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Chaos occurs widely in engineering and natural systems. Historically it has been noted only as irregular or unpredictable behaviour and often attributed to random external influences. Further studies have shown that chaotic phenomena are completely deterministic and characteristic for typical nonlinear systems. These studies posed the question of the practical applications of chaos. One of the possible answers is to control chaotic behaviour in such a way as to make it predictable. Recently there have been examples of the potential usefulness of chaotic behaviour and this has caused growing interest among engineers and applied scientists. In this book the new mathematical ideas in nonlinear dynamics are described such that engineers can apply them in real physical systems.

Inhaltsverzeichnis

Frontmatter

1. Response of a Nonlinear System

Abstract
This chapter briefly describes why the nonlinear phenomena are important to engineers. We show that during the investigations of nonlinear systems one can observe phenomena which are not known from the linear theory.
Tomasz Kapitaniak

2. Continuous Dynamical Systems

Abstract
In this chapter we describe in a simple way mathematical tools which are necessary in the analysis of dynamical systems. The fundamental term of attractor is introduced. We start from the fixed points, limit cycles and finally describe the properties of strange chaotic attractors. To complete this description we introduce Poincaré maps and Lyapunov exponents. Poincaré maps are tools which allow the system dimension reduction and whose idea for engineers is known from the stroboscopic lamp. Lyapunov exponents measure the divergence of trajectories starting from nearby initial conditions. These exponents are important as, in most engineering systems, initial conditions cannot be set or measured accurately. Additionally, we show that the analysis of the classical power spectrum can also be useful in the analysis of chaotic systems.
Tomasz Kapitaniak

3. Discrete Dynamical Systems

Abstract
A discrete dynamical system is a system which is discrete in time so we observe its dynamics not continuously but at the given moments of time like in the case of Poincaré maps introduced in the previous chapter. The dynamics of discrete dynamical systems is usually simple enough to be explained in details. We use these systems to describe the main phenomena of nonlinear dynamics.
Tomasz Kapitaniak

4. Fractals

Abstract
Fractals, objects with noninteger dimension at first sight look very unusual for any practical applications. In this chapter we introduce basic examples and properties of fractal sets starting with a classic example of the Cantor set and introduce different definitions of its dimension. Later we discuss the application of the fractal concept to the dynamics and show that it is very useful in the description of strange chaotic attractors.
Tomasz Kapitaniak

5. Routes to Chaos

Abstract
In the previous chapters we have introduced the methods of chaotic behaviour description. Here we will observe how the behaviour of our systems changes during the transition from periodic to chaotic states. The mechanism of the transition to chaos is of fundamental importance for understanding the phenomenon of chaotic behaviour. There are three main routes to chaos which can be observed in nonlinear oscillators.
Tomasz Kapitaniak

6. Applications

Abstract
Chaotic behaviour occurs in a great number of practical engineering and natural systems. In this chapter we briefly present several examples of chaotic behaviour in mechanical engineering, chemical reactions, electronic circuits, civil engineering problems, and fluid dynamics. Presented examples show the variety of possible applications of chaotic and fractal dynamics in different branches of engineering. They can be considered as starting points for readers’ own research in a chosen branch.
Tomasz Kapitaniak

7. Controlling Chaos

Abstract
As has been shown in previous chapter chaos occurs widely in engineering and natural systems; historically it has usually been regarded as a nuisance and designed out if possible. It has been noted only as irregular or unpredictable behaviour, and often attributed to random external influences. More recently there have been examples of the potential usefulness of chaotic behaviour, and we describe some of the potential usefulness of chaotic behaviour in this chapter.
Tomasz Kapitaniak

Backmatter

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