Abstract
This paper aims to study a class of differential equations with parametric Gaussian colored noise. We present the general framework to get the solvability conditions of the approximate stationary probability density function, which is determined by the Fokker-Planck-Kolmogorov (FPK) equations. These equations are derived using the stochastic averaging method and the operator theory with the perturbation technique. An illustrative example is proposed to demonstrate the procedure of our proposed method. The analytical expression of approximate stationary probability density function is obtained. Numerical simulation is carried out to verify the analytical results and excellent agreement can be easily found. The FPK equation for the probability density function of order ε 0 is used to examine the almost-sure stability for the amplitude process. Finally, the stability in probability of the amplitude process is investigated by Lin and Cai’s method.
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References
Bountis, T., Mahmoud, G.M.: Synchronized periodic orbits in beam–beam interaction models of one and two spatial dimensions. Part. Accel. 22, 124–147 (1987)
Li, R.: Selfconsistent simulation of the CSR effect. In: Proceedings of the European Particle Accelerator Conference, Stockholm, Sweden, 1998. Institute of Physics, London (1998)
Month, M., Herrera, J.C. (eds.): Nonlinear Dynamics and the Beam–Beam Interaction. AIP Conference Proceedings, vol. 57. AIP, New York (1979)
Dowell, D.H., O’Shea, P.G.: Coherent synchrotron radiation included emittance growth in a chicane buncher. In: Proceedings of the Particle Accelerator Conference, Vancouver, Canada, 1997. IEEE, Piscataway (1998)
Braun, H.H., Corsini, R. et al.: Emmitance growth and energy loss due to coherent synchrotron radiation in the bunch compressor of the CLIC Test Facility (CTFII). In: Proceedings of the European Particle Accelerator Conference, Vienna, Austria, 2000. http://accelconf.web.cern.ch/AccelConf/e00/index.html
Mahmoud, G.M.: On periodic orbits of nonlinear dynamical systems with many degrees of freedom. Physica A 181, 385–395 (1992)
Wedig, W.V.: Dynamic stability of beams under axial forces: Lyapunov exponents for general fluctuating loads. In: Kratzig, W.B., et al. (eds.) Structural Dynamics, pp. 141–148. Balkema, Rotterdam (1990)
Cveticanin, L., Maretic, R.: A van der Pol absorber for rotor vibrations. J. Sound Vib. 173(2), 145–155 (1994)
Cveticanin, L.: Vibrations of a textile machine rotor. J. Sound Vib. 97, 181–187 (1984)
Cveticanin, L.: Dynamical behavior of a rotor with time dependent parameters. JSME Int. J. Ser. C 37(1), 41–48 (1994)
Muszynska, A.: Whirl and whip-rotor/bearing stability problems. J. Sound Vib. 110(3), 443–462 (1986)
Genin, J., Maybee, J.S.: Stability in the three dimensional whirling problem. Int. J. Non-Linear Mech. 4, 205–215 (1969)
Tondl, A.: Determination of the limit of initiation of self-excited vibration of rotors. Int. J. Non-Linear Mech. 15, 417–428 (1980)
Mahmoud, G.M.: Beam–beam interaction models with a small stochastic perturbation. Physica A 216, 445–451 (1995)
Xu, Y., Xu, W., Mahmoud, G.M.: On a complex beam–beam interaction model with random forcing. Physica A 336, 347–360 (2004)
Xu, Y., Xu, W., Mahmoud, G.M., Lei, Y.M.: Beam–beam interaction models under narrow-band random excitation. Physica A 346, 372–386 (2005)
Xu, Y., Zhang, H.Q., Xu, W.: On stochastic complex beam–beam interaction models with Gaussian colored noise. Physica A 384, 259–272 (2007)
Mahmoud, G.M.: Stability regions for coupled Hill’s equations. Physica A 242, 239–249 (1997)
Stratonovich, R.L.: Topics in the Theory of Random Noise. Gordon and Breach, New York (1963)
Roberts, J.B.: The energy envelope of a randomly excited non-linear oscillator. J. Sound Vib. 60, 177–185 (1978)
Zhu, W.Q., Lin, Y.K.: On stochastic averaging of energy envelope. ASCE J. Eng. Mech. 117, 1890–1905 (1991)
Lin, Y.K.: Some observations on the stochastic averaging method. Prob. Eng. Mech. 1, 23–27 (1986)
Zhu, W.Q.: Stochastic averaging in random vibration. Appl. Mech. Rev. 41, 189–199 (1988)
Blankenship, G., Papanicolaou, G.C.: Stability and control of stochastic systems with wide-band noise disturbances. SIAM J. Appl. Math. 34, 437–476 (1978)
Roy, R.V.: Stochastic averaging of oscillators excited by colored Gaussian processes. Int. J. Non-Linear Mech. 29, 463–475 (1994)
Klosek-Dygas, M.M., Matkowsky, B.J., Schuss, Z.: Stochastic stability of nonlinear oscillators. SIAM J. Appl. Math. 48, 1115–1127 (1988)
Klosek-Dygas, M.M., Matkowsky, B.J., Schuss, Z.: Colored noise in dynamical systems. SIAM J. Appl. Math. 48, 425–441 (1988)
Liu, X.B., Liew, K.M.: On the stability properties of a Van der Pol-Duffing oscillator that is driven by a real noise. J. Sound Vib. 285, 27–49 (2005)
Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, Berlin (1983)
Hänggi, P., Jung, P.: Colored noise in dynamical systems. Adv. Chem. Phys. 89, 239–326 (1995)
Jung, P., Hänggi, P.: Dynamical systems: a unified colored-noise approximation. Phys. Rev. A 35, 4464–4466 (1987)
Haken, H.: The Fokker–Planck Equation. Springer, Berlin (1989)
Lin, Y.K., Cai, G.Q.: Probabilistic is Structural Dynamics, Advanced Theory and Applications. McGraw-Hill, New York (1995)
Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)
Zhu, W.Q.: Random Vibration. Science, Bei**g (1992)
Bartosh, L.: Generation of colored noise. Int. J. Mod. Phys. C 12(6), 851–855 (2001)
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Zhang, H., Xu, Y. & Xu, W. Approximate stationary solution and stochastic stability for a class of differential equations with parametric colored noise. Nonlinear Dyn 56, 213–221 (2009). https://doi.org/10.1007/s11071-008-9393-8
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DOI: https://doi.org/10.1007/s11071-008-9393-8