Classical method of Lyapunov exponents spectrum estimation for a n-th-order continuous-time, smooth dynamical system involves Gram–Schmidt orthonormalization and calculations perturbations lengths logarithms. In this paper, we have shown that using new, simplified method, it is possible to estimate full n by integration $$(n-1)$$ only. particular, enough integrate just one perturbation obtain two largest exponents, which enables search hyperchaos. Moreover, in the presented algorithm, only...

This paper presents a novel, simple method of Lyapunov Exponents (LEs) spectrum estimation for non-smooth and discontinuous systems. The presented algorithm works continuous-time dynamical systems, as well discrete maps. Its simplicity flexibility enables to estimate LEs complex, systems in an easy manner. starts with comprehensive review the state-of-art methods computation Then, novel is introduced. Further on, examples its use are provided. applications include maps, mechanical...

Controlling system dynamics with use of the Largest Lyapunov Exponent (LLE) is employed in many different areas scientific research. Thus, there still need to elaborate fast and simple methods LLE calculation. This article second part one presented Dabrowski (Nonlinear Dyn 67:283–291, 2012). It develops method LLEDP estimation shows that from time series two identical systems, can simply extract value stability parameter which be treated as largest LLE. Unlike part, developed (LLEDPT)...

Abstract In this paper the construction of a neural-network based closed-loop control discontinuous capsule drive is analyzed. The foundation designed controller an optimized open-loop function. A neural network used to determine dependence between output and state system. Robustness with respect variation parameters controlled system analyzed compared original control. It expected that presented method can facilitate controllers for which alternative methods are not effective, such as...

The behavior of systems coupled nonlinear oscillators and, connected with it, the synchronization phenomena are significant interest in many areas science. One most important problems this field is stability synchronous state. often applied tool which allows one to quantify largest Transversal Lyapunov Exponent (TLE) Master Stability Function (MSF) theory (Pecora and Carroll Phys. Rev. Lett. 80:2109, 1998). Thus there still need elaborate fast simple methods TLE calculation. new method...

Existing methods of determining vehicle velocity based on measurements body deformation have been linear models. This means that the relationship between equivalent energy speed (EES), which is a measure kinetic lost during collision cars, and value ratio Cs, representing average deformation, has sought. The aim this publication to present nonlinear innovative quick method, or its simpler form, using linearisation in seven consecutive ranges, for EES basis averaged Cs. most critical factor...

This paper covers application of the novel method Lyapunov exponents (LEs) spectrum estimation in non smooth mechanical systems. In presented method, LEs are obtained from a Poincaré map. By analysing map instead full trajectory, problems with transition perturbations through discontinuities can be avoided. However, explicit formula is usually not known. Therefore, Jacobi matrix estimated using small initial point. such manner, direct calculation The article provides detailed description...

Abstract Controlling dynamics of complex systems is one the most important issues in science and engineering. Thus, there continuous need to study develop numerical algorithms control methods. In this paper, we would like present our introductory a new simple method investigations such based on vector field properties reduced amount applied information. Firstly, basis approach for extraction nonlinear indicators two-dimensional systems. We show that basing simplified analyses exploiting half...

This paper presents a practical application of new, simplified method Lyapunov exponents estimation. The has been applied to optimization real, nonlinear inverted pendulum system. Authors presented how the algorithm Largest Exponent (LLE) estimation can be evaluate control systems performance. new LLE-based performance index proposed. Equations system fourth order have found. friction regulation object identified by means least squares method. Three different models tested: linear, cubic and...

Abstract Controlling dynamics of nonlinear systems is one the most important issues in science and engineering. Thus, there continuous need to study develop numerical algorithms control methods. Among frequently applied invariants characterizing different aspects a systems’ are Lyapunov exponents, fast index, angles small deviations, fractal dimension or entropy. There exist many methods estimation these indicators. In this paper, modification our novel method presented. We have shown that...