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Jia-Yuan Dai

Publications and source records attributed to Jia-Yuan Dai.

8 recordsLinked to original sources
arXiv2026

Global Continuation of Stable Periodic Orbits in Systems of Competing Predators

We develop a continuation technique to obtain global families of stable periodic orbits, delimited by transcritical bifurcations at both ends. To this end, we formulate a zero-finding problem whose zeros correspond to families of periodic orbits. We then define a Newton-like fixed-point operator and establish its contraction near a numerically computed approximation of the family. To verify the contraction, we derive sufficient conditions expressed as inequalities on the norms of the fixed-point operator, and involving the numerical approximation. These inequalities are then rigorously checked by the computer via interval arithmetic. To show the efficacy of our approach, we prove the existence of global families in an ecosystem with Holling's type II functional response, and thereby solve a stable connection problem proposed by Butler and Waltman in 1981. Our method does not rely on restricting the choice of parameters and is applicable to many other systems that numerically exhibit global families.

math.DS↗
arXiv2025

Transient rebellions in the Kuramoto oscillator: Morse-Smale structural stability and connection graphs of finite 2-shift type

The celebrated 1975 Kuramoto model of $N$ identical oscillators with phase angle vector $\boldsymbol{\vartheta}=(\vartheta_1,\ldots,\vartheta_N)$ and all-to-all coupling reads \begin{equation} \label{*} \dot\vartheta_j\,=\tfrac{1}{N}\sum_{k=1}^N \sin(\vartheta_k-\vartheta_j). \tag{*} \end{equation} Here we have passed to co-rotating coordinates in normalized time scale. The model is highly accessible to rigorous mathematical analysis, and has been studied as a paradigm for effects like total and partial synchronization. Most initial conditions $\boldsymbol{\vartheta}$ lead to total synchronization. The plethora of $2^N-1$ (circles of) partially synchronized states, however, is unstable. The precise behavior of transitions to synchrony seems to have eluded description. In the present paper, we address this gap. By the gradient structure of (*), the global dynamics decompose into equilibria and heteroclinic orbits between them. Except for the extremes of total instability and total synchrony, all equilibria are 2-cluster solutions: their phase angles $\vartheta_j$ attain only two values, with a phase difference of $π$ between them. Any heteroclinic orbit between 2-cluster equilibria, or towards synchrony, can be realized as a 3-cluster rebellion. Cluster rebellions split the smaller, "slim", minority cluster of the source equilibrium and send the rebellious part to join the bigger, "fat", majority cluster. Heteroclinic transversality identifies the Kuramoto model as a structurally stable Morse-Smale system. In particular, heteroclinic orbits can be concatenated in finite time. The options involved in successive cluster rebellions amount to finite symbol sequences of 2-shift type. The paper is dedicated to Professor Yoshiki Kuramoto, with admiration.

math.DS↗
arXiv2024

Hybrid Bifurcations: Periodicity from Eliminating a Line of Equilibria

We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations: Such bifurcations occur when a line of equilibria with an exchange point of normal stability vanishes. Our main result is the existence and stability criteria of periodic orbits that bifurcate from breaking a line of equilibria. As an application, we obtain stable periodic coexistent solutions in an ecosystem for two competing predators with Holling's type II functional response.

math.DS↗
arXiv2023

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

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.

math.DS↗
arXiv2022

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

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.

math.DS↗
arXiv2021

Ginzburg-Landau patterns in circular and spherical geometries: vortices, spirals and attractors

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.

math.DS↗
arXiv2020

Ginzburg-Landau Spiral Waves in Circular and Spherical Geometries

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.

math.AP↗