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.

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Abstract. This paper covers application of the novel method of Lyapunov exponents (LEs) spectrum estimation in non smooth mechanical systems. In the presented method, LEs are obtained from a Poincaré map. By analysing the map instead of the full trajectory, problems with transition of perturbations through discontinuities can be avoided. However, the explicit formula of the map is usually not known. Therefore, the Jacobi matrix of the map is estimated using small perturbations of the initial point. In such a manner, direct calculation of the Jacobi matrix can be avoided. The article provides a detailed description of the method accompanied by clear schemes. The algorithm of Jacobi matrix estimation is elaborated and an example is given. Efficiency of the method is confirmed by a numerical experiment. The mechanical oscillator with impact has been simulated. Bifurcation diagrams and Lyapunov exponents graphs have been generated. It has been shown that the method provides values of the whole Lyapunov exponents spectrum with high accuracy.

For an X = ∅ and a given family F ⊂ P(X) \ {∅}, we consider the Marczewski field S(F) which consists of sets A ⊂ X such that each set U ∈ F contains a set V ∈ F with V ⊂ A or V ∩ A = ∅. We also study the respective ideal S 0 (F). We show general properties of S(F) and certain representation theorems. For instance we prove that the interval algebra in [0, 1) is a Marczewski field. We are also interested in situations where S(F) = S(τ \ {∅}) for a topology τ on X. We propose a general method which establishes S(F) and S 0 (F) provided that F is the family of perfect sets with respect to τ , and τ is a certain ideal topology on R connected with measure or category.

Controlling dynamics of nonlinear systems is one of the most important issues in science and engineering. Thus, there is continuous need to study and develop numerical algorithms of control methods. Among the most frequently applied invariants characterizing different aspects of a systems’ dynamics are Lyapunov exponents, fast Lyapunov index, angles of small deviations, fractal dimension or entropy. There exist many different methods of estimation of these indicators. In this paper, modification of our novel method is presented. We have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. Results of efficiency measurements for typical mechanical, electrical and random systems were discussed. We have proved that discussed modification introduced to our method makes the LLE estimation 17–53% faster than using classical algorithms. In addition, unlike the results presented in our previous publication, an improvement in performance was achieved for each of the analyzed cases. As such, the new approach lends to prospective application of LLE not only in dynamical systems' stability investigations, but also in real-time control of systems since the basic calculations and fast, effective method of LLE estimation can be applied even in simple microcontrollers. Our approach could be also applied in investigations of vector field properties, global stability or basins of attraction analyses, allowing for huge time savings.