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Branched Hamiltonians for a quadratic type Li\'enard oscillator
Abstract
We point out that when a quadratic type Li\'enard equation is suitably interpreted shows branching due to the double valuedness of the governing Hamiltonian. Under certain approximation of the guiding coupling constant we derive its quantum counterpart that is guided by a momentum-dependent mass function.
... In the context of nonlinear models, certain Liénard-class systems present an intriguing feature of the Hamiltonian in which the roles of the position and momentum variables are exchanged with the emergence of the notion of a momentum-dependent mass [23,[26][27][28][29][30][31]. Naturally, the presence of the damping as is the case for Liénard systems poses a problem whenever one tries to contemplate a quantization of the model. ...
... which, like L(v), is also a multi-valued function (with cusps) in the conjugate momentum p, since each given p corresponds to one or three values of v, as shown in (29). For systems with a non-convex Lagrangian as sampled by (28), the routine construction of the corresponding Hamiltonian in the conjugate momentum variable is not unique. ...
... In electrodynamics, the field vectors ⃗ E and ⃗ B can be determined given such a potential function when the trajectories of a charged particle's motion are specified. In the present context, we proceed to set up an extended scheme where the Lagrangian depends on a velocity-dependent potential V (x, v) in the manner as given by [22,29]: ...
Time and again, non-conventional forms of Lagrangians with non-quadratic velocity dependence have received attention in the literature. For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics including specifically the types of the Liénard class; for another, very often, the problem of their quantization opens up multiple branches of the corresponding Hamiltonians, ending up with the presence of singularities in the associated eigenfunctions. In this article, we furnish a brief review of the classical theory of such Lagrangians and the associated branched Hamiltonians, starting with the example of Liénard-type systems. We then take up other cases where the Lagrangians depend on velocity with powers greater than two while still having a tractable mathematical structure, while also describing the associated branched Hamiltonians for such systems. For various examples, we emphasize the emergence of the notion of momentum-dependent mass in the theory of branched Hamiltonians.
... In the context of nonlinear models, the Liénard system presents an intriguing feature of the Hamiltonian in which the roles of the position and momentum variables often get exchanged [23,25]. Thus, Liénard systems are of potential importance in optics [26] as well as in non-Hermitian quantum mechanics [23,27]. Naturally, the presence of the damping as is the case for Liénard systems poses to be a problem whenever one tries to contemplate a quantization of the model. ...
... In electrodynamics, the field vectors ⃗ E and ⃗ B can be determined given such a potential function when the trajectories of a charged particle's motion are specified. In the present context, we proceed to set up an extended scheme where the Lagrangian depends upon a velocity-dependent potential V(x, v) in the manner as given by [19,27] ...
Time and again, non-conventional forms of Lagrangians have found attention in the literature. For one thing, such Lagrangians have deep connections with several aspects of nonlinear dynamics including specifically the types of the Liénard class; for another, very often the problem of their quantization opens up multiple branches of the corresponding Hamiltonians, ending up with the presence of singularities in the associated eigenfunctions. In this article, we furnish a brief review of the classical theory of such Lagrangians and the associated branched Hamiltonians, starting with the example of Liénard-type systems. We then take up other cases where the Lagrangians depend upon the velocity with powers greater than two while still having a tractable mathematical structure, while also describing the associated branched Hamiltonians for such systems. For various examples, we emphasize upon the emergence of the notion of momentum-dependent mass in the theory of branched Hamiltonians.
... To demonstrate our results, we consider an ML type oscillator [49,50] with non-polynomial position-dependent mass terms, which has drawn considerable attention for a few decades from different perspectives. In particular, the subject of position dependent mass has received considerable importance in recent literature due to its appearence in classical and quantum mechanical systems [51][52][53][54][55]. Before introducing the model we study, we explain the basic model of the ML oscillator, which is associated with a mechanical oscillator model with position dependent mass [56]. ...
We explore the dynamics of a damped and driven Mathews–Lakshmanan oscillator type model with position-dependent mass term and report two distinct bifurcation routes to the advent of sudden, intermittent large-amplitude chaotic oscillations in the system. We characterize these infrequent and recurrent large oscillations as extreme events (EE) when they are significantly greater than the pre-defined threshold height. In the first bifurcation route, the system exhibits a bifurcation from quasiperiodic (QP) attractor to chaotic attractor via strange non-chaotic (SNA) attractor as a function of damping parameter. In the second route, the chaotic attractor in the form of EE has emerged directly from the QP attractor. Hence, to the best of our knowledge, this is the first study to report the birth of EE from these two distinct bifurcation routes. We also discuss that EE are emerged due to the sudden expansion of the chaotic attractor via interior crisis in the system. Regions of different dynamical states are distinguished using the Lyapunov exponent spectrum. Further, SNA and QP dynamics are determined using the singular spectrum analysis and 0–1 test. The region of EE is characterized using the threshold height.
... The subject of PDM [3] has evinced a lot of interest in the literature and has developed into a separate branch of research over the past few decades [4][5][6][7][8][9][10]. In this connection, the relevance of PDM has been noteworthy in such areas as that of compositionally graded crystals [11], quantum dots [12], liquid crystals [13] etc. and some classical problems possessing quantum analogs, for example, in branched Hamiltonian systems [14]. ...
A classical double oscillator model, that includes in certain parameter limits, the standard harmonic oscillator and the inverse oscillator, is interpreted as a dynamical system. We study its essential features and make a qualitative analysis of orbits around the equilibrium points, period-doubling bifurcation, time series curves, surfaces of section and Poincare maps. An interesting outcome of our findings is the emergence of chaotic behavior when the system is confronted with a periodic force term like fcos{\omega}t.
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