Jan Sieber

Controlling Unstable Chaos: Stabilizing Chimera States by Feedback

We present a control scheme that is able to find and stabilize an unstable chaotic regime in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to classical delayed feedback control, the scheme is noninvasive, however only in an appropriately relaxed sense considering the chaotic regime as a statistical equilibrium displaying random fluctuations as a finite size effect. We demonstrate the control scheme for so-called chimera states, which are coherence-incoherence patterns in coupled oscillator systems. The control makes chimera states observable close to coherence, for small numbers of oscillators, and for random initial conditions.

Excitability and self-pulsations near homoclinic bifurcations in semiconductor laser systems

Many laser systems show self-pulsations with a large amplitude that are born suddenly in a homoclinic bifurcation. Just before the onset of these self-pulsations the laser is excitable where the excitability threshold is formed by the stable manifold of a saddle point. We show that there exists a special configuration, a codimension-two bifurcation called a non-central saddle-node homoclinic orbit, that acts as an organising centre of excitability in lasers. It is the key to understanding excitability in laser systems as diverse as lasers with saturable absorbers, lasers with optical injection and lasers with optical feedback.

Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity

We investigate a delay differential equation that models a pendulum stabilized in the upright position by a delayed linear horizontal control force. Linear stability analysis reveals that the region of stability of the origin (the upright position of the pendulum) is bounded for positive delay. We find that a codimension-three triple-zero eigenvalue bifurcation acts as an organizing centre of the dynamics. It is studied by computing and then analysing a reduced three-dimensional vector field on the centre manifold. The validity of this analysis is checked in the full delay model with the continuation software DDE-BIFTOOL. Among other things, we find stable small-amplitude solutions outside the region of linear stability of the pendulum, which can be interpreted as a relaxed form of successful control.

High-frequency pulsations in DFB lasers with amplified feedback

We describe the basic ideas behind the concept of distributed feedback (DFB) lasers with short optical feedback for the generation of high-frequency self-pulsations and show the theoretical background describing realized devices. It is predicted by theory that the self-pulsation frequency increases with increasing feedback strength. To provide evidence for this, we propose a novel device design which employs an amplifier section in the integrated feedback cavity of a DFB laser. We present results from numerical simulations and experiments. It has been shown experimentally that a continuous tuning of the self-pulsation frequency from 12 to 45 GHz can be adjusted via the control of the feedback strength. The numerical simulations, which are in good accordance with experimental investigations, give an explanation for a self-stabilizing effect of the self-pulsations due to the additional carrier dynamic in the integrated feedback cavity.

Dynamics of the nearly parametric pendulum

Dynamically stable periodic rotations of a driven pendulum provide a unique mechanism for generating a uniform rotation from bounded excitations. This paper studies the effects of a small ellipticity of the driving, perturbing the classical parametric pendulum. The first finding is that the region in the parameter plane of amplitude and frequency of excitation where rotations are possible increases with the ellipticity. Second, the resonance tongues, which are the most characteristic feature of the classical bifurcation scenario of a parametrically driven pendulum, merge into a single region of instability.

Numerical Bifurcation Analysis for Multisection Semiconductor Lasers

We investigate the dynamics of a multisection laser implementing a delayed optical feedback experiment where the length of the cavity is comparable to the length of the laser. First, we reduce the traveling-wave model with gain dispersion (a hyperbolic system of PDEs) to a system of ODEs describing the semiflow on a local center manifold. Then we analyze the dynamics of the system of ODEs using numerical continuation methods (AUTO). We explore the plane of the two parameters---feedback phase and feedback strength---to obtain a bifurcation diagram for small and moderate feedback strength. This diagram permits us to understand the roots of a variety of nonlinear phenomena observed numerically and experimentally such as, e.g., self-pulsations, excitability, hysteresis, or chaos, and to locate them in the parameter plane.

Extending the permissible control loop latency for the controlled inverted pendulum

A pendulum can be stabilized in its upright position by proportional-plus-derivative (PD) feedback control only if the latency in the control loop is smaller than a certain critical delay. This critical delay is determined by the presence of a fully symmetric triple-zero eigenvalue singularity, a bifurcation of codimension three. We investigate three possible modifications of the PD scheme with the aim of extending the range of permissible delays. Effectively, these modifications introduce another parameter. This additional parameter can be used to continue the triple-zero singularity in four parameters until it gains a higher-order degeneracy imposing a new limit on the permissible delay. It turns out that the most effective modification is to feed back the value of the position with a small (intentional) additional delay on top of the control loop latency.

Dynamics of delayed relay systems

Let M be a smooth compact manifold of any dimension. We consider the set of C1 maps f : M → M which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual set in the C1 topology. Mathematics Subject Classification: 37C40The paper studies the dynamics near periodic orbits in dynamical systems with relays (switches) that switch only after a fixed delay. As a motivating application, we study the problem of stabilizing an unstable equilibrium by feedback control in the presence of a delay in the control loop. We show that saddle-type equilibria can be stabilized to a periodic orbit by a switch even if this switch is subject to an arbitrarily large delay. This is in contrast to linear static feedback control, which fails when the delay is larger than a problem-dependent critical value. Our analysis is based on the reduction of the return map near a generic periodic orbit to a finite-dimensional map. This map is smooth if the periodic orbit satisfies two genericity conditions. A violation of any of these two conditions causes a discontinuity-induced bifurcation of the periodic orbit. We derive asymptotic expressions for the piecewise smooth return map for each of these two codimension-one bifurcations. This analysis shows that the introduction of a small delay into the switching decision can induce chaos in a relay system that had a single stable periodic orbit without delay. This small-delay behaviour is fundamentally different from smooth dynamical systems.

Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations

In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays, transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an extension of the Hopf bifurcation theorem by Eichmann (2006), along with an alternative proof.

Nonlinear softening as a predictive precursor to climate tipping

Approaching a dangerous bifurcation, from which a dynamical system such as the Earths climate will jump (tip) to a different state, the current stable state lies within a shrinking basin of attraction. Persistence of the state becomes increasingly precarious in the presence of noisy disturbances. We argue that one needs to extract information about the nonlinear features (a ‘softening’) of the underlying potential from the time series to judge the probability and timing of tipping. This analysis is the logical next step if one has detected a decrease of the linear decay rate. If there is no discernible trend in the linear analysis, nonlinear softening is even more important in showing the proximity to tipping. After extensive normal-form calibration studies, we check two geological time series from palaeo-climate tipping events for softening of the underlying well. For the ending of the last ice age, where we find no convincing linear precursor, we identify a statistically significant nonlinear softening towards increasing temperature.