19,091 research outputs found
The Steady-State Response of a Class of Dynamical Systems to Stochastic Excitation
In this paper a class of coupled nonlinear dynamical systems subjected to stochastic excitation is considered. It is shown how the exact steady-state probability density function for this class of systems can be constructed. The result is then applied to some classical oscillator problems
Time Discrete Approximation of Weak Solutions for Stochastic Equations of Geophysical Fluid Dynamics and Applications
As a first step towards the numerical analysis of the stochastic primitive
equations of the atmosphere and oceans, we study their time discretization by
an implicit Euler scheme. From deterministic viewpoint the 3D Primitive
Equations are studied with physically realistic boundary conditions. From
probabilistic viewpoint we consider a wide class of nonlinear, state dependent,
white noise forcings. The proof of convergence of the Euler scheme covers the
equations for the oceans, atmosphere, coupled oceanic-atmospheric system and
other geophysical equations. We obtain the existence of solutions weak in PDE
and probabilistic sense, a result which is new by itself to the best of our
knowledge
A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise
We develop a moment equation closure minimization method for the inexpensive
approximation of the steady state statistical structure of nonlinear systems
whose potential functions have bimodal shapes and which are subjected to
correlated excitations. Our approach relies on the derivation of moment
equations that describe the dynamics governing the two-time statistics. These
are combined with a non-Gaussian pdf representation for the joint
response-excitation statistics that has i) single time statistical structure
consistent with the analytical solutions of the Fokker-Planck equation, and ii)
two-time statistical structure with Gaussian characteristics. Through the
adopted pdf representation, we derive a closure scheme which we formulate in
terms of a consistency condition involving the second order statistics of the
response, the closure constraint. A similar condition, the dynamics constraint,
is also derived directly through the moment equations. These two constraints
are formulated as a low-dimensional minimization problem with respect to
unknown parameters of the representation, the minimization of which imposes an
interplay between the dynamics and the adopted closure. The new method allows
for the semi-analytical representation of the two-time, non-Gaussian structure
of the solution as well as the joint statistical structure of the
response-excitation over different time instants. We demonstrate its
effectiveness through the application on bistable nonlinear
single-degree-of-freedom energy harvesters with mechanical and electromagnetic
damping, and we show that the results compare favorably with direct Monte-Carlo
Simulations
Mean field games based on the stable-like processes
In this paper, we investigate the mean field games with K classes of agents who are weakly coupled via the empirical measure. The underlying dynamics of the representative agents is assumed to be a controlled nonlinear Markov process associated with rather general integro-differential generators of LĀ“evy-Khintchine type (with variable coefficients), with the major stress on applications to stable and stable- like processes, as well as their various modifications like tempered stable-like processes or their mixtures with diffusions. We show that nonlinear measure-valued kinetic equations describing the dynamic law of large numbers limit for system with large number N of agents are solvable and that their solutions represent 1/N-Nash equilibria for approximating systems of N agents
Volterra-series approach to stochastic nonlinear dynamics: linear response of the Van der Pol oscillator driven by white noise
The Van der Pol equation is a paradigmatic model of relaxation oscillations.
This remarkable nonlinear phenomenon of self-sustained oscillatory motion
underlies important rhythmic processes in nature and electrical engineering.
Relaxation oscillations in a real system are usually coupled to environmental
noise, which further enriches their dynamics, but makes theoretical analysis of
such systems and determination of the equation's parameter values a difficult
task. In a companion paper we have proposed an analytic approach to a similar
problem for another classical nonlinear model, the bistable Duffing oscillator.
Here we extend our techniques to the case of the Van der Pol equation driven by
white noise. We analyze the statistics of solutions and propose a method to
estimate parameter values from the oscillator's time series. We use
experimental data of active oscillations in a biological system to demonstrate
how our method applies to real observations and how it can be generalized for
more complex models.Comment: 12 pages, 6 figures, 1 tabl
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