Effect of Gaussian white noise on the dynamical behaviors of an extended Duffing-Van der Pol oscillator. (English) Zbl 1142.34023
Summary: This paper discusses the influence of Gaussian white noise on the dynamics of a self-excited triple well extended Duffing-Van der Pol oscillator already subjected to harmonic excitation. Firstly, the condition for the rise of hom/heteroclinic chaos is derived by random Melnikov’s technique under its corresponding mean-square criterion and the result indicates that the threshold amplitude of harmonic excitation is lowered by the appearance of Gaussian white noise. Moreover, the threshold is decreased as the noise intensity increases. Since the Melnikov’s criterion is only a necessary condition for the occurrence of chaotic motion, this prediction is tested against numerical simulations of the basins of attraction and the Lyapunov exponents. By vanishing the largest Lyapunov exponents, another criterion for the onset of chaos is obtained which is accorded with the theoretical one. Finally, how the noise effects the structure of periodic or chaotic attractor is investigated by simulating Poincaré maps of the original system and rich transition states displayed by the considered extended Duffing-Van der Pol oscillator are observed.
MSC:
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 34F05 | Ordinary differential equations and systems with randomness |
| 70L05 | Random vibrations in mechanics of particles and systems |
| 34C28 | Complex behavior and chaotic systems of ordinary differential equations |
References:
| [1] | DOI: 10.1016/S0167-2789(02)00565-1 · Zbl 1008.37011 · doi:10.1016/S0167-2789(02)00565-1 |
| [2] | DOI: 10.1006/jsvi.1995.0448 · Zbl 1049.70644 · doi:10.1006/jsvi.1995.0448 |
| [3] | DOI: 10.1016/0020-7462(88)90034-0 · Zbl 0667.70028 · doi:10.1016/0020-7462(88)90034-0 |
| [4] | DOI: 10.1016/0167-2789(93)90114-G · Zbl 0768.60039 · doi:10.1016/0167-2789(93)90114-G |
| [5] | DOI: 10.1007/978-1-4612-1140-2 · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2 |
| [6] | DOI: 10.1098/rsta.1979.0068 · Zbl 0423.34049 · doi:10.1098/rsta.1979.0068 |
| [7] | Lansiti M., Phys. Rev. E 55 pp 746– |
| [8] | DOI: 10.1016/0022-460X(90)90934-R · Zbl 1235.74096 · doi:10.1016/0022-460X(90)90934-R |
| [9] | DOI: 10.1115/1.2788897 · Zbl 0875.70115 · doi:10.1115/1.2788897 |
| [10] | DOI: 10.1103/PhysRevLett.55.1439 · doi:10.1103/PhysRevLett.55.1439 |
| [11] | DOI: 10.1016/j.physleta.2004.01.016 · Zbl 1118.81375 · doi:10.1016/j.physleta.2004.01.016 |
| [12] | DOI: 10.1016/0960-0779(92)90036-M · Zbl 0768.58032 · doi:10.1016/0960-0779(92)90036-M |
| [13] | DOI: 10.1016/0020-7462(80)90024-4 · Zbl 0447.70028 · doi:10.1016/0020-7462(80)90024-4 |
| [14] | DOI: 10.1016/j.chaos.2003.12.014 · Zbl 1046.70012 · doi:10.1016/j.chaos.2003.12.014 |
| [15] | DOI: 10.1016/j.physa.2005.06.070 · doi:10.1016/j.physa.2005.06.070 |
| [16] | DOI: 10.1016/S0378-4371(01)00500-3 · Zbl 1010.70017 · doi:10.1016/S0378-4371(01)00500-3 |
| [17] | DOI: 10.1016/S0167-2789(02)00447-5 · Zbl 1071.70015 · doi:10.1016/S0167-2789(02)00447-5 |
| [18] | Ueda Y., IEEE Trans. Circuits Syst. 28 pp 217– · doi:10.1109/TCS.1981.1084975 |
| [19] | DOI: 10.1103/PhysRevE.56.6321 · doi:10.1103/PhysRevE.56.6321 |
| [20] | Wei J. G., Appl. Math. Comput. 88 pp 77– |
| [21] | DOI: 10.1007/978-1-4612-1042-9 · doi:10.1007/978-1-4612-1042-9 |
| [22] | DOI: 10.1007/978-1-4757-4067-7 · doi:10.1007/978-1-4757-4067-7 |
| [23] | DOI: 10.1016/0167-2789(85)90011-9 · Zbl 0585.58037 · doi:10.1016/0167-2789(85)90011-9 |
| [24] | Xie W. C., Nonlin. Stoch. Dyn. 192 pp 215– |
| [25] | DOI: 10.1142/S0218127404010989 · Zbl 1074.34068 · doi:10.1142/S0218127404010989 |
| [26] | DOI: 10.1016/j.chaos.2004.12.005 · Zbl 1092.37532 · doi:10.1016/j.chaos.2004.12.005 |
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.
