Deterministic Chaos

The property of unstable manifold of the Lorenz-system zero equilibrium is proved. This permited to prove the sufficient condition of homoclinic orbit existence.

By the way of combined integrating of the motion and variation equations we calculated the maximal characteristic Lyapunov exponents in the wide limits of energy and time for the Henon–Heiles problem. It follows from the fitting procedure that the best approximate function is the exponential one with the parameter values, which are different from the earlier obtained parameter values (Benettin et al.).

In the frame of a model given before for simulation of chaotic dynamics of continuum medium the Lorenz attractor is represented. The simulation is given with the help of the structures that define the geometry of a fiber bundle associated with 3-dimensional regime of velocity pulsations. Lorenz dynamics appears as time dependence of pulsations along the lines of average flow.

The paper studies the chaotic dynamics in circular asymmetric billiard with beams reflection and refraction. Phase dynamics is characterized by a variety of dynamics modes, which is connected with the effect of traditional chaotization mechanisms as well as with the complicacy of allowable motion laws. In the multisheet symmetric phase space, the circular billiard reconstructions have been analysed its asymmetry degrees changes.

Chaotic RF generator with bipolar transistor is proposed. Mathematical model of  the generator, oscillator with 2.5 degrees of freedom, is investigated. Generator dynamics is analyzed with Advanced Design System (ADS) software using parameters of a real transistor, properties of the board substrate are taken into account by simulation. ADS simulation results are compared with experimental data. It is shown that the use of ADS software for analysis of generator dynamics and account for the board properties and topology allow to get simulation results closer to the experimental one.

Results of experimental investigation of chaotic dynamics of the ensemble of three cascade-coupled phase systems (phase-locked loops) are presented. The possibility of dynamical regimes control by means of coupling parameters changing without changing of inner parameters of elements is demonstrated. Spectral and correlation properties of different chaotic regimes are presented.

The work is devoted to anti-phase controlled synchronization of chaos in diffusivelly coupled cubic maps. Influence of asymmetry of controlling feed-back coupling on mechanizms of the synchronization loss is considered. A new bifurcational scenarium which includes a sequence of transcritical and saddle-repeller bifurcation has been found.

The synchronization by external periodic force of Lorenz system is under both numeric and analytical investigation in this paper. Properly studied the changes in synchronization caused by alteration of parameter value, which is responsible for arising of chaotic attractor in autonomous system.

Effective method of signals analysis based on the continuous wavelet transform is proposed in this paper. Application of this method to estimation of mean value both of laminar and turbulent phase durations corresponding to different types of intermittent behavior is considered including analysis of time series produced by living systems. It is shown that the proposed method is stable to noise and fluctuations distorting the initial time series.

The method of detection of the unstable periodic spatio-temporal states of spatial extended chaotic systems dynamics is proposed. The application of this method is illustrated by the consideration of the fluid model of Pierce diode which is one of the base system of plasma physics and of microwave electronics.