Bifurcation of Safe Basins and Chaos in Nonlinear Vibroimpact Oscillator under Harmonic and Bounded Noise Excitations

The erosion of the safe basins and chaoticmotions of a nonlinear vibroimpact oscillator under both harmonic and bounded random noise is studied. Using the Melnikov method, the system’s Melnikov integral is computed and the parametric threshold for chaotic motions is obtained. Using the Monte-Carlo and Runge-Kutta methods, the erosion of the safe basins is also discussed.The sudden change in the character of the stochastic safe basins when the bifurcation parameter of the system passes through a critical value may be defined as an alternative stochastic bifurcation. It is founded that random noise may destroy the integrity of the safe basins, bring forward the occurrence of the stochastic bifurcation, and make the parametric threshold for motions vary in a larger region, hence making the system become more unsafely and chaotic motions may occur more easily.

[1]  D. V. Iourtchenko,et al.  ENERGY BALANCE FOR RANDOM VIBRATIONS OF PIECEWISE-CONSERVATIVE SYSTEMS , 2001 .

[2]  Haiwu Rong,et al.  Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations , 2010 .

[3]  Jinqian Feng,et al.  Response probability density functions of Duffing–Van der Pol vibro-impact system under correlated Gaussian white noise excitations , 2013 .

[4]  Satish Nagarajaiah,et al.  Numerical investigation of coexisting high and low amplitude responses and safe basin erosion for a coupled linear oscillator and nonlinear absorber system , 2014 .

[5]  D. V. Iourtchenko,et al.  Numerical investigation of a response probability density function of stochastic vibroimpact systems with inelastic impacts , 2006 .

[6]  H. Zhu,et al.  Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises , 2014 .

[7]  Haiwu Rong,et al.  Subharmonic response of a single-degree-of freedom nonlinear vibroimpact system to a randomly disordered periodic excitation , 2009 .

[8]  Oleg Gaidai,et al.  Random vibrations with strongly inelastic impacts: Response PDF by the path integration method , 2009 .

[9]  Shui-Nee Chow,et al.  Bifurcations of subharmonics , 1986 .

[10]  Yu-Kweng Michael Lin,et al.  Probabilistic Structural Dynamics: Advanced Theory and Applications , 1967 .

[11]  Q. Feng,et al.  Modeling of the mean Poincaré map on a class of random impact oscillators , 2003 .

[12]  Xiao-Song Yang,et al.  A Simple Method for Finding Topological Horseshoes , 2010, Int. J. Bifurc. Chaos.

[13]  Zhengdong Du,et al.  Melnikov method for homoclinic bifurcation in nonlinear impact oscillators , 2005 .

[14]  E. Simiu,et al.  Chaotic Transitions in Deteffi 1 inistic and Stochastic Dynamical Systems : Applications of the Melnikov Method in Engineering , Physics , and Neuroscience , 2002 .

[15]  Steven W. Shaw,et al.  The transition to chaos in a simple mechanical system , 1989 .

[16]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[17]  B. Brogliato.  Nonsmooth Mechanics: Models, Dynamics and Control , 1999 .

[18]  D. V. Iourtchenko,et al.  Subharmonic Response of a Quasi-Isochronous Vibroimpact System to a Randomly Disordered Periodic Excitation , 1998 .

[19]  D. V. Iourtchenko,et al.  Random Vibrations with Impacts: A Review , 2004 .

[20]  R. G. Medhurst,et al.  Topics in the Theory of Random Noise , 1969 .

[21]  M. Shinozuka,et al.  Digital simulation of random processes and its applications , 1972 .

[22]  Hung-Sying Jing,et al.  Random response of a single‐degree‐of‐freedom vibro‐impact system with clearance , 1990 .

[23]  Xiao-Song Yang,et al.  Topological Horseshoes and Computer Assisted Verification of Chaotic Dynamics , 2009, Int. J. Bifurc. Chaos.

[25]  C. Gan.  Noise-induced chaos and basin erosion in softening Duffing oscillator , 2005 .