A novel embedding method for characterization of low-dimensional nonlinear dynamical systems

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.

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Characterization of Fat Fractals in Nonlinear Dynamical Systems

Physical Review Letters ◽

10.1103/physrevlett.57.2333 ◽

1986 ◽

Vol 57 (19) ◽

pp. 2333-2336 ◽

Cited By ~ 25

Author(s):

R. Eykholt ◽

D. K. Umberger

Keyword(s):

Dynamical Systems ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical

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Characterization of Fat Fractals in Nonlinear Dynamical Systems

Physical Review Letters ◽

10.1103/physrevlett.58.961.2 ◽

1987 ◽

Vol 58 (9) ◽

pp. 961-961

Author(s):

R. Eykholt ◽

D. K. Umberger

Keyword(s):

Dynamical Systems ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical

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

Brain Sciences ◽

10.3390/brainsci10080536 ◽

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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