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.

GENERALIZED SEMIFLOWS AND CHAOS IN MULTIVALUED DYNAMICAL SYSTEMS

2012 ◽  
Vol 26 (25) ◽  
pp. 1246016
Author(s):  
ZDENĚK BERAN ◽  
SERGEJ ČELIKOVSÝ
Keyword(s):  
Dynamical Systems ◽  
Differential Inclusion ◽  
Chaotic Behavior ◽  
Existence Of A Solution ◽  
Right Hand ◽  
Multivalued Dynamical Systems ◽  
The One ◽  
The Right ◽  
Generalized Semiflows

This contribution addresses a possible description of the chaotic behavior in multivalued dynamical systems. For the multivalued system formulated via differential inclusion the practical conditions on the right-hand side are derived to guarantee existence of a solution, which leads to the chaotic behavior. Our approach uses the notion of the generalized semiflow but it does not require construction of a selector on the set of solutions. Several applications are provided by concrete examples of multivalued dynamical systems including the one having a clear physical motivation.

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.

scholarly journals Bounded Motions of the Dynamical Systems Described by Differential Inclusions

2009 ◽  
Vol 2009 ◽  
pp. 1-9
Author(s):  
Nihal Ege ◽  
Khalik G. Guseinov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Sufficient Conditions ◽  
Invariant Sets ◽  
Control Vector ◽  
Upper Semicontinuous ◽  
Right Hand ◽  
The Right

The boundedness of the motions of the dynamical system described by a differential inclusion with control vector is studied. It is assumed that the right-hand side of the differential inclusion is upper semicontinuous. Using positionally weakly invariant sets, sufficient conditions for boundedness of the motions of a dynamical system are given. These conditions have infinitesimal form and are expressed by the Hamiltonian of the dynamical system.

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.

On Numerical Integration of Discontinuous Dynamical Systems

2017 ◽  
Vol 27 (14) ◽  
pp. 1750218 ◽  
Author(s):  
Marius-F. Danca ◽  
Michal Fec̆kan
Keyword(s):  
Dynamical Systems ◽  
Numerical Methods ◽  
Numerical Integration ◽  
Vector Fields ◽  
Continuous Functions ◽  
Numerical Integrations ◽  
Right Hand ◽  
Discontinuous Vector Fields ◽  
The Right ◽  
Continuous Approximations

This paper addresses an important issue in numerical integration of dynamical systems, integer- or fractional-order, with discontinuous vector fields. It is shown that these systems cannot be solved using numerical methods designed for ODEs with continuous functions on the right-hand side, therefore we have to resort to special schemes and procedures in numerical integrations such as continuous approximations of the right-hand sides of the ODEs.

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