Deterministic Chaos in One-Dimensional Continuous Systems

The following sections are included:

• Preface
• Contents

There are numerous papers and books devoted to dynamics of structural members, i.e. beams, plates, panels and shells. The aim of this chapter is to overview the literature and state-of-the art research devoted to dynamics of the aforementioned structural members and their interplay with bifurcation and chaotic phenomena. This is a novel challenging research track, and hence not many papers and books are published in this topic. In general, from a mathematical point of view, the book deals with nonlinear PDEs and the developed methods for analysis of their solutions with respect to stability, bifurcations, buckling, as well as regular and chaotic dynamics of the modeled continuous objects. On the other hand, the problem is always reduced (though by different ways) to a study of a set of large amount of nonlinear ODEs. In the case of a few first-order nonlinear ODEs, they may govern dynamics of simple nonlinear autonomous and non-autonomous oscillators (two or three first-order ODEs) or dynamics of coupled oscillators. This is why in the first four chapters a background about bifurcational and chaotic dynamics of difference and differential equations has been given to introduce the reader with basic knowledge devoted to dynamics of lumped mechanical systems and beyond. This effort allows for a smooth transition from nonlinear dynamics exhibited by simple dynamical systems to that of simple continuous systems, which can be also understood as 1D or 2D chains of nonlinear oscillators.

In this chapter, in order to get a deeper understanding of nonlinear dynamical phenomena covered by this book, the fundamental concepts of one-dimensional (1D) maps and fractal sets are briefly reviewed and illustrated. First notions of Cantor's set, Cantor's dust and Koch's snowflake are presented. Then 1D maps are considered putting emphasis on their regular and chaotic dynamics, the cobweb diagrams and the period doubling bifurcation routes to chaos including estimation of the Feigenbaum constant are also briefly revisited. The Sharkovsky theorem is addressed with presentation of its advantages and limitations. Next, the Julia, Fabout and Mandebrot sets are shortly described and illustrated.

This chapter is devoted to definitions of chaos and description of various routes to chaos including the Landau–Hopf scenario, the Ruelle–Takens–Newhouse scenario, the Feigenbaum scenario and the Pomeau–Manneville scenario. The synchronization of chaos is briefly addressed. Quantification of chaotic dynamics via Fourier analysis, Poincaré maps, Lyapunov exponents is described. The Melnikov method to detect strange chaotic attractors is presented. The remaining chapter part is focused on advantages of the wavelet analysis versus the classical Fourier analysis. Properties of different wavelets are illustrated and discussed.

This chapter presents a short study of simple autonomous and non-autonomous systems exhibiting strange chaotic attractors. In particular, the Lorenz and Rössler equations are discussed and the geometrical properties of the chaotic attractors governed by these equations are described. Other examples of autonomous chaotic systems include modified generators with inertial nonlinearity, Chua circuit, systems with quadratic nonlinearities, labyrynth chaos, jerk equations, two-scroll and three-scroll attractors, and Rikitake chaotic attractor.

Examples of simple non-autonomous systems generating chaos include forced van der Pol equation, Rayleigh equation, Duffing oscillator and single-well oscillator, and externally and parametrically excited oscillators.

The background given in this chapter is useful while exploring the next chapters since many of the presented and discussed chaotic attractors can also be found in the systems with infinite dimension.

The following sections are included:

• Introduction
• Linear Friction
• Nonlinear Friction
• Hysteretic Friction
• Impact Damping
• Damping in Continuous 1D Systems

This chapter is devoted to studying chaotic dynamics of the Euler–Bernoulli beams. At first, numerical approaches being further applied including both Benettin and Wolf algorithms, neural networks and classical and modified Gramm–Schmidt orthonormalization procedures are described and validated using the standard maps. Then the planar beams are investigated in a standard way, i.e. the governing PDEs are reduced to nonlinear ODEs, and the latter are analyzed using time histories, Fourier frequency spectra, phase curves, 3D beam deflections, as well as the first four Lyapunov exponents (LES) and Morlet wavelets. Numerous novel examples of beam chaotic and hyperchaotic dynamics are illustrated and discussed…

This chapter deals with the Timoshenko and Sheremetev–Pelekh beam theories and the chaotic vibrations of the beams within the mentioned theories. In both cases the applied assumptions and hypotheses are formulated first, and then the governing PDEs are studied. Numerous examples of chaotic dynamics with the help of classical tools of numerical analysis supported by wavelet transforms and Lyapunov exponents (LEs) computations are reported.

In Section 8.1, the rectangular plate-strip under uniformly distributed harmonic load taking into account physical and elastic-plastic deformations is studied. Various stress–strain relations while cyclic loading are applied. Then a computational algorithm including reduction of PDEs to ODEs via either the Bubnov–Galerkin Method (BGM) or the Finite Difference Method (FDM) is described. Different numerical methods for computation of ODEs are reviewed and discused…

Section 9.1 deals with shallow shells and plates with initial imperfection. The mathematical model is derived, and governing PDEs are reduced to ODEs. Reliability of the obtained numerical results is studied. Chaotic vibrations of spherical and conical shells with constant and variable thicknesses have been examined…

The following sections are included:

• Bibliography
• Index