Abstract
Classical method of Lyapunov exponents spectrum estimation for a n-th-order continuous-time, smooth dynamical system involves Gram–Schmidt orthonormalization and calculations of perturbations lengths logarithms. In this paper, we have shown that using a new, simplified method, it is possible to estimate full spectrum of n Lyapunov exponents by integration of (n-1) perturbations only. In particular, it is enough to integrate just one perturbation to obtain two largest Lyapunov exponents, which enables to search for hyperchaos. Moreover, in the presented algorithm, only very basic mathematical operations such as summation, multiplication or division are applied, which boost the efficiency of computations. All these features together make the new method faster than any other known by the authors if the order of the system under consideration is low. Correctness the method has been tested for three examples: Lorenz system, Duffing oscillator and three Duffing oscillators coupled in the ring scheme. Moreover, efficiency of the method has been confirmed by two practical tests. It has been revealed that for low-order systems, the presented method is faster than any other known by authors.
Highlights
Depending on a dynamical system type and a kind of information that is useful for its investigations, different types of invariants characterizing system dynamics are applied
In order to verify our method of the Lyapunov exponents spectrum estimation, three systems have been analyzed: Lorenz equations, Duffing oscillator with external periodic driving forcing and three Duffing systems coupled in a ring scheme
The presented article shows that from dot products of perturbations vectors and their derivatives, one can extract the full spectrum of Lyapunov exponents
Summary
Depending on a dynamical system type and a kind of information that is useful for its investigations, different types of invariants characterizing system dynamics are applied. In order to estimate Lyapunov exponents from a scalar time series, the Takens procedure [13] can be applied This approach can be utilized in the cases when discontinuities or time delays exist in the analyzed system. Has been revealed that application of our new algorithm increases the efficiency of the calculations compared to the classical method [47] when the order of the system is low: 2, 4 or 6 Such acceleration has been achieved due to integration of (n − 1) perturbations instead of n, and owing to the simplicity of the algorithm, which involves only the basic mathematical operations: addition, multiplication, division. Authors claim that the method presented in this paper is the fastest one in the assumed range of applications if the order of the system is low enough
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