Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application

Meccanica ◽  
1980 ◽  
Vol 15 (1) ◽  
pp. 21-30 ◽  
Author(s):  
Giancarlo Benettin ◽  
Luigi Galgani ◽  
Antonio Giorgilli ◽  
Jean-Marie Strelcyn
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian Systems ◽  
Numerical Application ◽  
Characteristic Exponents ◽  
Lyapunov Characteristic Exponents

scholarly journals A New Way to Compute the Lyapunov Characteristic Exponents for Non-Smooth and Discontinues Dynamical Systems

2022 ◽  
Author(s):  
Mikhail E. Semenov ◽  
Sergei V. Borzunov ◽  
Peter A. Meleshenko
Keyword(s):  
Dynamical Systems ◽  
Classical Method ◽  
Van Der Pol Oscillator ◽  
Characteristic Exponents ◽  
Right Hand ◽  
Hysteresis Nonlinearity ◽  
Lyapunov Characteristic Exponents ◽  
Preisach Operator ◽  
Proposed Modification ◽  
The Right

Abstract One of the most important problems of nonlinear dynamics is related to the development of methods concerning the identification of the dynamical modes of the corresponding systems. The classical method is related to the calculation of the Lyapunov characteristic exponents ( LCEs ). Usually, to implement the classical algorithms for the LCEs calculation the smoothness of the right-hand sides of the corresponding equations is required. In this work, we propose a new algorithm for the LCEs computation in systems with strong nonlinearities (these nonlinearities can not be linearized ) including the hysteresis. This algorithm uses the values of the Jacobi matrix in the vicinity of singularities of the right-hand sides of the corresponding equations. The proposed modification of the algorithm is also can be used for systems containing such design hysteresis nonlinearity as the Preisach operator (thus, this modification can be used for all members of the hysteresis family). Moreover, the proposed algorithm can be successfully applied to the well-known chaotic systems with smooth nonlinearities . Examples of dynamical systems with hysteresis nonlinearities , such as the Lorentz system with hysteresis friction and the van der Pol oscillator with hysteresis block, are considered. These examples illustrate the efficiency of the proposed algorithm.

Approach to Compute the Lyapunov Characteristic Exponents for Continuous Dynamical Systems of Dimension Four

2009 ◽  
Author(s):  
Hubertus von Bremen
Keyword(s):  
Dynamical Systems ◽  
Numerical Example ◽  
Characteristic Exponents ◽  
Lyapunov Characteristic Exponents ◽  
General Method

In this paper an existing general method for accurately computing the Lyapunov Characteristic Exponents of continuous dynamical systems will be implemented for systems of dimension four. Previously, this approach has only been implemented for systems up to dimension three. A numerical example illustrating the accuracy of the implementation is presented.

ON THE CALCULATION OF LYAPUNOV CHARACTERISTIC EXPONENTS FOR CONTINUOUS-TIME LTV DYNAMICAL SYSTEMS USING DYNAMIC EIGENVALUES

2012 ◽  
Vol 22 (01) ◽  
pp. 1250019
Author(s):  
MIGUEL ÁNGEL GUTIÉRREZ DE ANDA
Keyword(s):  
Dynamical Systems ◽  
Continuous Time ◽  
Linear Time ◽  
Analytic Solutions ◽  
Mean Value ◽  
Time Varying ◽  
Characteristic Exponents ◽  
Lyapunov Characteristic Exponents ◽  
The Mean ◽  
Time Linear

The concept of the dynamic eigenvalues may be used, in principle, to formulate in a general way analytic solutions of continuous-time linear time-varying dynamical (LTV) systems. It has also been suggested that the mean value of these quantities may be used to calculate Lyapunov characteristic exponents for the aforementioned systems. In this article, it will be demonstrated that this conjecture is not necessarily valid.

scholarly journals Characteristic exponents of dynamical systems in metric spaces

1983 ◽  
Vol 3 (1) ◽  
pp. 119-127 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Dynamical Systems ◽  
Metric Spaces ◽  
Invariant Sets ◽  
Characteristic Exponents ◽  
Smooth Case

AbstractWe introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.

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