arXiv subjects

Dongmei Xiao

Publications and source records attributed to Dongmei Xiao.

8 recordsLinked to original sources

arXiv2026

iSTAR: an algebraic-collapse framework for variational reduction in quantum-inspired continuous Ising solvers

Continuous Ising solvers embed a discrete optimization problem into a continuous dynamical system and recover the spin configuration by sign readout, but dense interaction evaluation gives an $O(N^2)$-per-step cost. We show that this cost is not intrinsic: during late-stage simulated bifurcation the trajectory collapses onto a lower-dimensional active subspace, and saturated coordinates can be eliminated exactly by a variational frozen-set identity whose couplings fold into an induced field on the unresolved subsystem. We prove large-parameter recovery for the external-field quartic model, the hard-box limit of ballistic confinement, and a robust-margin freezing criterion. The resulting algorithm, iSTAR (Ising Stable-set Tail-Aware Reduction), exploits this collapse by detecting stabilized coordinates and continuing only on the active tail. An online certified implementation on the G-set benchmark preserves the same-seed baseline in all runs and removes on average 64.4% of the dense interaction work.

math.NA↗

arXiv2026

On the cyclicity of the period annulus of quasi-homogeneous polynomial vector fields

In this article, we study the number of limit cycles, bifurcating from the period annulus of any quasi-homogeneous polynomial vector fields with a center, under a one-parameter polynomial perturbation. We first recharacterize quasi-homogeneous polynomial vector fields and its global center, then we establish an upper bound formula for the number of the isolated zeros of the $k$th order Melnikov function in terms of $k$, $max \{s_1,s_2\}$ and the degree $n$ of the perturbation by applying adapted Francoise's algorithm in conjunction with combinatorial techniques, where $(s_1,s_2)$ is the weight exponent of the quasi-homogeneous polynomial vector field. This extends relevant results presented in the literature [JDDE,21(2009)133-152] and [JDE, 276(2021)1-24]. As an application, we completely solve the limit cycle bifurcation problem of a perturbated quasi-homogeneous polynomial vector field.

math.DS↗

arXiv2025

Linear multiplicative noise destroys a two-dimensional attractive compact manifold of three-dimensional Kolmogorov systems

In the paper we first characterize three-dimensional Kolmogorov systems possessing a two-dimensional invariant sphere in $\mathbb{R}^3$, then establish a global attracting criterion for this invariant sphere in $\mathbb{R}^3$ except the origin, and give global dynamics with isolated equilibria on the sphere. Finally, we consider the persistence of the attractive invariant sphere under the perturbation induced by linear multiplicative Wiener noise. It is shown that suitable noise intensity can destroy the sphere and lead to bifurcation of stationary measures.

math.DS↗

arXiv2024

Stochastic bifurcation of a three-dimensional stochastic Kolmogorov system

In this paper we systematically investigate the stochastic bifurcations of both ergodic stationary measures and global dynamics for stochastic Kolmogorov differential systems, which relate closely to the change of the sign of Lyapunov exponents. It is derived that there exists a threshold $σ_0$ such that, if the noise intensity $σ\geqσ_0$, the noise destroys all bifurcations of the deterministic system and the corresponding stochastic Kolmogorov system is uniquely ergodic. On the other hand, when the noise intensity $σ<σ_0$, the stochastic system undergoes bifurcations from the unique ergodic stationary measure to three different types of ergodic stationary measures: (I) finitely many ergodic measures supported on rays, (II) infinitely many ergodic measures supported on rays, (III) infinitely many ergodic measures supported on invariant cones. Correspondingly, the global dynamics undergo similar bifurcation phenomena, which even displays infinitely many Crauel random periodic solutions in the sense of \cite{ELR21}. Furthermore, we prove that as $σ$ tends to zero, the ergodic stationary measures converge to either Dirac measures supported on equilibria, or to Haar measures supported on non-trivial deterministic periodic orbits.

math.DS↗

arXiv2023

Centers and invariant straight lines of planar real polynomial vector fields and its configurations

In the paper, we first give the least upper bound formula on the number of centers of planar real polynomial Hamiltonian vector fields. This formula reveals that the greater the number of invariant straight lines of the vector field and the less the number of its centers. Then we obtain some rules on the configurations of centers of planar real polynomial Hamiltonian Kolmogorov vector fields when the number of centers is exactly the least upper bound. As an application of these results, we give an affirmative answer to a conjecture on the topological classification of configurations for the cubic Hamiltonian Kolmogorov vector fields with four centers. Moreover, we discuss the relationship between the number of centers of planar real polynomial vector fields and the existence of limit cycles, and prove that cubic real polynomial Kolmogorov vector fields have no limit cycles if the number of its centers reaches the maximum. More precisely, it is shown that the cubic real polynomial Kolmogorov vector field must have an elementary first integral in $\mathbb{R}^2\setminus\{xy=0\}$ if it has four centers, and the number of configurations of its centers is one more than that of the cubic polynomial Hamiltonian Kolmogorov vector fields.

math.DS↗

arXiv2020

Mathematical Mechanism on Dynamical System Algorithms of the Ising Model

Various combinatorial optimization NP-hard problems can be reduced to finding the minimizer of an Ising model, which is a discrete mathematical model. It is an intellectual challenge to develop some mathematical tools or algorithms for solving the Ising model. Over the past decades, some continuous approaches or algorithms have been proposed from physical, mathematical or computational views for optimizing the Ising model such as quantum annealing, the coherent Ising machine, simulated annealing, adiabatic Hamiltonian systems, etc.. However, the mathematical principle of these algorithms is far from being understood. In this paper, we reveal the mathematical mechanism of dynamical system algorithms for the Ising model by Morse theory and variational methods. We prove that the dynamical system algorithms can be designed to minimize a continuous function whose local minimum points give all the candidates of the Ising model and the global minimum gives the minimizer of Ising problem. Using this mathematical mechanism, we can easily understand several dynamical system algorithms of the Ising model such as the coherent Ising machine, the Kerr-nonlinear parametric oscillators and the simulated bifurcation algorithm. Furthermore, motivated by the works of C. Conley, we study transit and capture properties of the simulated bifurcation algorithm to explain its convergence by the low energy transit and capture in celestial mechanics. A detailed discussion on $2$-spin and $3$-spin Ising models is presented as application.

math.OC↗

arXiv2019

Hilbert's 16th problem on a period annulus and Nash space of arcs

This article introduces an algebro-geometric setting for the space of bifurcation functions involved in the local Hilbert's 16th problem on a period annulus. Each possible bifurcation function is in one-to-one correspondence with a point in the exceptional divisor $E$ of the canonical blow-up $B_I{\mathbb C}^n$ of the Bautin ideal $I$. In this setting, the notion of essential perturbation, first proposed by Iliev, is defined via irreducible components of the Nash space of arcs $ Arc(B_I\mathbb C^n,E)$. The example of planar quadratic vector fields in the Kapteyn normal form is further discussed.

math.DS↗

arXiv2017

Dynamics of epidemic models with asymptomatic infection and seasonal succession

In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and valuated directly for this model, and the uniformly persistent of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0\le 1$, and as $\mathcal{R}_0>1$ the disease-free equilibrium is unstable and the model has an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infective and symptomatic infective are close. These theoretical results provide an intuitive basis for understanding that the asymptomatic infective individuals and the disease seasonal transmission promote the evolution of epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.

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