Lyapunov exponent

Topic and aim. The aim of the work is design of rough chaos generator, whose attractor implements dynamics close to Anosov flow on a manifold of negative curvature, as well as constructing and analyzing mathematical model, and
conducting circuit simulation of the dynamics using the Multisim software.

Investigated models. A mathematical model is considered that is a set of ordinary differential equations of the ninth order with algebraic nonlinearity, and a circuit representing the chaos generator is designed.

Topic and aim. In this work, we study the dynamics of the kicked van der Pol oscillator with the amplitude of kicks depending nonlinearly on the dynamic variable. We choose the expansions of the function cos x in a Taylor series near zero, as functions describing this dependence.

Virtual cathode nonstationary dynamics has been numerically studied for the two counterstreaming electron beams. The variety of the virtual cathode oscillatory regimes has been discovered from regular to wide band chaotic oscillations. Connection between value of the largest Lyapunov exponent and output signal power has been revealed.

In this paper we investigate modified «Arnold cat» map with dissipative terms, in which a hyperbolic chaos exists for small perturbation magnitudes, and in a certain range a hyperbolic chaotic attractor with Cantor transversal structure takes place, collapsing with a further perturbation amplitude increase.

The transition to chaos through the destruction of three-dimensional torus is studied in a nonautonomous system with quasi-periodic impact as example. Analysis is carried out of the influence both of additive noise and frequency fluctuations impact on the stability of three-dimensional torus. It is shown that under the influence of additive noise and frequency fluctuations impact Lyapunov exponent remains negative. This allows to conclude that in this model three-dimensional torus is structurally stable in contrast to the autonomous system. 

A simple autonomous three-dimensional system is introduced that demonstrates quasiperiodic self-oscillations and has as attractor a two-dimensional torus. The computing illustrations of quasiperiodic dynamics are presented: phase portraits, Fourie spectrums, graphics of Lyapunov exponents. The existing of Arnold tongues on the parametric plane and transition from quasiperiodic dynamics to chaos through destruction of invariant curve in the Poincare section are shown.

Dynamics and space of сontrol parameters for a nonlinear oscillator under quasi­periodic driving are investigated experimentally by using a nonlinear circuit with p­n junction diode and numerically by using maps and differential equations. The dynamics of the systems under quasiperiodic driving is invariant due to initial driving phases, as a result the plane of the driving amplitudes is symmetrical.

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.

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.

Topic and aim. The aim of the work is to consider an easy-to-implement system demonstrating the Smale–Williams hyperbolic attractor based on the Bonhoeffer–van der Pol oscillator, alternately manifesting a state of activity or suppression due to periodic modulation of the parameter by an external control signal, and supplemented with a delayed feedback circuit. Investigated models. A mathematical model is formulated as a non-autonomous second-order equation with delay. The scheme of the electronic device that implements this type of chaotic behavior is proposed.