By Yurii Bolotin, Anatoli Tur, Vladimir Yanovsky

This ebook bargains a brief and concise advent to the various elements of chaos theory.

While the learn of chaotic habit in nonlinear, dynamical platforms is a well-established examine box with ramifications in all components of technological know-how, there's a lot to be learnt approximately how chaos will be managed and, less than applicable stipulations, can really be positive within the experience of changing into a keep an eye on parameter for the procedure lower than research, stochastic resonance being a major example.

The current paintings stresses the latter facets and, after recalling the paradigm adjustments brought by way of the concept that of chaos, leads the reader skillfully throughout the fundamentals of chaos keep an eye on by means of detailing the appropriate algorithms for either Hamiltonian and dissipative platforms, between others.

The major a part of the ebook is then dedicated to the problem of synchronization in chaotic platforms, an advent to stochastic resonance, and a survey of ratchet types. during this moment, revised and enlarged version, extra chapters discover the numerous interfaces of quantum physics and dynamical structures, analyzing in flip statistical homes of power spectra, quantum ratchets, and dynamical tunneling, between others.

This textual content is especially appropriate for non-specialist scientists, engineers, and utilized mathematical scientists from similar components, wishing to go into the sphere speedy and efficiently.

From the studies of the 1st edition:

This e-book is a wonderful creation to the major thoughts and keep watch over of chaos in (random) dynamical platforms [...] The authors locate a good stability among major actual rules and mathematical terminology to arrive their viewers in a powerful and lucid demeanour. This publication is perfect for anyone who want to seize speedy the most concerns on the topic of chaos in discrete and non-stop time. *Henri Schurz, Zentralblatt MATH, Vol. 1178, 2010.*

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**Additional resources for Chaos: Concepts, Control and Constructive Use**

**Example text**

We start from an ordinary square Q. It is evident that its usual (topological) dimension equals 2. Let us now determine its Hausdorff dimension. First of all we cover the square Q with the same quadratic neighborhood of dimensionless size ". "/ is the number of square neighborhoods of size ", covering the square Q (see Fig. 5). The number of such squares (see Fig. Q/ D "p 2 : To estimate the Hausdorff measure we need to use the limit " ! 0. It is clear p that when p > 2 the sum m" is proportional to the positive power of " and correspondingly this sum limit is equal to zero.

0 D a1 a2 . 14). n D a1 1 a2 2 D en. 0 : It is easy to see that for conservative mappings, which preserve the phase volume, 1 C 2 D 0. In such systems, one of the Lyapunov exponents is positive and the other is negative: 2 D 1 . Thus, the presence of chaos in such systems is determined by the maximum positive Lyapunov exponent. We should note that there is a divergence of close trajectories in one direction and a convergence in the other. 32 3 Main Features of Chaotic Systems Fig. 8 xn For dissipative dynamical mappings 1 C 2 < 0.

2 Embedding Dimension Before discussing embedding dimension let us recall what is understood as manifold embedding in space. Let the space dimension or X manifold be smaller than the space dimension Y. If the map f W X ! X/, then the Jacobian rank @x is everywhere equal to the manifold dimension X dim X, and is called the embedding of X in Y. For instance, the embedding of a circle into two-dimensional space is shown in Fig. 2 at the left. Mapping f W X ! Y which does not need one-to-one correspondence is called immersion.