H. J. Van Roessel

Moment Lyapunov Exponent and Stochastic Stability of Two Coupled Oscillators Driven by Real Noise

In a recent paper an asymptotic approximation for the moment Lyapunov exponent, g(p), of two coupled oscillators driven by a small intensity real noise was obtained. The utility of that result is limited by the fact that it was obtained under the assumption that the moment p is small, a limitation which precludes, for example, the determination of the stability index. In this paper that limitation is removed and an asymptotic approximation valid for arbitrary p is obtained. The results are applied to study the moment stability of the stationary solutions of structural and mechanical systems subjected to stochastic excitation.

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.

Moment Lyapunov exponent for two coupled oscillators driven by real noise

In this paper, an approximation for the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response of two coupled oscillators driven by real noise, is constructed. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noise. The results are applied to study the moment stability of the stationary solutions of structural and mechanical systems subjected to stochastic excitation.

A centre-manifold analysis of variable speed machining

The mathematical models representing 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 model 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. We make used to the centre-manifold reduction and the method of normal forms to determine the periodic solutions and analyse the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions close to the new stability boundary.The mathematical models representing 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 model 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. We make used to the centre-manifold reduction and the method of normal forms to determine the periodic solutions and analyse the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions close to the new stability boundary.

Stochastic stability of coupled oscillators in resonance: A perturbation approach

A perturbation approach is used to obtain an approximation for the moment Lyapunov exponent of two coupled oscillators with commensurable frequencies driven by a small intensity real noise with dissipation. The generator for the eigenvalue problem associated with the moment Lyapunov exponent is derived without any restriction on the size of pth moment. An orthogonal expansion for the eigenvalue problem based on the Galerkin method is used to derive the stability results in terms of spectral densities. These results can be applied to study the moment and almost-sure stability of structural and mechanical systems subjected to stochastic excitation.

Some results on the coagulation equation

Eq. (1) represents the evolution of particles c( ; t) of size ≥ 0 at time t≥ 0 undergoing a change in size governed by reaction kernel K . A physical interpretation of the terms in (1) can be found in Melzak [5] or the review article by Drake [2]. Global existence and uniqueness of solutions to (1) has been proven by Aizenman and Bak [1] for constant kernels with suitable conditions imposed on the initial particle distribution c0. For the product kernel K( ; )= , McLeod [4] has shown, again for a suitably restricted initial particle distribution, that a mass conserving solution exists for some nite time. That solutions in this case need not conserve mass for all time was demonstrated by Stewart [6]. The phenomenon whereby conservation of mass breaks down in nite time is known as gelation and is physically interpreted as being caused by the appearance of an in nite “gel” or “superparticle” [3].

The mass-conserving solutions of Smoluchowski's coagulation equation: the general bilinear kernel

SummaryIt is proved that for the general bilinear kernel with arbitrary initial conditions, the solutions to the discrete coagulation equation can exhibit one of the following types of behaviour: conservation of mass for all time, conservation of mass for a finite time only, or instantaneous gelation.

Asymptotic stability of structural systems based on Lyapunov exponents and moment Lyapunov exponents

An asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response, have been obtained for systems driven by a small intensity real noise process. The systems under consideration are general four-dimensional dynamical systems with two critical modes. Almost-sure and moment stability conditions are obtained provided the natural frequencies of these critical modes are non-commensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. In this paper, the results obtained are applied to a thin rectangular beam under the action of a stochastic follower force and a model of a vehicle traveling over a rough road. The stability regions predicted by the two different criteria are then compared.

Maximal Lyapunov exponent and almost-sure stability for coupled two-degree-of-freedom stochastic systems

A perturbation approach is used to calculate the asymptotic growth rate of stochastically coupled two degree of freedom systems. The noise is assumed to be white and of small intensity in order to calculate the explicit asymptotic formulas for the maximum Lyapunov exponent. The Lyapunov exponents and rotation number for each degree of freedom are obtained in the appendix. The almost sure stability or instability of the four dimensional stochastic system depends on the sign of the maximum Lyapunov exponent

Scaling behaviour in a coagulation?annihilation model and Lotka?Volterra competition systems

FPC, JTP e RS foram parcialmente financiados pelo CAMGSD-LARSyS atraves do financiamento plurianual atribuido pela Fundacao para a Ciencia e Tecnologia (Portugal)