Narrow Results

We establish global-in-time averaging for the $L^2$-critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in $L^2$.

In systems which combine slow and fast motions the averaging principle says that a good approximation of the slow motion can be obtained by averaging its parameters in fast variables. This setup arises, for instance, in perturbations of Hamiltonian systems where motions on constant energy manifolds are fast and across them are slow. When these perturbations are deterministic Anosov's theorem says that the averaging principle works except for a small in measure set of initial conditions while Neistadt's theorem gives error estimates in the case of perturbations of integrable Hamiltonian systems. These results are extended here to the case of fast and slow motions given by stochastic differential equations.

Recently an averaging principle for diffusion processes with a null-recurrent fast component without a drift term was obtained in [4]. In this note this result is widened to allow a drift in the fast component. As a corollary a new result on the homogenization for parabolic PDE's is obtained.

Assuming that the fast motion in averaging is sufficiently well mixing we show that the slow motion can be approximated in the L²-sense by a diffusion solving Hasselmann's nonlinear stochastic differential equation and which provides a much better approximation than the one suggested by the averaging principle. Previously, only weak limit theorems in averaging were known which cannot justify, in principle, a nonlinear diffusion approximation of the slow motion.

Quasi-linear perturbations of a two-dimensional flow with a first integral and the corresponding parabolic PDEs with a small parameter at the second-order derivatives are considered in this paper.

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

We extend the Erdős–Rényi law of large numbers to the averaging setup both in discrete and continuous time cases. We consider both stochastic processes and dynamical systems as fast motions whenever they are fast mixing and satisfy large deviations estimates. In the continuous time case we consider flows with large deviations estimates which allow a suspension representation and it turns out that fast mixing of corresponding base transformations suffices for our results.

In this paper, we consider a fully-coupled slow–fast system of McKean–Vlasov stochastic differential equations with full dependence on the slow and fast component and on the law of the slow component and derive convergence rates to its homogenized limit. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. In the course of the proof we obtain related ergodic theorems and we gain results on the regularity of Poisson type of equations and of the associated Cauchy problem on the Wasserstein space that are of independent interest.

We study semi-linear evolution inclusion in a Banach space E with uniformly convex dual E *. Averaging and partial averaging results on a finite interval is proved, when A is linear m-dissipative and F, G are one sided Lipschitz. When F(t, ·) is one sided Lipschitz with negative constant we prove such results on an infinite interval.

Has applied mathematics disappeared behind the computer screen? High level computer languages and high capacity computers for simulation followed by data mining give powerful tools for studying complex systems. The two examples presented here are a system of difference equations from population genetics and a system of Volterra integral equations from demography, both perturbed by random noise. We analyze these systems using averaging methods and then compare the derived solution with computer simulations of the system. In the first case, mathematical analysis guides mining of the simulated solution data base. In the second, mathematical analysis reveals deterministic chaotic behavior in the averaged system that confounds the simulation/data-mining approach.

We overview several analytic methods of predicting the emergence of chaotic motion in nonlinear oscillatory systems. A special attention is given to the second method of Lyapunov, a technique that has been widely used in the analysis of stability of motion in the theory of dynamical systems but received little attention in the context of chaotic systems analysis. We show that the method allows formulating a necessary condition for the appearance of chaos in nonlinear systems. In other terms, it provides an analytic estimate of an area in the space of control parameters where the largest Lyapunov exponent is strictly negative. A complementary area thus comprises the values of controls, where the exponent can take positive values, and hence the motion can become chaotic. Contrary to other commonly used methods based on perturbation analysis, such as e.g., Melnikov criterion, harmonic balance, or averaging, our approach demonstrates superior performance at large values of the parameters of dissipation and nonlinearity. Several classical examples including mathematical pendulum, Duffing oscillator, and a system of two coupled oscillators, are analyzed in detail demonstrating advantages of the proposed method compared to other existing techniques.

We study Melnikov conditions predicting appearance of chaos in Duffing oscillator with hardening type of non-linearity under two-frequency excitation acting in the vicinity of the principal resonance. Since Hamiltonian part of the system contains no saddle points, Melnikov method cannot be applied directly. After separating the external force into two parts, we use a perturbation analysis that allows recasting the original system to the form suitable for Melnikov analysis. At the initial step, we perform averaging at one of the frequencies of the external force. The averaged equations are then analyzed by traditional Melnikov approach, considering the second frequency component of the external force and the dissipation term as perturbations. The numerical study of the conditions for homoclinic bifurcation found by Melnikov theory is performed by varying the control parameters of amplitudes and frequencies of the harmonic components of the external force. The predictions from Melnikov theory have been further verified numerically by integrating the governing differential equations and finding areas of chaotic behavior. Mismatch between the results of theoretical analysis and numerical experiment is discussed.