely Wannier exciton mas...

We consider two one dimensional nonlinear oscillators, namely (i) Higgs oscillator and (ii) a k-dependent nonpolynomial rational potential, where k is the constant curvature of a Riemannian manifold. Both the systems are of position dependent mass (PDM) form, (x)=\frac{1}{{(1+k{x}^{2})}^{2}}, belonging to the quadratic Li \acute{e} nard type nonlin...

In this paper, we point out that the two-component long wave-short wave resonance interaction (LSRI) system can admit a more general form of nondegenerate fundamental soliton solution than the one that is known in the literature and consequently its higher-order generalized soliton solutions as well. To derive this class of soliton solutions throug...

We study the existence of localized one-spin excitation in the Heisenberg one-dimensional ferromagnetic spin chain in the presence of perpendicular and parallel external magnetic fields and current with spin-transfer torque and field-like torque. The Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation is exactly solved for the one spin excitation i...

In the present article, we demonstrate the emergence and existence of the spiral wave chimera-like transient pattern in coupled ecological systems, composed of prey-predator patches, where the patches are connected in a three-dimensional medium through local diffusion. We explore the transition scenarios among the several collective dynamical behav...

We study the existence of localized one-spin excitation in the Heisenberg one-dimensional ferromagnetic spin chain in the presence of perpendicular and parallel external magnetic fields and current with spin-transfer torque and field-like torque. The Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation is exactly solved for the one spin excitation i...

We study the existence of localized one-spin excitation in the Heisenberg one-dimensional ferromagnetic spin chain in the presence of perpendicular and parallel external magnetic fields and current with spin-transfer torque and field-like torque. The Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation is exactly solved for the one spin excitation i...

In the present article, we demonstrate the emergence and existence of the spiral wave chimera-like transient pattern in coupled ecological systems, composed of prey–predator patches, where the patches are connected in a three-dimensional medium through local diffusion. We explore the transition scenarios among several collective dynamical behaviors...

Nonlinear dynamics of an optical pulse or a beam continue to be one of the active areas of research in the field of optical solitons. Especially, in multi-mode fibers or fiber arrays and photorefractive materials, the vector solitons display rich nonlinear phenomena. Due to their fascinating and intriguing novel properties, the theory of optical ve...

Nonlinear dynamics of an optical pulse or a beam continue to be one of the active areas of research in the field of optical solitons. Especially, in multi-mode fibers or fiber arrays and photorefractive materials, the vector solitons display rich nonlinear phenomena. Due to their fascinating and intriguing novel properties, the theory of optical ve...

In this paper, we investigate the quantum dynamics of underlying two one-dimensional quadratic Liénard type nonlinear oscillators which are classified under the category of maximal (eight parameter) Lie point symmetry group [Tiwari A K, Pandey S N, Senthilvelan M and Lakshmanan M 2013 J. Math. Phys. 54, 053 506]. Classically, both the systems were...