Ginzburg--Landau Patterns in Circular and Spherical Geometries: Vortices, Spirals, and Attractors

Dai, Jia-Yuan; Lappicy, Phillipo

doi:10.1137/20m1378739

Cited by 6 publications

(10 citation statements)

References 33 publications

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“…Remark 1.2. We not only greatly extend the existence results in [26,34], but also the bifurcation structure established in this paper is useful in studying other problems related to the dynamics of spiral waves, for instance, the hyperbolicity in [9] and feedback control stabilization by the method in [32].…”

Section: Resultsmentioning

confidence: 68%

“…Based on our bifurcation approach we are able to obtain solutions with sign-changing amplitude; see Lemma 3.5 and Fig. 4 in this paper, and also [9].…”

Section: Resultsmentioning

confidence: 91%

“…Remark 1.6. Based on the setting and approach established in [8], we successfully extend all other bifurcation curves globally by careful analysis on shooting manifolds; see [9].…”

Section: Resultsmentioning

confidence: 99%

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Ginzburg-Landau Spiral Waves in Circular and Spherical Geometries

Dai

2020

Preprint

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We prove the existence of m-armed spiral wave solutions for the complex Ginzburg-Landau equation in the circular and spherical geometries. We establish a new global bifurcation approach and generalize the results of existence for rigidly-rotating spiral waves. Moreover, we prove the existence of two new patterns: frozen spirals in the circular and spherical geometries, and 2-tip spirals in the spherical geometry.

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Section: Resultsmentioning

confidence: 68%

“…Based on our bifurcation approach we are able to obtain solutions with sign-changing amplitude; see Lemma 3.5 and Fig. 4 in this paper, and also [9].…”

Section: Resultsmentioning

confidence: 91%

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Ginzburg-Landau Spiral Waves in Circular and Spherical Geometries

Dai

2020

Preprint

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“…We aim to understand pattern formation, dynamical behavior, and feedback controls of Ginzburg-Landau spiral waves on the surface M. To this end we present a trilogy of research: existence, stability analysis, and feedback stabilization. The first two episodes regarding existence and stability analysis have been investigated extensively in [3,19,33] and also by Dai in [4,5]. This article serves as the third episode in which we stabilize certain classes of unstable spiral waves by introducing noninvasive symmetry-breaking feedback controls with spatio-temporal delays.…”

Section: Introductionmentioning

confidence: 98%

“…It has been proved in [10,16] that spiral waves of (1.1) exist on the plane R 2 . Since in experiments and numerical simulations the underlying domain is bounded, in [4,5] Dai carried out a global bifurcation analysis and proved the existence of spiral waves in circular and spherical geometries.…”

Section: Introductionmentioning

confidence: 99%

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

Schneider¹,

Wolff²,

Dai³

2022

Preprint

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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, most 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.

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Pattern-Selective Feedback Stabilization of Ginzburg–Landau Spiral Waves

Schneider

Wolff

Dai

2022

Arch Rational Mech Anal

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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.

show abstract

Ginzburg--Landau Patterns in Circular and Spherical Geometries: Vortices, Spirals, and Attractors

Cited by 6 publications

References 33 publications

Ginzburg-Landau Spiral Waves in Circular and Spherical Geometries

Ginzburg-Landau Spiral Waves in Circular and Spherical Geometries

Pattern-Selective Feedback Stabilization of Ginzburg--Landau Spiral Waves

Pattern-Selective Feedback Stabilization of Ginzburg–Landau Spiral Waves

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