Stochastic Stability of Coupled Oscillators With One Asymptotically Stable and One Critical Mode

An analytic explanation is given for the experimental results reported by Popp and Romberg (2001, “Influence of Stochastic Effects on Flow Induced Vibrations in Tube Bundles,” IUTAM Symposium on Nonlinearity and Stochastic Structural Dynamics (Solid Mechanics and Its Applications), S. Narayanan and R. N. Iyengar, eds., Kluwer Academic, Dordrecht, Vol. 85, pp. 197–208) on fluid flow over tube bundles by using the concept of the maximal Lyapunov exponent. The motion of one tube in the bundle is modeled as a two-degree-of-freedom (four dimensional) system with one critical mode and one asymptotically stable mode driven by a small intensity stochastic process. We obtain a general asymptotic approximation for the maximal Lyapunov exponent for this four dimensional system and explain how the stochastic components that couple the critical and stable modes play an important role in determining whether a noisy excitation can stabilize or destabilize the oscillatory critical mode.