Analysis of Generalized Lienard Oscillators

1.1 INTRODUCTION

Dynamics is the analysis of transition, and a dynamic system is just a recipe for telling how a system interacts and changes over time (state variables). It is possible to consider a dynamic system as an entity of some kind, the state of which evolves in time according to some dynamic law. The theory of dynamic systems provides a wide variety of analytical, geometrical, topological, and computational techniques for analyzing differential equations and iterated mappings. It is possible to classify any system that changes over time as a dynamical system. More specifically, the system evolves over time in such a way that dynamical systems are called the states of the system at time t depend on the states of the system at earlier times. In other words, a dynamic system consists of a number of potential states along with a law that, in terms of past states, defines the present states ([1]). Variable dynamical systems may be categorized as a discrete dynamical system or as a continuous dynamical system according to the character of the time. The time variable in discrete dynamical system is discrete i.e, t or N and the evolution is determined by a difference equation. For example xt+1 = f (xt,xt-1, ..,xt-n) ; is the form of a discrete dynamical system. A continuous dynamical system is represented by a differential equation possessing a unique solution x(t,t0) = x(t) satisfying the condition x(t0) = x0 . A flow is a deterministic continuous system on a manifold which is continuously differentiable with respect to time. In general a complete flow t(x) is a one parameter, differentiable mapping , such that (a) 0xx and (b) for all t and , where M is the manifold of phase space for a dynamical system and the composition symbol o means .

xz, z = xy – bz, where , r, b >0 represent a continuous dynamical system.

There is an enormous obsession with uncertainty today. The science of surprises, the non-linear and the unexpected, is chaos. This instructs us to expect the unexpected. While most traditional science deals with supposedly predictable phenomena such as gravity, electricity or chemical impossible to predict or regulate. If it has the following properties, a dynamic system is said to be chaotic [1].: 1) Initial situations must be sensitive to it. Sensitivity to initial conditions implies that other points with substantially different future paths or trajectories are arbitrarily closely approximated by each point in a chaotic system. An arbitrary minor change or disruption of the current trajectory may then lead to dramatically different future behaviour. A consequence of sensitivity to initial conditions is that the system is no longer predictable if we start with only a finite amount of knowledge about the system after a certain time. In the case of weather that is normally only predictable about a week ahead, this is most familiar. 2) It must be transitive topologically. Topological transitivity or topological mixing means that the system evolves over time such that its phase space ultimately overlaps with any other given region in any given region or open set. Such a complex structure can therefore not be broken down into two disjoint sets of non-empty interiors that do not interact under the transition. 3) Its periodic orbit must be dense. Dense periodic orbit means that arbitrary close to every point there is a periodic orbit in the phase space.

Nonlinear autonomous differential equations of second order have proved to be of much interest in the investigation of dynamical systems and nonlinear analysis [2-9]. which is a to quadratic Lienard class. where an overdot indicates a derivative with respect to the time variable t and r(x) and s(x) are two continuously differentiable functions of the spatial coordinate x. The following specific forms of r and s which are odd functions of x, namely where α and λ are nonzero real numbers, lead to a nonlinear equation defined by The Lagrangian relevant to (1.2.3) was studied by Mathews and Lakshmanan (ML) [8] long time ago in the search of a one-dimensional analogue of some quantum field theoretic model. One can observe that (2.4) speaks of a A-dependent deformation of the standard harmonic oscillator Lagrangian. Carmena et al [10] also pointed out that the kinetic term in this Lagrangian is invariant under the tangent lift of the vector field . The corresponding Hamiltonian represents a position-dependent effective mass system guided by the mass function of the specific type . In the literature dynamics of several types of nonlinear systems have been found to be influenced by a position-dependent effective mass [11-13]. From a physical point of view problems of position-dependent effective mass have relevance in describing the flow of electrons in problems of compositionally graded crystal, quantum dots and liquid crystals [14-17]. As is well known the nonlinear dynamics described by (1.2.3) admits, in particular, a periodic solution for x(t) given by the simple harmonic form but with the restriction that the frequency  is related to the amplitude A by the constraint . The amplitude thus depends on the frequency. From the form of the Lagrangian (1.2.4) it is easy to identify the corresponding potential V (x) as given by Recently Quesne [18] extended the above potential to two different types of generalizations by introducing a two-parameter deformation of the harmonic oscillator potential by bringing in an additional phenomenological parameter  in the manner:

while keeping the kinetic part of (1.2.4) unchanged. The respective Euler-Lagrange equations that follow from (1.2.7) and (1.2.8) read Let us emphasize that the functional form corresponding to s(x) in the above equations is not odd but of a mixed type consisting of a term that is odd and a term that is even. With an extra parameter ft at hand, quite expectedly, the potentials V1 and V11 allow dealing with richer behavior patterns of the solutions of the corresponding Euler-Lagrange equations than the ones provided by (4) for the ML case. One of the guiding factors in this regard is the sign of the deformation parameter  that corresponds to different asymptotic behavior of the two potentials V1 and V11 and the locations of the minimum for the latter. The first integral of the Euler-Lagrange equation when suitably confronted with the integration constant and the underlying discriminant also give crucial restrictions on the domains of the energy of the system for a physically viable solution. In this paper, we interpret the generalized nonlinear oscillators that are guided by V1 and V11 as examples of dynamical systems and perform a qualitative analysis to determine some interesting local properties for them. We also take up the issue of chaos in the presence of a periodic force termcosft for both the systems by fine-tuning some of the coupling parameters.

Let us rewrite the Euler-Lagrange equation in the presence of Vj as the following system of coupled equations The resulting fixed points are readily identified to be located at the points (x*1,0) and (x*2, 0) where

The positivity of the discriminant leads to the restriction . Evaluating the Jacobian matrix at the fixed points we obtain Where For  > 0, the respective eigenvalues of J at x1* and x2* are where implying . We therefore conclude that the point is a center while the other one, namely , is a saddle point. The phase diagram of the system (2.11) is plotted in Figure 2.1a taking the parameter values . In case of  < 0 the two eigenvalues of the Jacobian matrix (2.13) at x1* and x2*) are given by Where Here too the equilibrium points and correspond to centers since the eigenvalues of the Jacobian matrix are purely imaginary conjugate pairs for  < 0. Figure 2.1b represents the phase portrait of the system (2.11) for . These Figure 1.2a and Figure 1.2b are consistent with the linear analysis results.

The dynamical system for the potential V11 (x) is described by the following set of equations The two equilibrium points correspond to (x1*, 0) and (x2*, 0) where subject to the positivity condition. At the equilibrium points the Jacobian matrix takes the form Where A21 is given by and xc stands for xc = x1,2* . The eigenvalues of J represent a pair of degenerate conjugate complex quantities namely, While the linear stability analysis indicates the nonhyperbolic nature of the fixed points, as is well known such tests are not always in conformity with the nonlinear analysis. Indeed, the numerical simulation of the nonlinear system of 2.17, as shown in Figure 1.2, confirms that the linear stability results correctly predict the behavior of one of the equilibrium points namely x1* but it fails in the case of the other equilibrium point, x2*.

It is of considerable interest to study the dynamics of systems (1.4.1) and (1.4.2), under the influence of the external periodic forcing in the presence of additional damping, so that the respective equation of motion become The above nonautonomous systems can be rewritten as the following three dimensional autonomous nonlinear dynamical systems respectively. And We have done numerical simulations of the system (1.4.3) only. The results of simulation of the system (1.4.4) are qualitatively similar and hence not discussed.

We start with the dynamics of the ML-model in the absence of dissipation. Thus we set  = 0 and consider a set of sample parameter values  = –0.5 ,  = 2.0 ,  = 0.0 , w = 1.0 , f = 5.0 to plot the phase portrait in the xy-plane. It reveals the quasi-periodic nature of the system whose character survives for a wide range of inputs for the external periodic forcing term.

The phase diagram in the system's xy plane (2.24) was plotted in Figure 1.3 for the parameter  = –0.5 ,  = 2.0 ,  = 0.0 ,  = 1.0 , f = 5.0 corresponding to the test values of f = 5.0 as given by  = 0.001 and  = 0.1. These  values represent small and moderate deviations from the ML-model. The case of  = 0.001 is shown in Figure 1.3a, while the first return map of Poincare is shown in Figure 1.3b. We note that on 3 smooth closed curves, the first return map data confirms the presence of the system's quasi-periodic behaviour. In Figure 1.3c, the phase diagram for  = 0.001 keeping the other parameters fixed is shown. The first return map, as shown in Figure 1.3d, shows a dramatic shift, however. In the above figure, the irregular scattering of points points to the presence of disorder in the system. Therefore, as shown by the transformation from quasi-periodicity to chaos, we note the sensitivity of the system (1.4.4) to the choice of the x-values as the value of x is modified from β = 0.001 by two orders of magnitude.

Next, the phase diagrams and those for Poincar'e return maps for various values of forcing amplitude f are discussed for the system (2.24) for the data set λ = −0.5, α = 2.0 , β = 0.1, γ = 0.0, ω = 1.0. In Figure 1.4, these are summarised. We have plotted the phase portrait for f = 0.0 in Figure 1.4a and the corresponding first return map for Poincare is shown in Figure 1.4b. In the absence of any external forcing, the presence of just one point in Figure 1.4b proves the existence of a periodic orbit. The phase diagram of the system is presented in Figure 1.4c for f = 3.0 and the corresponding first return map of Poincare is shown in Figure 1.4d. The return map data is located on smooth closed curves and thus supports the system's quasi-periodic behaviour for f = 3.0.0. In Figure 1.5e, we have plotted the phase portrait for f = 5.0 and the corresponding return map information is plotted in Figure 2.4f. The abnormal set of points in the entire plane ensures the presence of chaos for f = 5.0. in the method. As in the previous case, therefore, the increase of the forcing amplitude also generates chaotic motion via the quasi-periodic path.

Now, we turn to the inclusion in the model of the dissipation effects (2.24). The following fixed sample values of  = –0.5,  = 2.0,  = 0.1,  = 1.0 and f = 5.0 are retained towards this end and the dissipation parameter varies. An assorted set of values ranging from very small to fairly large values is considered for 7. In Figure 1.5a, we represent the phase portrait of 7 = 0.002 in the xy plane and the corresponding Poincare return map in Figure 1.5b. We find a collection of randomly distributed points that inform us about the system's chaotic character (2.24). Next to  = 0.02, from the presence of quasi-periodic oscillations are noted. The phase portrait and time evolution of the variable y is shown in figures 1.5e and 1.5f for an order of magnitude higher value of  = 0.1. The time evolution of the corresponding phase portrait of Figure 1.5f ensures that a periodic behaviour is present for = 0.1. The bifurcation diagram of the variable y is given in Figure 1.6 in relation to parameter 7. It is evident that the system needs to undergo a transformation from chaos to quasi periodicity with the increase of dissipation, and then settles to a periodic behaviour.

In this chapter we studied the dynamical behavior of two quadratic schemes of generalized oscillators recently proposed by Quesne. We performed a local analysis of the governing potentials to demonstrate that while the first potential admits a typically center-like equilibrium point for both signs of the coupling analysis, only a center for both the signs of A. We have extended Quesne‘s scheme to include the effects of a linear dissipative term and shown how inclusion of an external periodic force term changes the qualitative behavior of the underlying systems drastically leading to the onset of chaos.

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Research Scholar, Department of Mathematics, Sri Satya Sai University of Technology & Medical Sciences, Sehore, M.P.