Delayed feedback control of unstable steady states with high-frequency modulation of the delay.

We analyze the stabilization of unstable steady states by delayed feedback control with a periodic time-varying delay in the regime of a high-frequency modulation of the delay. The average effect of the delayed feedback term in the control force is equivalent to a distributed delay in the interval of the modulation, and the obtained distribution depends on the type of the modulation. In our analysis we use a simple generic normal form of an unstable focus, and investigate the effects of phase-dependent coupling and the influence of the control loop latency on the controllability. In addition, we have explored the influence of the modulation of the delays in multiple delay feedback schemes consisting of two independent delay lines of Pyragas type. A main advantage of the variable delay is the considerably larger domain of stabilization in parameter space.

Experimental time-delayed feedback control with variable and distributed delays.

We report on several improvements of the classical time-delayed feedback control method for stabilization of unstable periodic orbits or steady states. In an electronic circuit experiment, we were able to realize time-varying and distributed delays in the control force leading to successful control for large parameter sets, including large time delays. The presented techniques make advanced use of the natural torsion of the orbits, which is also necessary for the original control method to work.

Kuramoto model with asymmetric distribution of natural frequencies.

We analyze the Kuramoto model of phase oscillators with natural frequencies distributed according to a unimodal asymmetric function g(omega) . It is obtained that besides a second-, also a first-order phase transition can appear if the distribution of natural frequencies possesses a sufficiently large flat section. It is derived analytically that for the first-order transitions the characteristic exponents describing the order parameter and synchronizing frequency near the critical point are equal to those for the order parameter in the corresponding symmetric case. Stability analysis of the incoherent phase shows that the synchronizing frequency at the onset of synchronization equals the perturbation rotation velocity at the border of stability. The analytic and numerical results are in agreement with numerical simulations.

Phase transitions in the Kuramoto model.

We consider the Kuramoto model of phase oscillators with natural frequencies distributed according to a unimodal function with the plateau section in the middle representing the maximum and symmetric tails falling off predominantly as |omega-omega0|m, m>0, in the vicinity of the flat region. It is found that the phase transition is of first order as long as there is a finite flat region and that in the vicinity of the critical coupling the following scaling law holds r-rc proportional, variant(K-Kc)2/(2m+3), where r is the order parameter and K is the coupling strength of the interacting oscillators.

Diffusion on Archimedean lattices.

We consider random diffusive motion of classical particles over the edges of Archimedean lattices. The diffusion coefficient is obtained by using periodic orbit theory. We also study deterministic motion over a honeycomb lattice without the possibility for an immediate return to the preceding node, controlled by a tent map with the golden ratio slope. Numerical analysis is performed to confirm the theoretical results.