Nonlinear dynamics of damped and driven velocity-dependent systems

Abstract

In this paper, the nonlinear dynamics of certain damped and forced versions of velocity-dependent potential systems, namely, ͑i͒ the motion of a particle on a rotating parabola and ͑ii͒ a nonlinear harmonic oscillator, is considered. Various bifurcations such as symmetry breaking, period doubling, intermittency, crises, and antimonotonicity are reported. We also investigate the transition from two-frequency quasiperiodicity to chaotic behavior in a model for the quasiperiodically driven rotating parabola system. As the driving parameter is increased, the route to chaos takes place in four distinct stages. The first stage is a torus doubling bifurcation. The second stage is a merging of doubled torus. The third stage is a transition from the merged torus to a strange nonchaotic attractor. The final stage is a transition from the strange nonchaotic attractor to a geometrically similar chaotic attractor. ͓S1063-651X͑97͒00505-9͔PACS number͑s͒: 05.45.ϩb

Position-dependent mass Lagrangians: nonlocal transformations, Euler–Lagrange invariance and exact solvability

“…In connection with the Mathews-Lakshmanan nonlinear oscillators I and II, we observe that the PDM-function (26) subjected to move in the set of four potential force fields (28) (One at a time) admits/feels exactly similar dynamical effects as documented in the corresponding Euler-Lagrange equations of motion (19) and (25) and follows exactly similar trajectories. This tendency of similar dynamics, similar trajectories and similar total energies…”

“…For more details on this issue the reader may refer to [17,22]. Obviously, moreover, equation (2) is a quadratic Liénard-type differential equation (quadratic in terms ofẋ 2 in (2)) which serves as a very interesting model in both physics and mathematics (cf., e.g., the sample of references [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] and related references cited therein).…”

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with a simple nonlinear element

“…But dynamically, they do not show sensitive dependence on initial conditions as seen from negative Lyapunov exponents, that is, they are strange but nonchaotic. Following the pioneering work of Grebogi et al [4], SNAs have been extensively investigated numerically in dynamical systems, such as biological oscillators [5,6], driven Duffing type oscillators [7][8][9][10][11] and in certain maps, namely driven velocitydependent systems [12,13], two dimensional maps [14], quasiperiodically forced logistic map [15][16][17][18][19][20], one dimensional cubic map [21][22][23][24], Harper map [25], map representing driven damped superconducting quantum interference device [26][27][28] and SNAs in HH-neural oscillator [29]. In some physically relevant situations, the existence of SNAs have also been demonstrated experimentally such as in electronic circuits [30][31][32][33], in neon glow discharge experiment [34] and in quasiperiodically forced, buckled, magnetoelastic ribbon [35].…”

Experimental realization of strange nonchaotic attractors in a quasiperiodically forced electronic circuit

“…[1], several theoretical as well as experimental studies have been made pertaining to the existence and characterization of SNAs in different quasiperiodically driven nonlinear dynamical systems. In particular the SNAs have been reported to arise in many physically relevant situations such as the quasiperiodically forced pendulum [2], the quantum particles in quasiperiodic potentials [3], biological oscillators [4], the quasiperiodically driven Duffingtype oscillators [5,6,7,8], velocity dependent oscillators [9], electronic circuits [10,11,12] and in certain maps [13,14,15,16,17,18,19,20,21,22]. Also, these exotic attractors were confirmed by an experiment consisting of a quasiperiodically forced, buckled, magnetoelastic ribbon [23], in analog simulations of a multistable potential [24], and in a neon glow discharge experiment [25].…”

“…In this context, several routes have been identified in recent times and for a few of them typical mechanisms have been found for the creation of SNAs. The major routes by which the SNAs appear may be broadly classified as follows: torus doubling route to chaos via SNAs [22], gradual fractalization of torus [17], the appearance of SNAs via blowout bifurcation [6], the occurrence of SNAs through intermittent phenomenon [12,13,19,20,21,31], the formation of SNAs via homoclinic collision [27], remerging of torus doubling bifurcations and the birth of SNAs [9], the existence of SNAs in the transition from two-frequency to three-frequency quasiperiodicity [7], the transition from three-frequency quasiperiodicity to chaos via a SNA [4] and the transition to chaos via strange nonchaotic trajectories on the torus [32]. Different mechanisms have been identified for some of the above routes, which are summarised in Table I.…”