Synchronization of a nonlinear oscillator with a sum signal from equivalent oscillators

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

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A simple new iterative method for solving strongly nonlinear oscillator systems having a rational and an irrational force

Alexandria Engineering Journal ◽

10.1016/j.aej.2017.03.013 ◽

2018 ◽

Vol 57 (2) ◽

pp. 1099-1107 ◽

Cited By ~ 1

Author(s):

Md. Abdur Razzak

Keyword(s):

Iterative Method ◽

Nonlinear Oscillator ◽

Strongly Nonlinear ◽

Strongly Nonlinear Oscillator ◽

New Iterative Method

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Closed-Form Expression for the Exact Period of a Nonlinear Oscillator Typified by a Mass Attached to a Stretched Wire

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.11-m1141 ◽

2011 ◽

Vol 3 (6) ◽

pp. 689-701

Author(s):

Malik Mamode

Keyword(s):

Harmonic Balance ◽

Nonlinear Oscillator ◽

Harmonic Balance Method ◽

Elliptic Functions ◽

Oscillatory System ◽

Balance Method ◽

Closed Form Expression ◽

Form Expression ◽

Polynomial Potential ◽

Exact Analytical Expression

AbstractThe exact analytical expression of the period of a conservative nonlinear oscillator with a non-polynomial potential, is obtained. Such an oscillatory system corresponds to the transverse vibration of a particle attached to the center of a stretched elastic wire. The result is given in terms of elliptic functions and validates the approximate formulae derived from various approximation procedures as the harmonic balance method and the rational harmonic balance method usually implemented for solving such a nonlinear problem.

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