Chapter 5
Analysis of Simple Nonlinear Dynamical
Systems
5.1 Introduction
Analysis of nonlinear oscillations of classical systems is carried out. It is devoted
to feasible methods for the computation of Lyapunov exponents since there is no
universal, verified and general method to compute their exact (in numerical sense)
values. This observation leads to the conclusion that there is a need to employ qualitatively different methods while checking the reliability of “true chaotic results”.
Furthermore, the analysis carried out in this chapter is a helpful tool for studying
systems with infinite dimensions. We show that the most perspective and useful is the
modified method of neural networks. It gives excellent convergence to the original
results and, as the only one (besides Benettin method), allows us to compute the
spectrum of all Lyapunov exponents. In addition, very good results were obtained
by Rosenstein method for all studied systems. However, the latter approach can be
used to estimate only the largest Lyapunov exponents.
5.2 Gauss Wavelets [1]
In some of the engineering problems, Fourier analysis is insufficient, since it deals
with the averaged spectrum of the whole studied vibration signal and presents only a
general picture of the signal. On the contrary, wavelets play the role of a microscope,
which allows one to observe the spectrum at each time instant, and hence to detect
a birth/death of the frequencies in time.
A wavelet transform of a 1D signal consists of its development with respect to a
basis being usually a soliton-like function with given properties. The basis is obtained
by displacement and tension/compression of a function called a wavelet.
In the present chapter, Gauss wavelets, defined as derivatives of Gauss function,
are used. Higher order derivatives have many zero moments, and hence they allow
© 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_5
113
Analysis of Simple Nonlinear Dynamical
Systems
5.1 Introduction
Analysis of nonlinear oscillations of classical systems is carried out. It is devoted
to feasible methods for the computation of Lyapunov exponents since there is no
universal, verified and general method to compute their exact (in numerical sense)
values. This observation leads to the conclusion that there is a need to employ qualitatively different methods while checking the reliability of “true chaotic results”.
Furthermore, the analysis carried out in this chapter is a helpful tool for studying
systems with infinite dimensions. We show that the most perspective and useful is the
modified method of neural networks. It gives excellent convergence to the original
results and, as the only one (besides Benettin method), allows us to compute the
spectrum of all Lyapunov exponents. In addition, very good results were obtained
by Rosenstein method for all studied systems. However, the latter approach can be
used to estimate only the largest Lyapunov exponents.
5.2 Gauss Wavelets [1]
In some of the engineering problems, Fourier analysis is insufficient, since it deals
with the averaged spectrum of the whole studied vibration signal and presents only a
general picture of the signal. On the contrary, wavelets play the role of a microscope,
which allows one to observe the spectrum at each time instant, and hence to detect
a birth/death of the frequencies in time.
A wavelet transform of a 1D signal consists of its development with respect to a
basis being usually a soliton-like function with given properties. The basis is obtained
by displacement and tension/compression of a function called a wavelet.
In the present chapter, Gauss wavelets, defined as derivatives of Gauss function,
are used. Higher order derivatives have many zero moments, and hence they allow
© 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_5
113