Noise-induced chaos and basin erosion in softening Duffing oscillator. (English) Zbl 1099.37065
The noise-induced chaotic responses and the effect of the Gaussian white or the bounded noise on the erosion of safe basins are discussed in some detail in the softening Duffing oscillator. Noise-induced chaotic responses are simulated and testified by employing the algorithm for the largest Lyapunov exponent, where the parameter values are chosen according to the given Melnikov condition. Moreover, the author presents the variations of safe basins numerically when one changes the amplitude of the harmonic excitation in the deterministic case, or the form or the strength of the random excitation in the stochastic case of the system, where one can see that fractal boundaries appear even the Duffing oscillator is excited only by the random excitation.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
| 37N05 | Dynamical systems in classical and celestial mechanics |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |
References:
| [1] | Bulsara, A. R.; Schieve, W. C.; Jacobs, E. W., Homoclinic chaos in systems perturbed by weak Langevin noise, Phys Rev A, 41, 2, 668-681 (1990) |
| [2] | Frey M, Simiu E. Equivalence between motions with noise-induced jumps and chaos with Smale horseshoes. Proc 9th Engrg Mech Conf, ASCE, 1992. p. 660-3; Frey M, Simiu E. Equivalence between motions with noise-induced jumps and chaos with Smale horseshoes. Proc 9th Engrg Mech Conf, ASCE, 1992. p. 660-3 |
| [3] | Frey, M.; Simiu, E., Noise-induced chaos and phase space flux, Physica D, 63, 321-340 (1993) · Zbl 0768.60039 |
| [4] | Xie, W. C., Effect of noise on chaotic motion of buckled column under periodic excitation, Nonlinear and Stochastic Dynamics, AMD Vol 192/DE Vol 73, 215-225 (1994) |
| [5] | Simiu, E.; Frey, M., Melnikov processes and noise-induced exits from a well, J Eng Mech, 122, 3, 263-270 (1996) |
| [6] | Lin, H.; Yim, S. C.S., Analysis of a nonlinear system exhibiting chaotic, noisy chaotic, and random behaviors, ASME J Appl Mech, 63, 509-516 (1996) · Zbl 0875.70115 |
| [7] | Wiggins, S., Global bifurcations and chaos: analytical methods (1988), Springer: Springer New York · Zbl 0661.58001 |
| [8] | Wolf, A.; Swift, J. R.; Swinney, H. L.; Vastano, J. A., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317 (1985) · Zbl 0585.58037 |
| [9] | Kantz, H.; Schreiber, T., Nonlinear time series analysis (1997), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0873.62085 |
| [10] | Sano, M.; Sawada, Y., Measurement of the Lyapunov spectrum from a chaotic time series, Phys Rev Lett, 55, 1082-1085 (1985) |
| [11] | Eckmann, J. P.; Kamphorts, S. O.; Rulle, D.; Cliliberto, S., Lyapunov exponents from a time series, Phys Rev A, 34, 4971-4979 (1986) |
| [12] | Rosenstein, M. T.; Collins, J. J.; Luca, C. J., A practical method for calculating largest Lyapunov exponents from small data sets, Physica D, 65, 117-134 (1993) · Zbl 0779.58030 |
| [13] | Soliman, M. S.; Thompson, J. M.T., Integrity measures quantifying the erosion of smooth and fractal basins of attraction, J Sound Vibr, 35, 453-475 (1989) · Zbl 1235.70106 |
| [14] | McDonald, S. W.; Grebogi, C.; Ott, E.; Yorke, J. A., Fractal basin boundaries, Physica D, 17, 125-153 (1985) · Zbl 0588.58033 |
| [15] | Moon, F. C.; Li, G. X., Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential, Phys Rev Lett, 55, 1439-1442 (1985) |
| [16] | Soliman, M. S., Fractal erosion of basins of attraction in coupled nonlinear systems, J Sound Vibr, 182, 727-740 (1995) · Zbl 1237.70089 |
| [17] | Senjanovic, I.; Parunov, J.; Cipric, G., Safety analysis of ship rolling in rough sea, Chaos, Solitons & Fractals, 4, 659-680 (1997) · Zbl 0963.70541 |
| [18] | Freitas, M. S.T.; Viana, R. L.; Grebogi, C., Erosion of the safe basin for the transversal oscillations of a suspension bridge, Chaos, Solitons & Fractals, 18, 829-841 (2003) · Zbl 1129.74317 |
| [19] | Gan, C. B.; Lu, Q. S.; Huang, K. L., Non-stationary effects on safe basins of a softening Duffing oscillator, ACTA Mech Sol Sin, 11, 3, 253-260 (1998) |
| [20] | Dimentberg, M. F., Statistical dynamics of nonlinear and time-varying systems (1988), Wiley: Wiley New York · Zbl 0713.70004 |
| [21] | Wedig, W. V., Analysis and simulation of nonlinear stochastic systems, (Schiehlen, W., Nonlinear dynamics in engineering systems (1989), Springer-Verlag: Springer-Verlag Berlin), 337-344 · Zbl 1135.37336 |
| [22] | Lin, Y. K.; Cai, G. Q., Probabilistic structural dynamics: advanced theory and applications (1995), McGraw-Hill, Inc: McGraw-Hill, Inc New York |
| [23] | Shinozuka, M., Digital simulation of random processes and its applications, J Sound Vibr, 25, 1, 111-128 (1972) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
