Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise

Asymptotic expansions for the exponential growth rate, known as the Lyapunov exponent, and rotation numbers for two coupled oscillators driven by real noise are constructed. Such systems arise naturally in the investigation of the stability of steady-state motions of nonlinear dynamical systems and in parametrically excited linear mechanical systems. Almost-sure stability or instability of dynamical systems depends on the sign of the maximal Lyapunov exponent. Stability conditions are obtained under various assumptions on the infinitesimal generator associated with real noise provided that the natural frequencies are noncommensurable. The results presented here for the case of the infinitesimal generator having a simple zero eigenvalue agree with recent results obtained by stochastic averaging, where approximate ItÔ equations in amplitudes and phases are obtained in the sense of weak convergence.Dedicated to Thomas K. Caughey on the occasion of his 65th birthday.     

Normal form for generalized Hopf bifurcation with non-semisimple 1 ∶ 1 resonance

The primary result of this research is the derivation of an explicit formula for the Poincaré-Birkhoff normal form of the generalized Hopf bifurcation with non-semisimple 1:1 resonance. The classical nonuniqueness of the normal form is resolved by the choice of complementary space which yields a unique equivariant normal form. The 4 leading complex constants in the normal form are calculated in terms of the original coefficients of both the quadratic and cubic nonlinearities by two different algorithms. In addition, the universal unfolding of the degenerate linear operator is explicitly determined. The dominant normal forms are then obtained by rescaling the variables. Finally, the methods of averaging and normal forms are compared. It is shown that the dominant terms of the equivariant normal form are, indeed, the same as those of the averaged equations with a particular choice for the constant of integration.Partially supported by NSF through grant MSS 90-57437, AFOSR through grant 91-0041 and NSERC of Canada.     

Stability of noisy nonlinear auto-parametric systems

The purpose of this work is to examine the stationary motion and stability properties of stationary motion of two degree-of-freedom noisy auto-parametric systems We shall use analytical techniques to extend the existing results to examine such multi-dimensional nonlinear systems with noise, and in particular additive white noise. We obtain an approximation for the top Lyapunov exponent, the exponential growth rate, of the response of the so-called single-mode stationary motion. We show analytically that the top Lyapunov exponent is positive, and for small values of noise intensity ɛ and dissipation ɛ² the exponent grows in proportion with ɛ^2/3.     

Averaging of Noisy Nonlinear Systems with Rapidly Oscillating and Decaying Components

We consider a noisy n-dimensional nonlinear dynamical system containing rapidly oscillating and decaying components. We extend the results of Papanicolaou and Kohler and Namachchivaya and Lin; these results state that as the noise becomes smaller, a lower dimensional Markov process characterizes the limiting behavior. Our approach springs from a direct consideration of the martingale problem and considers both quadratic and cubic nonlinearities.     

Stochastically Perturbed Linear Gyroscopic Systems

ABSTRACT

Dynamic stability of linear conservative gyroscopic systems under stochastic parametric excitations of small intensity is examined. Conditions for mean square stability of dynamic response are obtained. Results are shown to depend only on those values of the excitation spectral density near twice the natural frequencies and the combination frequencies of the system. These results are applied to the problem of flow induced vibration in a supported pipe conveying fluid with pulsating velocity. The effects of mean flow velocity and virtual mass on the extent of parametric instability regions are then discussed.     

Delay equations with fluctuating delay related to the regenerative chatter II: analysis of reduced bifurcation equation

In this article, we study the reduced bifurcation equations of the nonlinear delay differential equations with periodic delays, which models the machine tool chatter with continuously modulated spindle speed to determine the periodic solutions and analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions close to the new stability boundary.     

Delay equations with fluctuating delay related to the regenerative chatter I: reduced bifurcation equation

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. The suppression of regenerative chatter by spindle speed variation is attracting increasing attention. In this paper, we study nonlinear delay differential equations with periodic delays which models the machine tool chatter with continuously modulated spindle speed. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov–Schmidt Reduction method.     

Particle Filters with Nudging in Multiscale Chaotic Systems: With Application to the Lorenz ’96 Atmospheric Model

This paper presents reduced-order nonlinear filtering schemes based on a theoretical framework that combines stochastic dimensional reduction and nonlinear filtering. Here, dimensional reduction is achieved for estimating the slow-scale process in a multiscale environment by constructing a filter using stochastic averaging results. The nonlinear filter is approximated numerically using the ensemble Kalman filter and particle filter. The particle filter is further adapted to the complexities of inherently chaotic signals. In particle filters, an ensemble of particles is used to represent the distribution of the state of the hidden signal. The ensemble is updated using observation data to obtain the best representation of the conditional density of the true state variables given observations. Particle methods suffer from the “curse of dimensionality,” an issue of particle degeneracy within a sample, which increases exponentially with system dimension. Hence, particle filtering in high dimensions can benefit from some form of dimensional reduction. A control is superimposed on particle dynamics to drive particles to locations most representative of observations, in other words, to construct a better prior density. The control is determined by solving a classical stochastic optimization problem and implemented in the particle filter using importance sampling techniques.

     

Method of stochastic normal forms

—The method of normal forms, originally developed for deterministic non-linear dynamical systems, is extended to include stochastic excitations, with the objective of obtaining an optimal reduction of dimensionality of the system while retaining its essential dynamic characteristics. Similar to the deterministic case, the crucial step in the normal-form computation is to find the so-called resonant terms which cannot be eliminated through a non-linear change of variables. Subsequent to the reduction of dimensionality, the associated stochastic normal form is obtained using a Markovian approximation. It is shown that the second order stochastic terms must be retained, in order to capture the stochastic contributions of the stable modes to the drift terms of the critical modes. Furthermore, for a specific class of non-linear systems, the results obtained from the stochastic normal form analysis are the same as those obtained from an extended stochastic averaging procedure. Thus, for this particular class, the two methods are equivalent.