On diagrammatic technique for nonlinear dynamical systems

In this paper, we investigate phase flows over ℂn and ℝn generated by vector fields V = ∑ Pi∂i where Pi are finite degree polynomials. With the convenient diagrammatic technique, we get expressions for evolution operators ev {V|t} : x(0) ↦ x(t) through the series in powers of x(0) and t, represented as sum over all trees of a particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.

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Global vector-field reconstruction of nonlinear dynamical systems from a time series with SVD method and validation with Lyapunov exponents

Chinese Physics ◽

10.1088/1009-1963/12/12/005 ◽

2003 ◽

Vol 12 (12) ◽

pp. 1366-1373 ◽

Cited By ~ 10

Author(s):

Liu Wei-Dong ◽

K. F Ren ◽

S Meunier-Guttin-Cluzel ◽

G Gouesbet

Keyword(s):

Time Series ◽

Vector Field ◽

Dynamical Systems ◽

Lyapunov Exponents ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical ◽

Field Reconstruction ◽

Global Vector ◽

Svd Method

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Nonlinear Dynamical Systems and Decompositions of Vector Fields

Physica Scripta ◽

10.1238/physica.regular.072a00016 ◽

2005 ◽

Vol 72 (1) ◽

pp. 16-18 ◽

Cited By ~ 5

Author(s):

Willi-Hans Steeb ◽

Timothy Scholes

Keyword(s):

Dynamical Systems ◽

Vector Fields ◽

Nonlinear Dynamical Systems ◽

Nonlinear Dynamical

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Nonlinear Dynamical Systems, Their Stability, and Chaos

Applied Mechanics Reviews ◽

10.1115/1.4026864 ◽

2014 ◽

Vol 66 (2) ◽

Cited By ~ 1

Author(s):

Amol Marathe ◽

Rama Govindarajan

Keyword(s):

Fluid Flow ◽

Dynamical Systems ◽

Nonlinear Systems ◽

Fluid Mechanics ◽

Fixed Points ◽

Chaotic Systems ◽

Nonlinear Dynamical Systems ◽

Not Given ◽

Nonlinear Dynamical ◽

Detailed Explanation

This introduction to nonlinear systems is written for students of fluid mechanics, so connections are made throughout the text to familiar fluid flow systems. The aim is to present how nonlinear systems are qualitatively different from linear and to outline some simple procedures by which an understanding of nonlinear systems may be attempted. Considerable attention is paid to linear systems in the vicinity of fixed points, and it is discussed why this is relevant for nonlinear systems. A detailed explanation of chaos is not given, but a flavor of chaotic systems is presented. The focus is on physical understanding and not on mathematical rigor.

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SYNCHRONIZATION OF A CLASS OF SECOND-ORDER NONLINEAR SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408022500 ◽

2008 ◽

Vol 18 (11) ◽

pp. 3461-3471 ◽

Cited By ~ 1

Author(s):

A. P. MIJOLARO ◽

L. F. C. ABERTO ◽

N. G. BRETAS

Keyword(s):

Asymptotic Behavior ◽

Vector Field ◽

Dynamical Systems ◽

Nonlinear Dynamical Systems ◽

Second Order ◽

Coupled Systems ◽

Nonlinear Dynamical ◽

Unbounded Solutions ◽

Synchronization Region ◽

Coupling Parameters

The asymptotic behavior of a class of coupled second-order nonlinear dynamical systems is studied in this paper. Using very mild assumptions on the vector-field, conditions on the coupling parameters that guarantee synchronization are provided. The proposed result does not require solutions to be ultimately bounded in order to prove synchronization, therefore it can be used to study coupled systems that do not globally synchronize, including synchronization of unbounded solutions. In this case, estimates of the synchronization region are obtained. Synchronization of two-coupled nonlinear pendulums and two-coupled Duffing systems are studied to illustrate the application of the proposed theory.

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