Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

Abstract

Bifurcations and route to chaos of the Mathieu-Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.

A modified incremental harmonic balance method for rotary periodic motions

“…The IHB method was extended to deal systems with Coulomb damping [16,17] and systems with piecewise linearities [18][19][20]. By introducing a continuous parameter and appropriate path following procedure, the IHB method is also suitable for bifurcation and chaos analysis [21][22][23]. The IHB method is described briefly below.…”

Dynamic stability of a nonlinear multiple-nanobeam system

“…For stability of the periodic solution, Floquet multipliers must be centred within the unit circle with the origin in the centre of the complex plane. However, if the values of Floquet multipliers leave the unit circle, we can predict bifurcation according to [71]. For particular values of material parameters and initial conditions, we consider the iterative equation (20) and obtain the following stable periodic solutions wherein the stability of the trajectory is confirmed by using the Floquet theory and above-described iterative procedure [69,70].…”

An open-plus-closed-loop control for chaotic Mathieu-Duffing oscillator

“…According to Ref. [16], the dynamics of the Mathieu-Duffing oscillator can be summarized as follows x(t)…”

“…(1) Also, the analytical approximations for the periodic and higher-periodic orbits under the certain parameter values were explicitly calculated in Ref. [16]. For example, when β = 4.60, the analytical approximation for the period-1 orbit is given by …”

“…However, since the accuracy of the first-order approximation is somewhat low, the performances of their control are not good enough. Here, we utilize the higher-order approximation obtained in our previous paper [16] by the incremental balance method. For instance, we can use Eq.…”