Chaotic dynamics of the vibro-impact system under bounded noise perturbation. (English) Zbl 1352.70044
Summary: In this paper, chaotic dynamics of the vibro-impact system under bounded noise excitation is investigated by an extended Melnikov method. Firstly, the Melnikov method in the deterministic vibro-impact system is extended to the stochastic case. Then, a typical stochastic Duffing vibro-impact system is given to application. The analytic conditions for occurrence of chaos are derived by using the random Melnikov process in the mean-square-value sense. In addition, the numerical simulations confirm the validity of analytic results. Also, the influences of interesting system parameters on the chaotic dynamics are discussed.
MSC:
| 70H05 | Hamilton’s equations |
| 35R12 | Impulsive partial differential equations |
| 37M05 | Simulation of dynamical systems |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
References:
| [1] | Chin, W.; Ott, E.; Nusse, H. E.; Grebogi, C., Grazing bifurcations in impact oscillators, Phys Rev E, 50, 4427-4444 (1994) |
| [2] | Dimentberg, M. F.; Iourtchenko, D. V., Random vibrations with impacts: a review, Nonlinear Dyn, 36, 229-254 (2004) · Zbl 1125.70019 |
| [3] | de Souza, S. L.T.; Caldas, I. L., Controlling chaotic orbits in mechanical systems with impacts, Chaos, Solitons Fractals, 19, 171-178 (2004) · Zbl 1086.37045 |
| [4] | di Bernardo, M.; Budd, C. J., Piecewise-smooth dynamical systems: theory and applications (2007), Springer-Verlag: Springer-Verlag London |
| [5] | Wang, L.; Xu, W., Response of a stochastic Duffing-Van der Pol elastic impact oscillator, Chaos, Solitons Fractals, 41, 2075-2080 (2009) · Zbl 1198.60029 |
| [6] | Luo, G. W., Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops, Phys Lett A, 323, 210-217 (2004) · Zbl 1118.81404 |
| [7] | di Bernardo, M., Unified framework for the analysis of grazing and border-collisions in piecewise-smooth system, Phys Rev Lett, 86, 2553-2556 (2001) |
| [8] | Yue, Y.; Xie, J. H., Neimark-Sacker-pitchfork bifurcation of the symmetric period fixed point of the Poincaré map in a three-degree-of-freedom vibro-impact system, Int J Nonlinear Mech, 48, 51-58 (2013) |
| [9] | Gu, X. D.; Zhu, W. Q., A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations, J Sound Vib, 333, 2632-2642 (2014) |
| [10] | Makarenkov, O.; Lamb, J. S.W., Dynamics and bifurcations of non-smooth systems: a survey, Physica D, 241, 1826-1844 (2012) |
| [11] | Wang, Z.; Szolnoki, A.; Perc, M., Rewarding evolutionary fitness with links between populations promotes cooperation, J Theor Biol, 349, 50-56 (2014) · Zbl 1412.91016 |
| [12] | Boccaletti, S., The structure and dynamics of multilayer networks, Phys Rep, 544, 1-122 (2014) |
| [13] | Holmes, P. J., A nonlinear oscillator with a strange attractor, Philos Trans R Soc A, 292, 419-448 (1979) · Zbl 0423.34049 |
| [14] | Guchenheimer, J.; Holmes, P. J., Nonlinear oscillations, dynamical systems and bifurcations of vector fields (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0515.34001 |
| [15] | Simiu, E., Chaotic transitions in deterministic and stochastic dynamical systems (2002), Princeton University Press: Princeton University Press UK · Zbl 0999.37001 |
| [16] | Awrejcewicz, J., Routes to chaos in continuous mechanical systems. Part 1: Mathematical models and solution methods, Chaos, Solitons Fractals, 45, 687-708 (2012) · Zbl 1414.74013 |
| [17] | Liu, W. Y.; Zhu, W. Q.; Huang, Z. L., Effect of bounded noise on chaotic motion of Duffing oscillator under parameter excitation, Chaos, Solitons Fractals, 12, 527-537 (2001) · Zbl 1090.37523 |
| [18] | Du, Z.; Zhang, W., Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput Math Appl, 50, 445-458 (2005) · Zbl 1097.37043 |
| [19] | Xu, W.; Feng, J. Q.; Rong, H. W., Melnikov’s method for a general nonlinear vibro-impact oscillator, Nonlinear Anal-Theory, 71, 418-426 (2009) · Zbl 1176.34052 |
| [20] | Li, G. X.; Moon, F. C., Criteria for chaos of a triple-well potential oscillator with homoclinic and heteroclinic orbits, J Sound Vib, 136, 17-34 (1990) · Zbl 1235.74096 |
| [21] | Gan, C.; Lei, H., Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations, J Sound Vib, 330, 2174-2184 (2011) |
| [22] | Shinozuka, M., Digital simulation of random processes and its applications, J Sound Vib, 25, 111-128 (1972) |
| [23] | Moon, F. C.; Li, G.-X., Fractal basin boundary and homoclinic orbits for periodic motion in a two-well potential, Phys Rev E, 35, 1439-1442 (1985) |
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