Chapter 3
Lyapunov Exponents and Methods of
Their Analysis
3.1 Introduction
The book is devoted to study the nonlinear phenomena exhibited by the sizedependent structural members including bifurcations and chaotic processes, and
hence this chapter provides an overview of one of the main tools for identifying
the nonlinear dynamics of these objects. Namely, the concept of Lyapunov exponents is briefly revisited, which allows us to distinguish between regular (periodic
or quasi-periodic) and chaotic vibrations of the size-dependent beams, plates and
shells studied in this book. In particular, the methods of Benettin, Wolf, Rosenstein,
Kantz based on Jacobian estimation and the neural network method are presented
and discussed. As noted above, an important issue in solving problems of nonlinear
dynamics, especially at the nano-level, is the question of the reliability of chaotic
oscillations. This problem was first identified by René Lozi in 2013. In this monograph, in order to obtain reliable results, it is proposed to achieve a coincidence
not only of the basic functions during chaotic oscillations, but also of their second
derivatives with respect to time. This question was formulated by the authors of the
monograph in the book “Deterministic Chaos in One Dimensional Continuous Systems, World Scientific, Singapore, 2016”. In addition, various types of definitions of
chaos are given and a methodology for identifying the truth of chaos is presented.
Moreover, this chapter is devoted to identification of the truth of chaos and the reliability of the results using various methods for determining Lyapunov exponents.
This question was investigated using numerical experiments based on classical simple nonlinear systems: Hénon map, hyperchaotic Hénon map, logistic map, as well
as the Rössler and Lorenz systems. The case studies are analyzed using the Fourier
spectrum and wavelets of various types (Morlet, Mexican hat, Haar, Daubechies and
Gauss of various orders). Preference is given to the Morlet and Gaussian wavelet
32. It was shown that when analyzing Lyapunov exponents, neural network method
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_3
79
Lyapunov Exponents and Methods of
Their Analysis
3.1 Introduction
The book is devoted to study the nonlinear phenomena exhibited by the sizedependent structural members including bifurcations and chaotic processes, and
hence this chapter provides an overview of one of the main tools for identifying
the nonlinear dynamics of these objects. Namely, the concept of Lyapunov exponents is briefly revisited, which allows us to distinguish between regular (periodic
or quasi-periodic) and chaotic vibrations of the size-dependent beams, plates and
shells studied in this book. In particular, the methods of Benettin, Wolf, Rosenstein,
Kantz based on Jacobian estimation and the neural network method are presented
and discussed. As noted above, an important issue in solving problems of nonlinear
dynamics, especially at the nano-level, is the question of the reliability of chaotic
oscillations. This problem was first identified by René Lozi in 2013. In this monograph, in order to obtain reliable results, it is proposed to achieve a coincidence
not only of the basic functions during chaotic oscillations, but also of their second
derivatives with respect to time. This question was formulated by the authors of the
monograph in the book “Deterministic Chaos in One Dimensional Continuous Systems, World Scientific, Singapore, 2016”. In addition, various types of definitions of
chaos are given and a methodology for identifying the truth of chaos is presented.
Moreover, this chapter is devoted to identification of the truth of chaos and the reliability of the results using various methods for determining Lyapunov exponents.
This question was investigated using numerical experiments based on classical simple nonlinear systems: Hénon map, hyperchaotic Hénon map, logistic map, as well
as the Rössler and Lorenz systems. The case studies are analyzed using the Fourier
spectrum and wavelets of various types (Morlet, Mexican hat, Haar, Daubechies and
Gauss of various orders). Preference is given to the Morlet and Gaussian wavelet
32. It was shown that when analyzing Lyapunov exponents, neural network method
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_3
79