2126. Parameter-induced fractal erosion of the safe basin in a softening Duffing oscillator

Shanguo Yang

School of Mechatronic Engineering, China University of Mining and Technology,
Xuzhou 221116, P. R. China

Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment,
China University of Mining and Technology, Xuzhou 221116, P. R. China

E-mail: ysgcumt@163.com

Received 26 May 2016; received in revised form 11 July 2016; accepted 18 July 2016

DOI https://doi.org/10.21595/jve.2016.17209

Abstract. The parameter-induced fractal erosion of the safe basin is investigated in a softening Duffing system. For a fixed excitation, we make the linear stiffness, the nonlinear stiffness and the damping coefficient as the control parameter. At first, the necessary condition for the fractal erosion of the safe basin is obtained by the Melnikov method. Then, the analytical predications are verified by the numerical simulations. With the variation of the stiffness or the damping coefficient, the fractal erosion of the safe basin will appear or vanish. Both the linear and the nonlinear stiffness influence the topology of the safe basin. With the increase of the linear stiffness, the fractal erosion of the safe basin will appear at first and then disappear gradually. The area of the safe basin is an increasing function of the linear stiffness. With the increase of the nonlinear stiffness, the fractal erosion of the safe basin appears and the area of the safe basin turns smaller. The topology of the safe basin is independent of the damping coefficient. For small damping coefficient, the fractal erosion of the safe basin occurs much more easily. The damping coefficient suppresses the fractal erosion of the safe basin.

Keywords: safe basin erosion, fractal, chaos, Melnikov integral.


[1]        Thompson J. M. T., Soliman M. S. Fractal control boundaries of driven oscillators and their relevance to safe engineering design. Proceedings of the Royal Society a Mathematical Physical and Engineering Sciences, Vol. 319, Issue 428, 1991, p. 1‑13.

[2]        Thompson J. M. T., Rainey R. C. T., Soliman M. S. Ship stability criteria based on chaotic transients from incursive fractals. Philosophical Transactions of the Royal Society B Biological Sciences, Vol. 332, Issue 1624, 1990, p. 149‑167.

[3]        Soliman M. S., Thompson J. M. Global dynamics underlying sharp basin erosion in nonlinear driven oscillators. Physical Review A, Vol. 45, Issue 6, 1992, p. 3425‑3431.

[4]        Hu H. Y. Applied Nonlinear Dynamics. Aviation Industry Press, Beijing, 2000.

[5]        De Freitas M. S. T., Viana R. L., Grebogi C. Erosion of the safe basin for the transversal oscillations of a suspension bridge. Chaos Solitons and Fractals, Vol. 18, Issue 4, 2003, p. 829‑841.

[6]        Gonçalves P. B., Silva F. M. A., Prado Z. J. G. N. Global stability analysis of parametrically excited cylindrical shells through the evolution of basin boundaries. Nonlinear Dynamics, Vol. 50, Issue 1, 2007, p. 121‑145.

[7]        Xu J., Lu Q. Z., Huang K. L. Controlling erosion of safe basin in nonlinear parametrically excited systems. Acta Mechanica Sinica, Vol. 12, Issue 3, 1996, p. 281‑288.

[8]        Shang H., Xu J. Delayed feedbacks to control the fractal erosion of safe basins in a parametrically excited system. Chaos Solitons and Fractals, Vol. 41, Issue 4, 2009, p. 1880‑1896.

[9]        Naik R. D., Singru P. M. Resonance, stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback. Communications in Nonlinear Science and Numerical Simulation, Vol. 15, Issue 8, 2011, p. 3397‑3410.

[10]     Alsaleem F. M., Younis M. I. Stabilization of electrostatic MEMS resonators using a delayed feedback controller. Smart Materials and Structures, Vol. 19, Issue 19, 2010, p. 335‑341.

[11]     Shang H. L., Wen Y. P. Fractal erosion of the safe basin in a helmholtz oscillator and its control by linear delayed velocity feedback. Chinese Physics Letters, Vol. 28, Issue 11, 2011, p. 110503‑110503.

[12]     Simiu E., Frey M. R. Melnikov processes and noise-induced exits from a well. Journal of Engineering Mechanics, Vol. 122, Issue 3, 1996, p. 263‑270.

[13]     Lin H., Yim S. C. S. Chaotic roll motion and capsize of ships under periodic excitation with random noise. Applied Ocean Research, Vol. 17, Issue 3, 1995, p. 185‑204.

[14]     Bulsara A. R., Schieve W. C., Jacobs E. W. Homoclinic chaos in systems perturbed by weak Langevin noise. Physical Review A, Vol. 41, Issue 41, 1990, p. 668‑681.

[15]     Lin H., Yim S. C. S. Analysis of a nonlinear system exhibiting chaotic, noisy chaotic, and random behaviors. Journal of Applied Mechanics, Vol. 63, Issue 2, 1996, p. 509‑516.

[16]     Liu W. Y., Zhu W. Q., Huang Z. L. Effect of bounded noise on chaotic motion of duffing oscillator under parametric excitation. Chaos Solitons and Fractals, Vol. 12, Issue 3, 2011, p. 527‑537.

[17]     Gan C. Noise-induced chaos and basin erosion in softening Duffing oscillator. Chaos Solitons and Fractals, Vol. 25, Issue 5, 2005, p. 1069‑1081.

[18]     Gan C. Noise-induced chaos in duffing oscillator with double wells. Nonlinear Dynamics, Vol. 45, Issue 3, 2006, p. 305‑317.

[19]     Li X. C, Xu W., Li R. H. Chaotic motion of the dynamical system under both additive and multiplicative noise excitations. Chinese Physics B, Vol. 17, Issue 2, 2008, p. 557‑568.

[20]     Wei D. Q., Zhang B., Qiu D. Y., Luo X. S. Effect of noise on erosion of safe basin in power system. Nonlinear Dynamics, Vol. 61, Issue 3, 2010, p. 477‑482.

[21]     Li S., Li Q., Li J., Feng J. Chaos prediction and control of good win’s nonlinear accelerator model. Nonlinear Analysis Real World Applications, Vol. 12, Issue 4, 2011, p. 1950‑1960.

[22]     Gammaitoni L., Hänggi P., Jung P., Marchesoni F. Stochastic resonance. Reviews of Modern Physics, Vol. 70, Issue 1, 1998, p. 223‑287.

[23]     Duan F. B., Xu B. H. Parameter-induced stochastic resonance and baseband binary pam signals transmission over an awgn channel. International Journal of Bifurcation and Chaos, Vol. 13, Issue 2, 2011, p. 411‑425.

[24]     Jiang S., Guo F., Zhou Y., Gu T. Parameter-induced stochastic resonance in an over-damped linear system. Physica A Statistical Mechanics and Its Applications, Vol. 375, Issue 2, 2007, p. 483‑491.

Cite this article

Yang Shanguo Parameter‑induced fractal erosion of the safe basin in a softening Duffing oscillator. Journal of Vibroengineering, Vol. 18, Issue 5, 2016, p. 3329‑3336.


© JVE International Ltd. Journal of Vibroengineering. Aug 2016, Vol. 18, Issue 5. ISSN 1392-8716