Non-Linear Systems Under Parametric Alpha-Stable Le´vy White Noises

2005 ◽  
Author(s):  
Mario Di Paola ◽  
Antonina Pirrotta ◽  
Massimiliano Zingales
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Characteristic Function ◽  
Stochastic Analysis ◽  
Nonlinear Dynamical Systems ◽  
Parametrically Excited ◽  
Nonlinear Dynamical ◽  
Non Linear Systems ◽  
The Fourier Transform ◽  
White Noises

In this study stochastic analysis of nonlinear dynamical systems under a-stable, multiplicative white noise has been performed. Analysis has been conducted by means of the Itoˆ rule extended to the case of α-stable noises. In this context the order of increments of Levy process has been evaluated and differential equations ruling the evolutions of statistical moments of either parametrically and external dynamical systems have been obtained. The extended Itoˆ rule has also been used to yield the differential equation ruling the evolution of the characteristic function for parametrically excited dynamical systems. The Fourier transform of the characteristic function, namely the probability density function is ruled by the extended Einstein-Smoluchowsky differential equation to case of parametrically excited dynamical systems. Some numerical applications have been reported to assess the reliability of the proposed formulation.

Singularities in the complex time plane and exactly solvable dynamical systems

1994 ◽  
Vol 72 (3-4) ◽  
pp. 147-151 ◽  
Author(s):  
W.-H. Steeb ◽  
Assia Fatykhova
Keyword(s):  
Differential Equation ◽  
Quantum Mechanics ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Numerical Study ◽  
Complex Time ◽  
Nonlinear Dynamical ◽  
Singularity Structure ◽  
Exactly Solvable ◽  
Solvable Dynamical Systems

A powerful tool in the study of nonlinear dynamical systems is the investigation of the singularity structure in the complex time plane. In most cases the singularity structure can only be found numerically. Here we give two models that can be solved exactly, i.e., we can give the singularities in the complex time plane. The two models play a central role in quantum mechanics. Then we compare them with the numerical study of the nonlinear differential equation.

scholarly journals Convolutionless Nakajima–Zwanzig equations for stochastic analysis in nonlinear dynamical systems

2014 ◽  
Vol 470 (2166) ◽  
pp. 20130754 ◽  
Author(s):  
D. Venturi ◽  
G. E. Karniadakis
Keyword(s):  
Dynamical Systems ◽  
Stochastic Analysis ◽  
Evolution Equations ◽  
Nonlinear Dynamical Systems ◽  
Coarse Graining ◽  
Perturbation Series ◽  
Stochastic Nonlinear Systems ◽  
Nonlinear Dynamical ◽  
Major Interest ◽  
Low Dimensional

Determining the statistical properties of stochastic nonlinear systems is of major interest across many disciplines. Currently, there are no general efficient methods to deal with this challenging problem that involves high dimensionality, low regularity and random frequencies. We propose a framework for stochastic analysis in nonlinear dynamical systems based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g. functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima–Zwanzig–Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance and stochastic advection–reaction problems.

scholarly journals Aggregation of a class of large-scale, interconnected, nonlinear dynamical systems

2001 ◽  
Vol 7 (4) ◽  
pp. 379-392 ◽  
Author(s):  
Swaroop Darbha ◽  
K. R. Rajagopal
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Dynamical Systems ◽  
Large Scale ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Hierarchical Systems ◽  
Nonlinear Dynamical ◽  
Analysis And Design ◽  
Computational Perspective

In this paper, the authors consider the issue of the construction of a meaningful average for a collection of nonlinear dynamical systems. Such a collection of dynamical systems may or may not have well defined ensemble averages as the existence of ensemble averages is predicated on the specification of appropriate initial conditions. A meaningful “average” dynamical system can represent the macroscopic behavior of the collection of systems and allow us to infer the behavior of such systems on an average. They can also prove to be very attractive from a computational perspective. An advantage to the construction of the meaningful average is that it involves integrating a nonlinear differential equation, of the same order as that of any member in the collection. An average dynamical system can be used in the analysis and design of hierarchical systems, and will allow one to capture approximately the response of any member of the collection.

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