Chaos diagram of harmonically excited vibration absorber control duffing's oscillator

Abstract

"This study utilised positive Lyapunov exponents' criteria to develop chaos diagram on the parameters space of 4-dimensionalharmonically excited vibration absorber control Duffing's Oscillator. Relevant simulations were effected by choice combination of constant step Runge-Kutta methods and Grahm Schmidt Orthogonal rules. Simulations of 4-dimensional hyper-chaotic models of modified Lorenz and RÖsier were used for validation purposes. Lyapunov's spectrums were obtained at (197 x 301) mesh points of parameters space (µ,αa). Lyapunov's spectrum of modified Lorenz system by constant time step (NRK1) fourth order Runge-Kutta method (04208.01650. - 0.0807, -26.4603) compare correspondingly well with (0.4254, 0.1286, 0.0000, -26.5493) reported by Yuxia et et. Similarly, Lyapunovs spectrum of modified Rosier system by constant time step (NRK1) fourth order Runge-Kutta method (0.1424, 0.0051, -0.0041, -24.0831) compare correspondingly and qualitatively with (0.1287, 0.0149, -0.0056, -22.8617) reported by Marco (1996). The sum of Lyapunov exponents (-22.7237, -31.3107, -27.8797) in Rosier compare correspondingly and qualitatively with variation matrix measure -AVERT (- 24.0181, -30.9462, -28.1991) respectively for fourth, fifth and modified fifth order Runge-Kutta methods. The chaos diagram results suggested preferentially higher mass ratio for effective chaos control of Duffing's Oscillator main mass. The parameters space in the region of relative lower mass ratio suffered irregular boundaries. The practical applications of this chaos diagram plot include, by instance, walking in the parameters-space of vibration absorber control Duffing's Oscillator along suitable engineering paths. "