On the Moment Stability of Stochastic Parametrically Forced Equations with Rank One Forcing

We derive simplified formulas for analyzing the moment stability of stochastic parametrically forced linear systems. This analysis extends the results in [T. Blass and L. A. Romero, SIAM J. Control Optim., 51 (2013), pp. 1099--1127], where, under the assumption that the stochastic excitation is small, the stability of such systems was computed using a weighted sum of the extended power spectral density over the eigenvalues of the unperturbed operator. In this paper, we show how to convert this sum to a sum over the residues of the extended power spectral density. For systems where the parametric forcing term is a rank one matrix, this approach leads to an enormous simplification. We give two examples of systems with rank one forcing, including the problem of stochastically forced Faraday waves.

[1]  John William Strutt Rayleigh,et al.  The theory of sound. Vol. 2 , 2015 .

[2]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[3]  E. Tirapegui,et al.  Faraday's instability in viscous fluid , 1998, Journal of Fluid Mechanics.

[4]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[5]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[6]  Timothy Blass,et al.  Stability of Ordinary Differential Equations with Colored Noise Forcing , 2013, SIAM J. Control. Optim..

[7]  R. Khasminskii.  Stability of Stochastic Differential Equations , 2012 .

[8]  Kumar.  Mechanism for the faraday instability in viscous liquids , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[10]  Handbook of stochastic methods volume 13 of the Springer series in synergetics , 1984 .

[11]  J. Gollub,et al.  Phenomenological model of chaotic mode competition in surface waves , 1985 .

[12]  J. Casademunt,et al.  Study of the parametric oscillator driven by narrow‐band noise to model the response of a fluid surface to time‐dependent accelerations , 1993 .

[13]  Enrique Tirapegui,et al.  Faraday's Instability for Viscous Fluids , 1997 .

[14]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[15]  N. Namachchivaya,et al.  Application of stochastic averaging for nonlinear dynamical systems with high damping , 1988 .

[16]  Thomas Brooke Benjamin,et al.  The stability of the plane free surface of a liquid in vertical periodic motion , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[17]  George Papanicolaou,et al.  Asymptotic theory of mixing stochastic ordinary differential equations , 2010 .

[18]  R. Repetto,et al.  Finite amplitude Faraday waves induced by a random forcing , 2002 .

[19]  P. Spanos,et al.  Stochastic averaging: An approximate method of solving random vibration problems , 1986 .

[20]  K. Kumar.  Linear theory of Faraday instability in viscous liquids , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[21]  S. Fauve,et al.  Effect of multiplicative noise on parametric instabilities , 2003 .

[22]  Michael Faraday,et al.  XVII. On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces , 1831, Philosophical Transactions of the Royal Society of London.

[23]  E. M. Lifshitz,et al.  Course in Theoretical Physics , 2013 .