arXiv subjects

Haozhe Shu

Publications and source records attributed to Haozhe Shu.

3 recordsLinked to original sources

arXiv2026

Critical Three-Species Competition-Diffusion: Traveling-Wave Rigidity and the Fastest-Species Selection

We study the rigidity of traveling waves and long-time species selection in a one-dimensional critical three-species competition-diffusion system. We establish several rigidity results for traveling waves connecting distinct equilibria on the critical simplex. In the case where one species has a strictly larger Fisher-KPP spreading speed than the other two, an entropy method, combined with Gagliardo-Nirenberg and Nash inequalities yieldsuniform extinction of the slower species and convergence to the fastest-species equilibrium throughout every cone with speed below its KPP speed.

math.AP↗

arXiv2026

Asymptotic behavior of solutions to linear evolution equations with time delay via a spectral theory on Gelfand triples

In this paper, a class of linear evolution equations with time delay is studied in which the presence of continuous spectrum on the imaginary axis obstructs the analysis of long-time dynamics. To address it, a generalized spectral framework on a Gelfand triple is utilized. When the spectral measure of the unperturbed term (a skew-adjoint operator) admits some analyticity condition, the resolvent is extended to a generalized resolvent. Called generalized spectrum, the collection of singularities on the Riemann surface of the generalized resolvent may differ from the spectrum in the usual sense because of the change of topology via the Gelfand triple. It is shown that under some compactness assumption, the generalized spectrum consists only of isolated generalized eigenvalues (resonance poles). This structure allows contour deformation in the inverse Laplace representation and yields exponential decay in a weak topology. As an application, we analyze the continuum limit of the Kuramoto-Daido model with time delay and prove linear stability of the incoherent state in the weak coupling regime.

math.DS↗

arXiv2025

Analysis of pitchfork bifurcations and symmetry breaking in the elliptic restricted three-body problem

A unified framework is proposed to quantitatively characterize pitchfork bifurcations and associated symmetry breaking in the elliptic restricted three-body problem (ERTBP). It is known that planar/vertical Lyapunov orbits and Lissajous orbits near the collinear libration points undergo pitchfork bifurcations with varying orbital energy. These bifurcations induce symmetry breaking, generating bifurcated families including halo/quasi-halo orbits, axial/quasi-axial orbits, and their corresponding invariant manifolds. Traditional semi-analytical methods for constructing halo orbits, based on resonant bifurcation mechanisms, have obstacles in fully exploiting the intrinsic symmetry breaking characteristics in pitchfork bifurcations. In this paper, a unified trigonometric series-based framework is proposed to analyze these bifurcated families from the perspective of coupling-induced bifurcation mechanisms. By introducing a coupling coefficient and various bifurcation equations into the ERTBP, different symmetry breaking is achieved when the coupling coefficient is non-zero. This unified semi-analytical framework captures bifurcations of both periodic/quasi-periodic and transit/non-transit orbits. Furthermore, it reveals that pitchfork bifurcation solutions in the ERTBP fundamentally depend solely on the orbital eccentricity and three amplitude parameters of the system's degrees of freedom, governing both the elliptic direction and the hyperbolic one.

math.DS↗