We indicate a possibility of implementing hyperbolic chaos using a Froude pendulum that is able to produce self-oscillations due to the suspension on a shaft rotating at constant angular velocity, in the presence of time-delay feedback and of periodic braking by the application of additional frictional force. We formulate a mathematical model and carry out its numerical research. In the parameter space we reveal areas of chaotic and regular dynamics using the analysis of Lyapunov exponents and some other diagnostic tools. It is shown that there are regions in the parameter space where the Poincaré stroboscopic map has an attractor, which is a kind of Smale–Williams solenoid embedded in the infinite-dimensional state space. We confirm the hyperbolicity of the attractor by numerical calculations including the analysis of angles of intersections of stable and unstable invariant subspaces of vectors of small perturbations for trajectories on the attractor and verify the absence of tangencies between these subspaces.

The dynamics of a low-dimensional ensemble consisting of a network of five discrete phase oscillators is considered. A two-parameter synchronization picture, which appears instead of the Arnol'd tongues with an increase of the system dimension, is discussed. An appearance of the Arnol'd resonance web is detected on the "frequency–coupling" parameter plane. The cases of attractive and repulsive interactions are discussed.

No abstract

We propose a new three-dimensional map that demonstrates the two-and three-frequency quasi-periodicity. For this map all basic quasi-periodic bifurcations are possible. The study was realized by using method of Lyapunov charts completed by plots of Lyapunov exponents, phase portraits and bifurcation trees illustrating the quasiperiodic bifurcations. The features of the three-parameter structure associated with quasi-periodic Hopf bifurcation are discussed. The comparison with non-autonomous model is carried out.

Ensembles of several Rössler chaotic oscillators are considered. It is shown that a typical phenomenon for such systems is the emergence of different and sufficiently high dimensional invariant tori. The possibility of a quasi-periodic Hopf bifurcation and a cascade of such bifurcations based on tori of increasing dimension is demonstrated. The domains of resonance tori are revealed. Boundaries of these domains correspond to the saddle-node bifurcations. Inside the domains of resonance modes, torus-doubling bifurcations and destruction of tori are observed.MSC2010 numbers: 70K43, 65P20, 65P30, 34D08