Cascades of subharmonic stationary states in strongly non-linear driven planar systems

Abstract

The dynamics of a one-degree of freedom oscillator with arbitrary polynomial non-linearity subjected to an external periodic excitation is studied. The sequences (cascades) of harmonic and subharmonic stationary solutions to the equation of motion are obtained by using the harmonic balance approximation adapted for arbitrary truncation numbers, powers of non-linearity, and orders of subharmonics. A scheme for investigating the stability of the harmonic balance stationary solutions of such a general form is developed on the basis of the Floquet theorem. Besides establishing the stable/unstable nature of a stationary solution, its stability analysis allows obtaining the regions of parameters, where symmetry-breaking and period-doubling bifurcations occur. Thus, for perioddoubling cascades, each unstable stationary solution is used as a base solution for finding a subsequent stationary state in a cascade. The procedure is repeated until this stationary state becomes stable provided that a stable solution can finally be achieved. The proposed technique is applied to calculate the sequences of subharmonic stationary states in driven hardening Duffing's oscillator. The existence of stable subharmonic motions found is confirmed by solving the differential equation of motion numerically by means of a time-difference method, with initial conditions being supplied by the harmonic balance approximation.

Signatures and characterization of dominating Kerr nonlinearity between two driven systems with application to a suspended magnetic beam

“…This is in contrast to optomechanical coupling that generates all possible harmonics, including the first sidebands. While the first sidebands do appear if the system undergoes a subharmonic bifurcation [54,55], we expect that it does not occur in the steady state of a moderately driven system. Hence, the absence of the first harmonic indicates that there is no optomechanical coupling in the system.…”

“…with k and l integers. We assume that the system does not undergo a subharmonic bifurcation [54][55][56][57] and subharmonics are negligible. a and b can then be expanded in the form…”

“…This bifurcation was studied in Refs. [54,55] in a Duffing oscillator without damping. While our system is related to it [59], we expect that, due to the damping, subharmonics do not occur in the steady state of a moderately driven system [56].…”

Coexistence of Multiple Attractors, Hysteresis, and Vibrational Resonance in the Classical Morse Oscillator Driven by an Amplitude Modulated Signa

“…If = 1/ , where is an integer, then it refers to the superharmonic resonance. Recently, Lukomsky et al [19] proposed a consecutive scheme for studying the harmonic and subharmonic driven oscillations described by second-order differential equations with arbitrary polynomial nonlinearity.…”