V.V. Emel’yanov, S.P. Kuznetsov, and N.M. Ryskin* Saratov State University, , Saratov, Russia * GENERATION OF HYPERBOLIC CHAOS IN THE SYSTEM OF COUPLED KLYSTRONS

FUNDAMENTALS OF HYPERBOLIC THEORY Fig.1. The neighborhood of hyperbolic trajectory (red color) In the hyperbolic attractors all trajectories are saddle type, their stable and unstable manifolds are of the same dimension, and there no tangencies between the stable and unstable manifolds. Such hyperbolic attractors are robust or structurally stable, that means insensitivity of the type of dynamics and of the phase space structure in respect to the variations of the parameters and functions describing the system.

FUNDAMENTALS OF HYPERBOLIC THEORY Recently the principle design of radio frequency oscillator with hyperbolic attractor has been proposed.  Kuznetsov, S. P., Phys. Rev. Lett., vol. 95, ,  Kuznetsov S.P., Seleznev E.P., Journal of Experimental and Theoretical Physics, vol. 129, 2006 The operation principle of such system is alternating excitation of two coupled oscillators so that the transformation of the signal phase is described by the chaotic Bernoulli map Generators of the hyperbolic chaos attract considerable practical interest for chaos-based data-transmission and radar systems.

SCHEMATIC OF THE KLYSTRON GENERATOR OF HYPERBOLIC CHAOS Fig. 1. Schematic diagram of the proposed chaos generator based on coupled drift klystrons. 1 — electron guns, 2 — electron beams, 3 — collectors, 4 — variable attenuators, 5 — phase shifters

BASIC EQUATIONS  normalized complex amplitudes of fields in the cavities  delay parameter  normalized amplitude of the reference signal  excitation parameters - amount of feedback and phase shift respectively - unperturbed electron transit angle in a drift space

Assuming that the field oscillations in the cavities are established so rapidly that the derivatives of their complex amplitudes can be neglected we obtain an iterative map for the amplitude of the signal in the input cavity. For    retain only a term with number m=0 - Bernoulli map - defines only a constant component of a signal phase shift

NUMERICAL RESULTS FOR THE SIMPLIFIED ITERATIVE MAP Amplitude dynamics. Period doublingsIterative diagram for the phase is the same as for Bernoulli map. Hyperbolic chaos

NUMERICAL RESULTS FOR THE “COMPLETE” MAP The domain of the hyperbolic chaos at taking into account of two (circles) and five (solid lines) terms of series. Dashed lines — corresponding borders for a simplified map The iterative diagram for a phase

LYAPUNOV MAPS Charts of the largest Lyapunov exponent. In the domain of hyperbolic chaos  ≈ ln2

NUMERICAL RESULTS FOR THE TIME-DELAYED DIFFERENTIAL EQUATIONS Similar to the map, choose the following values of parameters The values of other parameters approximately correspond to the values of parameters of the W-bang oscillators described in  Y. M. Shin et al., Phys. Plasmas 13, (2006).  Y. M. Shin et al., Appl. Phys Lett., 88, (2006). The domains of hyperbolic chaos for the time-delayed differential equations (dashed lines) and for the map (solid lines)

The largest Lyapunov exponent of the hyperbolic attractor vs   Waveform of the field amplitude in the input cavity of the first klystron The iterative diagram for the phase of oscillations and the projection of the phase portrait

CONCLUSIONS  The schematic of the generator of hyperbolic chaos in the microwave band is proposed that consists of two klystrons closed in the ring.  The mathematical model of the generator, that represents a system of the time-delayed differential equations is obtained. Under the assumption of the instantaneous build-up of field oscillations in the cavities this model reduces to the two-dimensional map for a complex amplitude of a field in the input cavity of the first klystron.  The results of numerical simulation of both models are similar and show that in the enough large area of control parameters structurally stable hyperbolic chaos is established. The dynamics of a phase is described by a Bernoulli map, the attractor has a topology of the Smale–Williams type, that are typical for the systems with hyperbolic chaos.  The largest Lyapunov is almost independent from the parameters that allows to make an assumption about structural stability of chaotic attractor, the principal attribute of the hyperbolicity.