chaos

Topic and aim. The aim of the work is to review circuits of chaos generators, those described in the literature and some original ones, in a unified style basing on circuit simulations with the NI Multisim package, which makes the comparison of the various devices apparent.

If we allow non-smooth or discontinuous functions in definition of an evolution operator for dynamical systems, then situations of quasi-hyperbolic chaotic dynamics often occur like, for example, on attractors in model Lozi map and in Belykh map.

A new autonomous system with chaotic dynamics corresponding to Smale–Williams attractor in Poincare map is introduced. The system is constructed on the basis of the model with «figure-eight» separatrix on the phase plane discussed in former times by Y.I. Neimark. Our system is composed of two Neimark subsystems with generalized coordinates x and y. It is described by the equations with additional terms due to which the system becomes self-oscillating.

The article deals with the bases of fractal geometry and fates of its creators. The biographies and the discoveries of Felix Hausdorff and Abram Besicovitch – the main characters of the great play called fractal geometry – are presented with the possible degree of detail. There is no doubt that the author and director of this play was Benoit Mandelbrot.

A contemporary consideration of such concepts as dimension and entropy of dynamical systems is given. Description of these characteristics includes into the analysis the other notions and properties related to complicated behavior of nonlinear systems as embedding dimension, prediction horizon etc., which are used in the paper. A question concerning the application of these ideas to real observables of the economical origin, i.e. market prices of the companies Schlumberger, Deutsche Bank, Honda, Toyota, Starbucks, BP is studied.

Examples of model non­autonomous systems are constructed and studied possessing hyperbolic attractors of Smale–Williams type in their stroboscopic maps. The dynamics is determined by application of a periodic sequence of kicks, in such way that on one period of the external driving the angular coordinate, or the phase of oscillations, behaves in accordance with an expanding circle map with chaotic dynamics.

Different universal methods of calculation of index of chaotic synchronization are considered. One of the methods, which is based on mutual coherence function, is presented in more detail. Its advantages and disadvantages, sensibility to external noise and distortions are discussed. Application of the algorithm to process of destruction of complete chaotic synchronization in two coupled systems with discrete time are demonstrated.

The stabilization of chaos in the Rossler system by external signal is investigated. Different types of external action are considered: both of pulsed and harmonic signal. There are illustrations: charts of dynamical regimes, phase porters, stroboscopic section of Poincare, spectrum of Lyapunov exponents. Comparative analysis of efficiency of stabilization of band chaos and spiral chaos by different signal is carried out. The dependence of synchronization picture on direction of acting pulses is shown.

The work studies the effects of temperature on drift velocity of the electrons in semiconductor superlattices in electric and tilted magnetic fields. It is shown that a thermal distribution of the electrons can counter­intuitively enhance the phenomena related to resonances between the Bloch and the cyclotron frequencies of electron motion in superlattices. In particular, the increase of temperature leads to more prominent resonant maxima in the dependence of drift velocity of electrons on the strength of an electric field.

A modified radio­electronic analog of the nonlinear ring cavity is realized in laboratory. The device represents a special class of oscillations or waves sources. An operation principle of the sources is based on interference amplification of feedback signal by an input signal. A laboratory experiments are performed, the likeness of their results and simulation data is shown. An intermittency, chaos, regular, static modes are detected. A thesis on controlled nonlinearity of dynamical systems is suggested.