Isabelle Schneider

Time delay control of symmetry-breaking primary and secondary oscillation death

We show that oscillation death as a specific type of oscillation suppression, which implies symmetry breaking, can be controlled by introducing time-delayed coupling. In particular, we demonstrate that time delay influences the stability of an inhomogeneous steady state, providing the opportunity to modulate the threshold for oscillation death. Additionally, we find a novel type of oscillation death representing a secondary bifurcation of an inhomogeneous steady state.

Stable and transient multicluster oscillation death in nonlocally coupled networks.

In a network of nonlocally coupled Stuart-Landau oscillators with symmetry-breaking coupling, we study numerically, and explain analytically, a family of inhomogeneous steady states (oscillation death). They exhibit multicluster patterns, depending on the cluster distribution prescribed by the initial conditions. Besides stable oscillation death, we also find a regime of long transients asymptotically approaching synchronized oscillations. To explain these phenomena analytically in dependence on the coupling range and the coupling strength, we first use a mean-field approximation, which works well for large coupling ranges but fails for coupling ranges, which are small compared to the cluster size. Going beyond standard mean-field theory, we predict the boundaries of the different stability regimes as well as the transient times analytically in excellent agreement with numerical results.

Delayed feedback control of three diffusively coupled Stuart–Landau oscillators: a case study in equivariant Hopf bifurcation

The modest aim of this case study is the non-invasive and pattern-selective stabilization of discrete rotating waves (‘ponies on a merry-go-round’) in a triangle of diffusively coupled Stuart–Landau oscillators. We work in a setting of symmetry-breaking equivariant Hopf bifurcation. Stabilization is achieved by delayed feedback control of Pyragas type, adapted to the selected spatio-temporal symmetry pattern. Pyragas controllability depends on the parameters for the diffusion coupling, the complex control amplitude and phase, the uncontrolled super-/sub-criticality of the individual oscillators and their soft/hard spring characteristics. We mathematically derive explicit conditions for Pyragas control to succeed.

PHASE: a universal software package for the propagation of time-dependent coherent light pulses along grazing incidence optics

The software package PHASE includes routines for the propagation of coherent light within the stationary phase approximation (SPA). The code is based on a nonlinear analytic transformation of electric field arrays across longitudinally extended optical elements in normal and grazing-incidence geometries. Recently, the representation of the optical elements (OEs) has been extended to 8th-order polynomials in the OE-coordinates. Strongly curved mirror surfaces can be treated and systematic fabrication errors can be modeled up to 8th order. Each element is represented by an individual matrix and the combination of several elements is accomplished by simple matrix multiplications. The SPA-method can be interpreted as a thick lens approximation, whereas the Fourier Optics algorithm deals with thin lenses. Both methods have advantages and disadvantages. Recently, the PHASE package has been extended to Fourier Optics methods. The appropriate propagator or even a combination of different propagators can be selected from the same interface, which is running under IDL. This permits a one-by-one comparison of both methods via the same interface, which helps to evaluate the advantages and limitations of both methods.

Symmetry-Breaking Control of Rotating Waves

Our aim is the stabilization of time-periodic spatio-temporal synchronization patterns. Our primary examples are coupled networks of Stuart-Landau oscillators. We work in the spirit of Pyragas control by noninvasive delayed feedback. In addition we take advantage of symmetry aspects. For simplicity of presentation we first focus on a ring of coupled oscillators. We show how symmetry-breaking controls succeed in selecting and stabilizing unstable periodic orbits of rotating wave type. Standard Pyragas control at minimal period fails in this selection task. Instead, we use arbitrarily small noninvasive time-delays. As a consequence we succeed in stabilizing rotating waves—for arbitrary coupling strengths, and far from equilibrium.

Bistability in two simple symmetrically coupled oscillators with symmetry-broken amplitude- and phase-locking

In the model system of two instantaneously and symmetrically coupled identical Stuart-Landau oscillators, we demonstrate that there exist stable solutions with symmetry-broken amplitude- and phase-locking. These states are characterized by a non-trivial fixed phase or amplitude relationship between both oscillators, while simultaneously maintaining perfectly harmonic oscillations of the same frequency. While some of the surrounding bifurcations have been previously described, we present the first detailed analytical and numerical description of these states and present analytically and numerically how they are embedded in the bifurcation structure of the system, arising both from the in-phase and the anti-phase solutions, as well as through a saddle-node bifurcation. The dependence of both the amplitude and the phase on parameters can be expressed explicitly with analytic formulas. As opposed to the previous reports, we find that these symmetry-broken states are stable, which can even be shown analytically. As an example of symmetry-breaking solutions in a simple and symmetric system, these states have potential applications as bistable states for switches in a wide array of coupled oscillatory systems.

An Introduction to the Control Triple Method for Partial Differential Equations

We give an introduction to the control triple method, a new type of noninvasive spatio-temporal feedback control. The notion of a control triple defines how we transform the output signal, space, and time in the control term. This Ansatz, especially well suited for the control of partial differential equations, does not exist in the literature so far. It incorporates the spatio-temporal patterns of the equilibria and periodic orbits into the control term. We give linear examples to demonstrate the success of the control triple method.