Analysis of noise-induced transitions for Hopf system with additive and multiplicative random disturbances. (English) Zbl 1197.37069
Summary: An analysis of the classical Hopf differential system perturbed by multiplicative and additive noises is carried out. An explicit representation for the stationary probability density function is found as an analytical solution of related Fokker-Planck equation. The difference in the response of Hopf systems perturbed by additive and multiplicative random noises is investigated. That difference can be seen in the zone of the transition from the trivial equilibrium point to noisy limit cycle. A delaying shift of the Hopf bifurcation point induced by multiplicative noise is recognized. In fact, an explicit formula of the radius of the stochastic limit cycle as a function of the involved parameters is stated. The phenomenon of inverse stochastic bifurcation in which auto-oscillations are suppressed by multiplicative noise is clearly observed. Eventually, the analytical description of the probability density of the randomly forced Hopf system offers the excellent possibility to test and compare the accuracy of different numerical schemes with respect to the replication of stochastic limit cycles. The superiority of the linear-implicit trapezoidal-type method is demonstrated in this respect.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.
MSC:
| 37H10 | Generation, random and stochastic difference and differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34F05 | Ordinary differential equations and systems with randomness |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 60G35 | Signal detection and filtering (aspects of stochastic processes) |
References:
| [1] | Arnold, L., Random dynamical systems (1998), Springer: Springer Berlin · Zbl 0906.34001 |
| [2] | Arnold, L.; Bleckert, G.; Schenk-Hoppe, K., The stochastic Brusselator: parametric noise destroys Hopf bifurcation, (Stochastic dynamics 1997, Bremen (1999), Springer: Springer New York), 71-92 · Zbl 0939.60057 |
| [3] | Franzoni, L.; Mannella, R.; McClintock, P.; Moss, F., Postponement of Hopf bifurcations by multiplicative colored noise, Phys Rev A, 36, 834 (1987) |
| [4] | Gardiner, C. W., Handbook of stochastic methods (1996), Springer: Springer New York · Zbl 0862.60050 |
| [5] | Horsthemke, W.; Lefever, R., Noise-induced transitions (1984), Springer: Springer Berlin · Zbl 0529.60085 |
| [6] | Landa, P. S.; McClintock, P. V.E., Changes in the dynamical behavior of nonlinear systems induced by noise, Phys Rep, 323, 1-80 (2000) |
| [7] | Lefever, R.; Turner, J., Sensitivity of a Hopf bifurcation to external multiplicative noise, (Horsthemke, W.; Kondepudi, D. K., Fluctuations and sensitivity in equilibrium systems (1984), Springer: Springer Berlin), 143-149 |
| [8] | Lefever, R.; Turner, J., Sensitivity of a Hopf bifurcation to multiplicative colored noise, Phys Rev Lett, 56, 1631-1634 (1986) |
| [9] | Leung, H. K., Stochastic Hopf bifurcation in a biased van der Pol model, Physica A, 254, 146-155 (1998) |
| [10] | Malick, K.; Marcq, P., Stability analysis of noise-induced Hopf bifurcation, Eur Phys J, 36, 119 (2003) |
| [11] | Namachchivaya, N. Sri., Hopf bifurcation in the presence of both parametric and external stochastic excitations, J Appl Mech, 110, 923 (1988) · Zbl 0671.93055 |
| [12] | Schenk-Hoppe, K. R., Bifurcation scenarios of the noisy Duffing-van der Pol oscillator, Nonlinear Dyn, 11, 255-274 (1996) |
| [13] | Kurrer, C.; Schulten, K., Effect of noise and perturbations on limit cycle systems, Physica D, 50, 311-320 (1991) · Zbl 0749.58044 |
| [14] | Pontryagin, L. S.; Andronov, A. A.; Witt, A. A., On the statistical treatment of dynamical systems (in Russian), Zh Eksp Teor Fiz, 3, 165-180 (1933) |
| [15] | Stratonovich, R. L., Topics in the theory of random noise (1963), Gordon and Breach: Gordon and Breach New York · Zbl 0119.14502 |
| [16] | Ibrahim, R. A., Parametric random vibration (1985), John Wiley and Sons: John Wiley and Sons New York · Zbl 0652.70002 |
| [17] | Soong, T. T.; Grigoriu, M., Random vibration of mechanical and structural systems (1993), RTL Prentice-Hall, Englewood Cliffs: RTL Prentice-Hall, Englewood Cliffs New Jersey · Zbl 0788.73005 |
| [18] | Freidlin, M. I.; Wentzell, A. D., Random perturbations of dynamical systems (1984), Springer: Springer New York · Zbl 0522.60055 |
| [19] | Dembo, M.; Zeitouni, O., Large deviations techniques and applications (1995), Jones and Bartlett Publishers: Jones and Bartlett Publishers Boston |
| [20] | Naeh, T.; Klosek, M. M.; Matkowsky, B. J.; Schuss, Z., A direct approach to the exit problem, SIAM J Appl Math, 50, 595-627 (1990) · Zbl 0709.60059 |
| [21] | Roy, R. V., Asymptotic analysis of first passage problem, Int J Non-Linear Mech, 32, 173-186 (1997) · Zbl 0897.70018 |
| [22] | Smelyanskiy, V. N.; Dykman, M. I.; Maier, R. S., Topological features of large fluctuations to the interior of a limit cycle, Phys Rev E, 55, 2369-2391 (1997) |
| [23] | Bashkirtseva, I. A.; Ryashko, L. B., Stochastic sensitivity of 3D-cycles, Math Comput Simul, 66, 55-67 (2004) · Zbl 1090.34048 |
| [24] | Ryagin, M.; Ryashko, L., The analysis of the stochastically forced periodic attractors for Chua’s circuit, Int J Bifurcat Chaos, 14, 11, 3981-3987 (2004) |
| [25] | Bashkirtseva, I. A.; Ryashko, L. B., Sensitivity and chaos control for the forced nonlinear oscillations, Chaos, Solitons and Fractals, 26, 1437-1451 (2005) · Zbl 1122.37028 |
| [26] | Schurz, H., Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications (1997), Logos-Verlag: Logos-Verlag Berlin · Zbl 0905.60002 |
| [27] | Schurz, H., Preservation of probabilistic laws through Euler methods for Ornstein-Uhlenbeck process, Stoch Anal Appl, 17, 3, 463-486 (1999) · Zbl 0931.60047 |
| [28] | Schurz, H., The invariance of asymptotic laws of linear stochastic systems under discretization, Z Angew Math Mech, 79, 6, 375-382 (1999) · Zbl 0938.34072 |
| [29] | Schurz, H., Numerical analysis of SDEs without tears (invited chapter), (Lakshmikantham, V.; Kannan, D., Handbook of stochastic analysis and applications (2002), Marcel Dekker: Marcel Dekker Basel), 237-339 · Zbl 0995.60052 |
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.
