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Stabilization of periodic orbits by time-delayed feedback

In order to understand the fundamental mechanism of the feedback scheme a bifurcation analysis in the parameter space is required. Bifurcation diagrams allow for the systematic conclusion on general bifurcation mechanisms of the stabilization of periodic orbits by time-delayed feedback. Moreover, they help in predicting the effect of various control schemes in dependence on their parameters. We studied numericaly the Rössler model  as a prototype example of a simple dynamical system with no hyperbolic chaotic attractors. We were able to find a complicated, multi-leaf bifurcation diagram in the control parameter plane including Hopf bifurcations, Neimark-Sacker bifurcations of a torus, period doubling cascades and delay-induced chaos. The single leaves correspond to bifurcations of periodic solutions each of different period. If the time delay is different from an integer multiple of the period of the unperturbed unstable periodc orbit (UPO), a delay-induced stable orbit with another period can be generated through a supercritical Hopf bifurcation. Using an analytical approximation we were able to calculate this period for time delays  other than the period of the original UPO.

A widely accepted crucial limitation of the Pyragas method is claimed to be that periodic orbits with an odd number of real Floquet multipliers cannot be stabilized (Odd Number Limitation Theorem). The work of  Nakajima (1997) alone has been cited more than ninety times in this context. We were able to show that in the generic case of an unstable periodic orbit, generated by a subcritical Hopf bifurcation (and consequently having a single real Floquet multiplier larger than one), this may indeed be stabilized for appropriate control parameters.  A crucial feature here is that the control force should not be coupled by a unity matrix but rather with a phase (like in the case of optical systems, where this occurs naturally). Here, we considered the complex normal form of a subcritical Hopf bifurcation extended by the Pyragas control. The Hopf bifurcation generates unstable periodic solutions. The Hopf curve and the Pyragas-curve can be obtained analytically. The feedback term vanishes by construction on the Pyragas curve. If the Pyragas curve is steeper than the  Hopf curve then it is stable, something we could verify through a direct simulation as well as by calculating the Floquet multiplers. This condition can be analytically formulated as a condition on the complex control force and produces a stabilization regime.  The unique Floquet multiplier actually crosses the unity circle in the complex plane and is stabilized  for a certain control amplitude. Given that unstable periodic orbits without torsion, as generated by a supercritical Hopf bifurcation, are abundant in nonlinear systems in physics, chemistry and biology, our results are of great importance not only for the fundamental understanding of the Pyragas control, but for a wide range of applicationa as well.

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