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Disclosing the Complexity of Nonlinear Ship Rolling and Duffing Oscillators by a Signum Function

Chein-Shan Liu1
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail:

Computer Modeling in Engineering & Sciences 2014, 98(4), 375-407.


In this paper we study the nonlinear dynamical system x·=f(x,t) from a newly developed theory, viewing the time-varying function of sign(||f||2||x||2− 2(f·x)2) = −sign(cos 2θ) as a key factor, where θ is the intersection angle between x and f. It together with sign(cos θ) can reveal the complexity of nonlinear Duffing oscillator and a quadratic ship rolling oscillator. The barcode is formed by plotting sign(||f||2||x||2− 2(f·x)2) with respect to time. We analyze the barcode to point out the bifurcation of subharmonic motions and the range of chaos in the parameter space. The bifurcation diagram obtained by plotting the percentage of the first set of dis-connectivity A1 : = {sign(cos θ) = + 1 and sign(cos 2θ) = + 1} with respect to the amplitude of harmonic loading leads to a finer structure of a devil staircase for the ship rolling oscillator, as well as a cascade of subharmonic motions to chaos for the Duffing oscillator.


Nonlinear dynamical system, SO0(n, 1) Lie-group scheme, Signum function, Barcode, Duffing oscillator, Ship rolling equation, Jumping in Barcode, First set of dis-connectivity.

Cite This Article

Liu, C. (2014). Disclosing the Complexity of Nonlinear Ship Rolling and Duffing Oscillators by a Signum Function. CMES-Computer Modeling in Engineering & Sciences, 98(4), 375–407.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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