such a tool for studying nonlinear dynamics as wavelet analysis and the spectrum of
Lyapunov exponents is at the initial stage of its development. Furthermore, chaotic
vibrations, hyperchaotic vibrations, hyper-hyperchaotic and other types of vibrations are not sufficiently studied. The critical overview of the latest nonlinear
dynamics of micro/nanostructures presented in this chapter demonstrates the need
for innovative theoretical and numerical tools to achieve reliable and validated
results.
The second chapter (Chap. 2) concerns the latest literature review devoted to
micro/nano size-dependent mathematical models of beams, plates and shells. The
review includes nonlocal theory of elasticity, surface theory of elasticity, modified
couple stress theory and modified theory of deformation gradient taking into
account higher order shear deformation theory. Particular emphasis is placed on
nonlocal models of the Euler-Bernoulli and Timoshenko beams, Kirchhoff plates
and Kirchhoff-Love shells. Both the review and the real-world vibrational behaviour of the size-dependent structural member imply the need to consider physical,
geometric, material and design nonlinearity. It is expected that the novel approach
based on employment of the earlier developed concepts of nonlinear dynamics like
the Fourier and wavelet spectra, phase portraits, Poincaré maps, the Lyapunov
exponents’ computation (at least the largest one), the autocorrelation functions, etc.
will improve mathematical models of partial differential equations (PDEs) to be in a
good fit with experimentally observed nonlinear phenomena exhibited by the
size-dependent structural members.
Chapter 3, since the book is devoted to study the nonlinear phenomena exhibited
by the size-dependent structural members including bifurcations and chaotic processes, 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
and Kantz that 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 nanolevel, 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 the identification of 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
analysed using the Fourier spectrum and wavelets of various types (Morlet,
x
Introduction
Lyapunov exponents is at the initial stage of its development. Furthermore, chaotic
vibrations, hyperchaotic vibrations, hyper-hyperchaotic and other types of vibrations are not sufficiently studied. The critical overview of the latest nonlinear
dynamics of micro/nanostructures presented in this chapter demonstrates the need
for innovative theoretical and numerical tools to achieve reliable and validated
results.
The second chapter (Chap. 2) concerns the latest literature review devoted to
micro/nano size-dependent mathematical models of beams, plates and shells. The
review includes nonlocal theory of elasticity, surface theory of elasticity, modified
couple stress theory and modified theory of deformation gradient taking into
account higher order shear deformation theory. Particular emphasis is placed on
nonlocal models of the Euler-Bernoulli and Timoshenko beams, Kirchhoff plates
and Kirchhoff-Love shells. Both the review and the real-world vibrational behaviour of the size-dependent structural member imply the need to consider physical,
geometric, material and design nonlinearity. It is expected that the novel approach
based on employment of the earlier developed concepts of nonlinear dynamics like
the Fourier and wavelet spectra, phase portraits, Poincaré maps, the Lyapunov
exponents’ computation (at least the largest one), the autocorrelation functions, etc.
will improve mathematical models of partial differential equations (PDEs) to be in a
good fit with experimentally observed nonlinear phenomena exhibited by the
size-dependent structural members.
Chapter 3, since the book is devoted to study the nonlinear phenomena exhibited
by the size-dependent structural members including bifurcations and chaotic processes, 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
and Kantz that 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 nanolevel, 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 the identification of 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
analysed using the Fourier spectrum and wavelets of various types (Morlet,
x
Introduction