Abstract
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.
| Original language | English |
|---|---|
| Pages (from-to) | 631-658 |
| Number of pages | 28 |
| Journal | Archive for Rational Mechanics and Analysis |
| Volume | 246 |
| Issue number | 2-3 |
| Early online date | 15 Oct 2022 |
| DOIs | |
| Publication status | Published - Dec 2022 |
Bibliographical note
Funding Information:I. S. and B. d.W. have been partially supported by the Deutsche Forschungsgemeinschaft, SFB 910, Project A4 “Spatio-Temporal Patterns: Control, Delays, and Design”; B. d.W. was supported by the Berlin Mathematical School. J.-Y. D. has been supported by MOST Grant Number 110-2115-M-005-008-MY3. We are grateful to Bernold Fiedler, Alejandro López Nieto, and Jan Totz for many inspiring and fruitful discussions.
Publisher Copyright:
© 2022, The Author(s).
Funding
I. S. and B. d.W. have been partially supported by the Deutsche Forschungsgemeinschaft, SFB 910, Project A4 “Spatio-Temporal Patterns: Control, Delays, and Design”; B. d.W. was supported by the Berlin Mathematical School. J.-Y. D. has been supported by MOST Grant Number 110-2115-M-005-008-MY3. We are grateful to Bernold Fiedler, Alejandro López Nieto, and Jan Totz for many inspiring and fruitful discussions.
