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

作者：Liu Chein Shan^*

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

• 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

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(parallel to f parallel to(2)parallel to x parallel to(2) - 2(f.x)(2)) = -sign(cos 2 theta) as a key factor, where theta is the intersection angle between x and f. It together with sign(cos theta) can reveal the complexity of nonlinear Duffing oscillator and a quadratic ship rolling oscillator. The barcode is formed by plotting sign(parallel to f parallel to(2)parallel to x parallel to(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 A(1)(-) := {sign(cos theta) = +1 and sign(cos 2 theta) = + 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.

• 出版日期2014-4