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.

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.

scholarly journals NONLINEAR DYNAMICS OF MOVING CURVES AND SURFACES: APPLICATIONS TO PHYSICAL SYSTEMS

2005 ◽  
Vol 15 (01) ◽  
pp. 51-63 ◽  
Author(s):  
S. MURUGESH ◽  
M. LAKSHMANAN
Keyword(s):  
Dynamical Systems ◽  
Soft Matter ◽  
Evolution Equations ◽  
Nonlinear Dynamical Systems ◽  
Dimensional Space ◽  
Classical Mechanics ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Dynamical ◽  
Physical Systems ◽  
Curves And Surfaces

The subject of moving curves (and surfaces) in three-dimensional space (3-D) is a fascinating topic not only because it represents typical nonlinear dynamical systems in classical mechanics, but also finds important applications in a variety of physical problems in different disciplines. Making use of the underlying geometry, one can very often relate the associated evolution equations to many interesting nonlinear evolution equations, including soliton possessing nonlinear dynamical systems. Typical examples include dynamics of filament vortices in ordinary and superfluids, spin systems, phases in classical optics, various systems encountered in physics of soft matter, etc. Such interrelations between geometric evolution and physical systems have yielded considerable insight into the underlying dynamics. We present a succinct tutorial analysis of these developments in this article, and indicate further directions. We also point out how evolution equations for moving surfaces are often intimately related to soliton equations in higher dimensions.

scholarly journals Hopf Bifurcations in Complex Multiagent Activity: The Signature of Discrete to Rhythmic Behavioral Transitions

2020 ◽  
Vol 10 (8) ◽  
pp. 536
Author(s):  
Gaurav Patil ◽  
Patrick Nalepka ◽  
Rachel W. Kallen ◽  
Michael J. Richardson
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Limit Cycle ◽  
Nonlinear Dynamical Systems ◽  
Rhythmic Movements ◽  
Human Actions ◽  
Nonlinear Dynamical ◽  
Behavioral Transitions ◽  
Small Set ◽  
Low Dimensional

Most human actions are composed of two fundamental movement types, discrete and rhythmic movements. These movement types, or primitives, are analogous to the two elemental behaviors of nonlinear dynamical systems, namely, fixed-point and limit cycle behavior, respectively. Furthermore, there is now a growing body of research demonstrating how various human actions and behaviors can be effectively modeled and understood using a small set of low-dimensional, fixed-point and limit cycle dynamical systems (differential equations). Here, we provide an overview of these dynamical motorprimitives and detail recent research demonstrating how these dynamical primitives can be used to model the task dynamics of complex multiagent behavior. More specifically, we review how a task-dynamic model of multiagent shepherding behavior, composed of rudimentary fixed-point and limit cycle dynamical primitives, can not only effectively model the behavior of cooperating human co-actors, but also reveals how the discovery and intentional use of optimal behavioral coordination during task learning is marked by a spontaneous, self-organized transition between fixed-point and limit cycle dynamics (i.e., via a Hopf bifurcation).

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