arXiv subjects

Casey Crane

Publications and source records attributed to Casey Crane.

4 recordsLinked to original sources

arXiv2026

Global Hopf Bifurcation and Symmetric Periodic Solutions in Multi-Agent Systems with Neutral Distributed Delays

We study the emergence of symmetric oscillatory behavior in multi-agent systems where each agent incorporates a continuous memory of its past states and past rates of change, modeled by distributed retarded and neutral delays. The closed-loop dynamics are described by a system of nonlinear neutral functional differential equations (NFDEs) with a high degree of symmetry, arising from a network of homogeneous agents. By reformulating the problem as a fixed point operator equation, we apply equivariant degree theory to establish rigorous conditions for unbounded global Hopf bifurcation from the consensus equilibrium. Our main results provide sufficient conditions for the local asymptotic stability of consensus and for the existence of unbounded global branches of non-constant periodic solutions with prescribed spatio-temporal symmetries. The question of whether such periodic solutions are stable (and therefore constitute periodic multiconsensus) is not resolved by the degree method; we address it in an illustrative example via numerical simulation. The example, which models eight coupled asset markets with momentum traders and fundamentalists, demonstrates how memory-driven instability can generate periodic boom-bust cycles across clusters of assets. The numerical experiments confirm the bifurcation predictions and reveal the stability of the resulting oscillations, illustrating the power of combining symmetric bifurcation theory with targeted numerical analysis.

math.DS↗

arXiv2025

Unbounded Branches of Non-Radial Solutions to Semilinear Elliptic Systems on a Unit Ball in $\mathbb R^3$ and Their Patterns

We investigate symmetry-breaking phenomena in semilinear elliptic systems on the unit ball in $\mathbb{R}^3$, focusing on the emergence of non-radial solution branches with prescribed spatial and internal symmetries. Extending previous scalar results, we develop a framework for systems equivariant under $G := O(3) \times \Gamma \times \mathbb{Z}_2$, where $\Gamma$ is a finite group encoding coupling symmetries. Using the $G$-equivariant Leray--Schauder degree and Burnside ring techniques, we derive computable criteria for the existence of unbounded branches of non-radial solutions and classify their isotropy types. Our approach accommodates non-simple eigenvalue multiplicities and provides explicit bifurcation conditions in terms of spectral resonance between coupling eigenvalues and spherical Laplacian modes. Applications to coupled spherical oscillators illustrate how Platonic symmetries and internal permutations interact to produce complex patterns. These results establish a general method for detecting and characterizing symmetry-breaking in high-dimensional elliptic systems.

math.AP↗

arXiv2025

Unbounded Branches of Non-Radial Solutions to Semilinear Elliptic Systems on a Disc and their Patterns

In this paper, we leverage the $O(2) \times \mathbb Z$-equivariant Leray-Schauder degree and a novel characterization of the Burnside Ring $A(O(2) \times \mathbb Z_2)$ presented by Ghanem in \cite{Ghanem1} to obtain $(\rm i)$ an existence result for non-radial solutions to the problem $-\Delta u = f(z,u) + Au$, $u|_{\partial D} = 0$ and $(\rm ii)$ local and global bifurcation results for multiple branches of non-radial solutions to the one-parameter family of equations $-\Delta u = f(z,u) + \textbf{A}(\alpha)u$, $u|_{\partial D} = 0$, where $D$ is the planar unit disc, $u(z) \in \mathbb R^N$, $A : \mathbb R^N \rightarrow \mathbb R^N$ is an $N \times N$ matrix, $\textbf{A}: \mathbb R \rightarrow L(\mathbb R^N)$ is a continuous family of $N \times N$ matrices and $f: \overline D \times \mathbb R^N \rightarrow \mathbb R^N$ is a sublinear, $O(2) \times \mathbb Z_2$-equivariant function of order $o(|u|)$ as $u$ approaches the origin in $\mathbb R^N$.

math.AP↗

arXiv2024

Global Bifurcation of Non-Radial Solutions for Symmetric Sub-linear Elliptic Systems on the Planar Unit Disc

In this paper, we prove a global bifurcation result for the existence of non-radial branches of solutions to the paramterized family of $\Gamma$-symmetric problems $-\Delta u=f(\alpha,z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in \mathbb{C} : |z|<1\}$ with $u(z)\in \mathbb{R}^k$, where $\mathbb{R}^k$ is an orthogonal $\Gamma$-representation, $f: \mathbb{R} \times \overline{D} \times \mathbb{R}^k\to \mathbb{R}^k$ is a sub-linear $\Gamma$-equivariant continuous function, differentiable with respect to $u$ at zero and satisfying the conditions $f(\alpha, e^{i\theta}z,u)=f(\alpha, z,u)$ for all $\theta\in \mathbb{R}$ and $f(z,-u)=-f(z,u)$.

math.AP↗