Search

The aim of this paper is to investigate the use of Pyragas control on the stability of stationary, localised coherent structures in a general class of two-component, singularly perturbed, reaction-diffusion systems. We use noninvasive Pyragas-like proportional feedback control to stabilise a singular pulse solution to a two-component, singularly perturbed reaction-diffusion system. We show that in a significant region of parameter space, the control can be adjusted to stabilise an otherwise unstable pulse., Comment: 11 pages, 5 figures

Reaction-diffusion equations are ubiquitous in various scientific domains and their patterns represent a fascinating area of investigation. However, many of these patterns are unstable and therefore challenging to observe. To overcome this limitation, we present new noninvasive feedback controls based on symmetry groupoids. As a concrete example, we employ these controls to selectively stabilize unstable equilibria of the Chafee--Infante equation under Dirichlet boundary conditions on the interval. Unlike conventional reflection-based control schemes, our approach incorporates additional symmetries that enable us to design new convolution controls for stabilization. By demonstrating the efficacy of our method, we provide a new tool for investigating and controlling systems with unstable patterns, with potential implications for a wide range of scientific disciplines.

The complex Ginzburg--Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg--Landau $m$-armed spiral waves have been investigated extensively. However, many multi-armed spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this article we selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. Our tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.

We explore stability and instability of rapidly oscillating solutions $x(t)$ for the hard spring delayed Duffing oscillator $$x''(t)+ ax(t)+bx(t-T)+x^3(t)=0.$$ Fix $T>0$. We target periodic solutions $x_n(t)$ of small minimal periods $p_n=2T/n$, for integer $n\rightarrow \infty$, and with correspondingly large amplitudes. Note how $x_n(t)$ are also marginally stable solutions, respectively, of the two standard, non-delayed, Hamiltonian Duffing oscillators $$x''+ ax+(-1)^nbx+x^3=0.$$ Stability changes for the delayed Duffing oscillator. Simultaneously for all sufficiently large $n\geq n_0$, we obtain local exponential stability for $(-1)^nb<0$, and exponential instability for $(-1)^nb>0$, provided that $$0 \neq (-1)^{n+1}b\,T^2< \tfrac{3}{2}\pi^2.$$ We interpret our results in terms of noninvasive delayed feedback stabilization and destabilization for large amplitude rapidly periodic solutions of the standard Duffing oscillators. We conclude with numerical illustrations of our results for small and moderate $n$ which also indicate a Neimark-Sacker torus bifurcation at the validity boundary of our theoretical results., Comment: 27+ii pages, 6 figures

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 and 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 as well as 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 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.

We study scalar delay equations $$\dot{x} (t) = \lambda f(x(t-1)) + b^{-1} (x(t) + x(t -p/2))$$ with odd nonlinearity $f$, real nonzero parameters $\lambda, \, b$, and two positive time delays $1,\ p/2$. We assume supercritical Hopf~bifurcation from $x \equiv 0$ in the well-understood single-delay case $b = \infty$. Normalizing $f' (0)=1$, branches of constant minimal period $p_k = 2\pi/\omega_k$ are known to bifurcate from eigenvalues $i\omega_k = i(k+\tfrac{1}{2})\pi$ at $\lambda_k = (-1)^{k+1}\omega_k$, for any nonnegative integer $k$. The unstable dimension of these rapidly oscillating periodic solutions is $k$, at the local branch $k$. We obtain stabilization of such branches, for arbitrarily large unstable dimension $k$, and for, necessarily, delicately narrow regions of control amplitudes $b < 0$. For $p$:= $p_k$ the branch $k$ of constant period $p_k$ persists as a solution, for any $b\neq 0$. Indeed the delayed feedback term controlled by $b$ vanishes on branch $k$: the feedback control is noninvasive there. Following an idea of Pyragas (1992), we seek parameter regions $\mathcal{P} = (\underline{b}_k,\overline{b}_k)$ of controls $b \neq 0$ such that the branch $k$ becomes stable, locally at Hopf~bifurcation. We determine rigorous expansions for $\mathcal{P}$ in the limit of large $k$. Our analysis is based on a 2-scale covering lift for the slow and rapid frequencies involved. These results complement earlier results by Fiedler and Oliva (2016) which required control terms $$b^{-1} (x(t-\vartheta) + x(t-\vartheta -p/2))$$ with a third delay $\vartheta$ near 1., Comment: 49 pages, 7 figures

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.