Ljapunov Exponents, Hyperchaos and Hurst Exponent

2005 ◽  
Vol 60 (4) ◽  
pp. 252-254 ◽  
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.

On Global Transversality and Chaos in Two-Dimensional Nonlinear Dynamical Systems

2007 ◽  
Vol 2 (3) ◽  
pp. 242-248 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Approximate Expression ◽  
Melnikov Function ◽  
Two Dimensional ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Energy Increment ◽  
Exact Energy

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.

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

2014 ◽  
Vol 24 (07) ◽  
pp. 1430020 ◽  
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.

scholarly journals Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

2018 ◽  
Author(s):  
Geoff Boeing
Keyword(s):  
Dynamical Systems ◽  
Phase Diagrams ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Models ◽  
Logistic Map ◽  
Three Dimensional ◽  
Basins Of Attraction ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
One Dimensional Maps

Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

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