Mexican hat, Haar, Daubechies and Gauss of various orders). Preference is given to
the Morlet and Gaussian wavelet 32. It was shown that when analysing Lyapunov
exponents, neural network method (this method was proposed by the authors of this
monograph) makes it possible to calculate the spectrum of Lyapunov exponents and
hence to identify phenomena such as the transition of the system into chaos,
hyper-chaos, etc. in a fast and reliable way.
The fourth chapter deals with the methodology for detecting true chaos (in terms
of nonlinear dynamics) and is developed on the example of a structure composed of
two beams with a small clearance. The Euler-Bernoulli hypothesis is employed, and
the contact interaction between beams follows the Kantor model. The complex
nonlinearity results from the von Kármán geometric nonlinearity as well as the
nonlinearity implied by the contact interaction. The governing PDEs are reduced to
ODEs by the second-order finite difference method (FDM). The obtained system of
equations is solved by Runge-Kutta methods of different accuracy. To purify the
signal from errors introduced by numerical methods, the principal component
analysis is employed and the sign of the first Lyapunov exponent is estimated by the
Kantz, Wolf and Rosenstein methods.
Analysis of nonlinear oscillations of classical systems is carried out in Chap. 5. It
is devoted to feasible methods for 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 stands for 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 the Benettin
method), allows us to compute the spectrum of all Lyapunov exponents. In addition, very good results were obtained by the Rosenstein method for all studied
systems. However, the latter approach can be used to estimate only the largest
Lyapunov exponents.
In Chap. 6, mathematical models of nonlinear micro- and nanocylindrical panels
in temperature fields are introduced and studied. First, the application of the
modified couple stress theory of thermoelastic curvilinear panels based on the
third-order hypotheses has been described. Then, a technical theory of the
Sheremetev-Pelekh, Timoshenko and Bernoulli-Euler models is presented. The
method of solving static problems is outlined in Sect. 6.4. Chaotic dynamics of the
size-dependent flexible Bernoulli-Euler, Timoshenko and Sheremetev-Pelekh
beams based on the modified couple stress theory of elasticity is investigated in
Sect. 6.5. The two last sections are devoted to the construction of the so-called
charts of vibration character with regard to amplitude and frequency of the external
excitation and their study with the use of the first, second and third kinematic
hypotheses.
Chapter 7 is devoted to the analysis of nonlinear functionally graded material
(FGM) straight and curved beams’ behaviour based on the modified couple stress
theory. Defining the deflection curve in order to simplify the governing equations,
Introduction
xi
the Morlet and Gaussian wavelet 32. It was shown that when analysing Lyapunov
exponents, neural network method (this method was proposed by the authors of this
monograph) makes it possible to calculate the spectrum of Lyapunov exponents and
hence to identify phenomena such as the transition of the system into chaos,
hyper-chaos, etc. in a fast and reliable way.
The fourth chapter deals with the methodology for detecting true chaos (in terms
of nonlinear dynamics) and is developed on the example of a structure composed of
two beams with a small clearance. The Euler-Bernoulli hypothesis is employed, and
the contact interaction between beams follows the Kantor model. The complex
nonlinearity results from the von Kármán geometric nonlinearity as well as the
nonlinearity implied by the contact interaction. The governing PDEs are reduced to
ODEs by the second-order finite difference method (FDM). The obtained system of
equations is solved by Runge-Kutta methods of different accuracy. To purify the
signal from errors introduced by numerical methods, the principal component
analysis is employed and the sign of the first Lyapunov exponent is estimated by the
Kantz, Wolf and Rosenstein methods.
Analysis of nonlinear oscillations of classical systems is carried out in Chap. 5. It
is devoted to feasible methods for 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 stands for 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 the Benettin
method), allows us to compute the spectrum of all Lyapunov exponents. In addition, very good results were obtained by the Rosenstein method for all studied
systems. However, the latter approach can be used to estimate only the largest
Lyapunov exponents.
In Chap. 6, mathematical models of nonlinear micro- and nanocylindrical panels
in temperature fields are introduced and studied. First, the application of the
modified couple stress theory of thermoelastic curvilinear panels based on the
third-order hypotheses has been described. Then, a technical theory of the
Sheremetev-Pelekh, Timoshenko and Bernoulli-Euler models is presented. The
method of solving static problems is outlined in Sect. 6.4. Chaotic dynamics of the
size-dependent flexible Bernoulli-Euler, Timoshenko and Sheremetev-Pelekh
beams based on the modified couple stress theory of elasticity is investigated in
Sect. 6.5. The two last sections are devoted to the construction of the so-called
charts of vibration character with regard to amplitude and frequency of the external
excitation and their study with the use of the first, second and third kinematic
hypotheses.
Chapter 7 is devoted to the analysis of nonlinear functionally graded material
(FGM) straight and curved beams’ behaviour based on the modified couple stress
theory. Defining the deflection curve in order to simplify the governing equations,
Introduction
xi