On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems

In this paper, the global transversality and tangency in two-dimensional nonlinear dynamical systems are discussed, and the exact energy increment function (L-function) for such nonlinear dynamical systems is presented. The Melnikov function is an approximate expression of the exact energy increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are derived. Numerical simulations are carried out for illustrations of the analytical conditions. From analytical and numerical results, the simple zero of the energy increment (or the Melnikov function) may not imply that chaos exists. The conditions for the global transversality and tangency to the separatrix may be independent of the Melnikov function. Therefore, the analytical criteria for chaotic motions in nonlinear dynamical systems need to be further developed. The methodology presented in this paper is applicable to nonlinear dynamical systems without any separatrix.

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Cleaning the Periodicity in Discrete- and Continuous-Time Nonlinear Dynamical Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414300201 ◽

2014 ◽

Vol 24 (07) ◽

pp. 1430020 ◽

Cited By ~ 2

Author(s):

Paulo C. Rech

Keyword(s):

Dynamical Systems ◽

Continuous Time ◽

Nonlinear Dynamical Systems ◽

Angular Frequency ◽

Two Dimensional ◽

Periodic Forcing ◽

Dimensional Parameter ◽

Parameter Spaces ◽

Nonlinear Dynamical ◽

First Order

We investigate periodicity suppression in two-dimensional parameter-spaces of discrete- and continuous-time nonlinear dynamical systems, modeled respectively by a two-dimensional map and a set of three first-order ordinary differential equations. We show for both cases that, by varying the amplitude of an external periodic forcing with a fixed angular frequency, windows of periodicity embedded in a chaotic region may be totally suppressed.

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Ljapunov Exponents, Hyperchaos and Hurst Exponent

Zeitschrift für Naturforschung A ◽

10.1515/zna-2005-0406 ◽

2005 ◽

Vol 60 (4) ◽

pp. 252-254 ◽

Cited By ~ 1

Author(s):

Willi-Hans Steeb ◽

Eugenio Cosme Andrieu

Keyword(s):

Dynamical Systems ◽

Hurst Exponent ◽

Nonlinear Dynamical Systems ◽

Logistic Map ◽

Two Dimensional ◽

Nonlinear Dynamical

Abstract We consider nonlinear dynamical systems with chaotic and hyperchaotic behaviour.We investigate the behaviour of the Hurst exponent at the transition from chaos to hyperchaos. A two-dimensional coupled logistic map is studied.

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