-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.Introduction. -Time-delayed couplings arise naturally in many types of networks, for instance in coupled lasers [1], neural networks [2-4], electronic circuits [5], or genetic oscillators [6], due to finite signal transmission and processing times, and memory and latency effects. While investigating real-world systems, it is necessary to take time delay into account, since the presence of time delay is an inherent property of the vast majority of processes that occur in nature [7,8]. Moreover, time-delayed coupling and feedback represent an important aspect of control [9]. Previous theoretical and experimental works have shown that time delay can be treated as a control parameter and can stabilize initially unstable states. In particular, time-delayed feedback has been used to stabilize unstable periodic orbits embedded in a deterministic chaotic attractor [10,11], or generated by a Hopf bifurcation [12], unstable steady states [13], spatio-temporal patterns [14][15][16], or control the coherence and timescales of stochastic motion [17]. In coupled nonlinear systems and networks time-delayed couplings represent an ubiquitous feature [18] which can also be used to control stability.

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 multi-cluster 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.

One contribution of 15 to a Theme Issue 'Dynamics, control and information in delay-coupled systems' .

Pyragas control is a widely used time-delayed feedback control for the stabilization of periodic orbits in dynamical systems. In this paper we investigate how we can use equivariance to eliminate restrictions of Pyragas control, both to select periodic orbits for stabilization by their spatio-temporal pattern and to render Pyragas control possible at all for those orbits. Another important aspect is the optimization of equivariant Pyragas control, i.e. to construct larger control regions. The ring of n identical Stuart-Landau oscillators coupled diffusively in a bidirectional ring serves as our model.

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.

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We explore stability and instability of rapidly oscillating solutions x(t) for the hard spring delayed Duffing oscillatorFix T > 0. We target periodic solutions x n (t) of small minimal periods p n = 2T /n, for integer n → ∞, and with correspondingly large amplitudes. Simultaneously, for sufficiently large n ≥ n 0 , we obtain local exponential stability for (−1) n b < 0, and exponential instability for (−1) n b > 0, provided thatWe interpret our results in terms of noninvasive delayed feedback stabilization and destabilization of large amplitude rapidly periodic solutions in the standard Duffing oscillator. We conclude with numerical illustrations of our results for small and moderate n which also indicate a Neimark-Sacker-Sell torus bifurcation at the validity boundary of our theoretical results.

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