Olga Isaeva

Within the framework of the nonstationary model with nonfixed field structure, we investigate the model of a 3-mm band gyroklystron with delayed feedback. It is shown that both chaotic and hyperchaotic generation regimes are possible in this system. The chaotic regime due to a Feigenbaum period-doubling cascade is characterized by one positive Lyapunov exponent. Further transition to hyperchaos is determined by the appearance of another positive exponent in the Lyapunov spectrum. The correlation dimension of the corresponding attractors reaches values of about 3.5.

An abstract discrete time dynamical system, given by an implicit function of the values of a variable at successive moments of time, is presented. The dynamics of this system is defined ambiguously both in reverse and forward time. An example of a system of such type is described in the works of Bullett, Osbaldestin and Percival [Physica D, 1986, vol. 19, pp. 290–300; Nonlinearity, 1988, vol. 1, pp. 27–50]; it demonstrates some features of the behavior of Hamiltonian systems. The map under study allows a smooth transition from the case of the explicitly defined evolution operator to an implicit one and, further, to the “conservative” limit, corresponding to the symmetric evolution operator satisfying the unitarity condition. Being created on the basis of the complex Mandelbrot map, it demonstrates the transformation of the phenomena of complex analytical dynamics to “conservative” phenomena and allows us to identify the relationship between them.

We describe one possible scenario of destruction or of a birth of the hyperbolic attractors considering the Smale—Williams solenoid as an example. The content of the transition observed under variation of the control parameter is the pairwise merge of the orbits belonging to the attractor and to the unstable invariant set on the border of the basin of attraction, in the course of the set of bifurcations of the saddle-node type. The transition is not a single event, but occupies a finite interval on the control parameter axis. In an extended space of the state variables and the control parameter this scenario can be regarded as a mutual transformation of the stable and unstable solenoids one to each other. Several model systems are discussed manifesting this scenario e.g. the specially designed iterative maps and the physically realizable system of coupled alternately activated non-autonomous van der Pol oscillators. Detailed studies of inherent features and of the related statistical and scaling properties of the scenario are provided.

We outline a possibility of chaotic dynamics associated with a hyperbolic attractor of the Smale–Williams type in mechanical vibrations of a nonhomogeneous string with nonlinear dissipation arising due to parametric excitation of modes at the frequencies $\omega$ and $3\omega$, when the pump is supplied by means of the string tension variations alternately at frequencies of $2\omega$ and $6\omega$.