• NCTS Young Theoretical Scientists Award (2023) 

• Doctoral Fellowship, German Academic Exchange Service (DAAD) (2012)

• Government Scholarship to Study Abroad, Ministry of Education of Taiwan (2012)

4. Symmetry Groupoids for Pattern-Selective Feedback Stabilization of the Chafee–Infante Equation

    Chaos: An Interdisciplinary Journal of Nonlinear Science, 33, 073141 (2023) 

　(Special Issue: Control of Self-Organizing Nonlinear Systems) 

    with Isabelle Schneider 

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. 

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

    Archive for Rational Mechanics and Analysis, 246, no. 2-3, 631-658 (2022) 

    with Isabelle Schneider and  Babette de Wolff

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.

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

    SIAM Journal on Applied Dynamical Systems, 20, no. 4, p.1959-1984 (2021)

with Phillipo Lappicy

This paper consists of three results on pattern formation of Ginzburg-Landau m-armed vortex solutions and spiral waves in circular and spherical geometries. First, we completely describe the global bifurcation diagram of vortex equilibria. Second, we prove persistence of all bifurcation curves under perturbations of parameters, which yields the existence of spiral waves for the complex Ginzburg-Landau equation. Third, we explicitly construct the global attractor of m-armed vortex solutions. Our main tool is a new shooting method that allows us to prove hyperbolicity of vortex equilibria in the invariant subspace of vortex solutions.

1. Ginzburg-Landau Spiral Waves in Circular and Spherical Geometries

    SIAM Journal on Mathematical Analysis, 53, no. 1, p.1004-1028 (2021)

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.