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Zuohuan Zheng

Publications and source records attributed to Zuohuan Zheng.

At least 19 recordsLinked to original sources
arXiv2026

Countable periodic solutions of the Lorentz force equation under a time-dependent current

The resonant dynamics of a charged particle, governed by the Lorentz force equation in an electromagnetic field generated by a current-carrying wire with a small harmonic modulation, is considered in this study. When regarded as a Hamiltonian system with periodic perturbation, the resonance of periodic orbits in the unperturbed system is analyzed by the Melnikov method. The existence of exactly one harmonic radial periodic solution with period $T_1$ is confirmed, matching the period of the current. Moreover, it is established that any other radial periodic solution must be subharmonic with period $nT_1$ for some integer $n > 1$, with at most one such solution for each $n$. Dynamically, these surviving periodic orbits correspond to invariant cylinders that partition the phase space and globally confine the particle's radial motion.

math.DS↗
arXiv2022

Dynamics of COVID-19 models with asymptomatic infections and quarantine measures

Considering the propagation characteristics of COVID-19 in different regions, the dynamics analysis and numerical demonstration of long-term and short-term models of COVID-19 are carried out, respectively. The long-term model is devoted to investigate the global stability of COVID-19 model with asymptomatic infections and quarantine measures. By using the limit system of the model and Lyapunov function method, it is shown that the COVID-19-free equilibrium $V^0$ is globally asymptotically stable if the control reproduction number $\mathcal{R}_{c}<1$ and globally attractive if $\mathcal{R}_{c}=1$, which means that COVID-19 will die out; the COVID-19 equilibrium $V^{\ast}$ is globally asymptotically stable if $\mathcal{R}_{c}>1$, which means that COVID-19 will be persistent. In particular, to obtain the local stability of $V^{\ast}$, we use proof by contradiction and the properties of complex modulus with some novel details, and we prove the weak persistence of the system to obtain the global attractivity of $V^{\ast}$. Moreover, the final size of the corresponding short-term model is calculated and the stability of its multiple equilibria is analyzed. Numerical simulations of COVID-19 cases show that quarantine measures and asymptomatic infections have a non-negligible impact on the transmission of COVID-19.

math.DS↗
arXiv2022

A novel analysis approach of uniform persistence for a COVID-19 model with quarantine and standard incidence rate

A coronavirus disease 2019 (COVID-19) model with quarantine and standard incidence rate is first developed, then a novel analysis approach for finding the ultimate lower bound of COVID-19 infectious individuals is proposed, which means that the COVID-19 pandemic is uniformly persistent if the control reproduction number $\mathcal{R}_{c}>1$. This approach can be applied to other related biomathematical models, and some existing works can be improved by using it. In addition, the COVID-19-free equilibrium $V^0$ is locally asymptotically stable (LAS) if $\mathcal{R}_{c}<1$ and linearly stable if $\mathcal{R}_{c}=1$, respectively; while $V^0$ is unstable if $\mathcal{R}_{c}>1$.

q-bio.PE↗
arXiv2022

Large deviations principle for stationary solutions of stochastic differential equations with multiplicative noise

We study the large deviations principle (LDP) for stationary solutions of a class of stochastic differential equations (SDE) in infinite time intervals by the weak convergence approach, and then establish the LDP for the invariant measures of the SDE by the contraction principle. We further point out the equivalence of the rate function of the LDP for invariant measures induced by the LDP for stationary solutions and the rate function defined by quasi-potential. This fact gives another view of the quasi-potential introduced by Freidlin and Wentzell.

math.PR↗
arXiv2022

Random periodic solutions of nonautonomous stochastic feedback systems with multiplicative noise

We investigate the dynamical behavior of pull-back trajectories for nonautonomous stochastic feedback systems with multiplicative noise. We proved that there exists a random periodic solution of this system and all pull-back trajectories converge to this random periodic solution as time goes to infinitely almost surely. Our results can be applied to nonautonomous stochastic Goodwin negative feedback system, nonautonomous stochastic Othmer-Tyson positive feedback system and nonautonomous stochastic competitive systems etc.

math.DS↗
arXiv2021

Exact mobility edges for 1D quasiperiodic models

Mobility edges (ME), i.e. critical energies which separate absolutely continuous spectrum and purely point spectrum, is an important issue in quantum physics. So far there are two experimentally feasible 1D quasiperiodic models that have been discovered to have exact mobility edge. However, all the theoretical studies have remained at the numerical level. In this paper, we rigorously prove the existence and give the precise location of the MEs for these models.

math.DS↗
arXiv2021

Duality between two generalized Aubry-Andre models with exact mobility edges

A mobility edge (ME) in energy separating extended from localized states is a central concept in understanding various fundamental phenomena like the metal-insulator transition in disordered systems. In one-dimensional quasiperiodic systems, there exist a few models with exact MEs, and these models are beneficial to provide exact understanding of ME physics. Here we investigate two widely studied models including exact MEs, one with an exponential hopping and one with a special form of incommensurate on-site potential. We analytically prove that the two models are mutually dual, and further give the numerical verification by calculating the inverse participation ratio and Husimi function. The exact MEs of the two models are also obtained by calculating the localization lengths and using the duality relations. Our result may provide insight into realizing and observing exact MEs in both theory and experiment.

cond-mat.dis-nn↗
arXiv2021

Random periodic solutions of non-autonomous stochastic differential equations

In this paper, we study the existence of random periodic solutions for nonlinear stochastic differential equations with additive white noise. We extend the input-to-state characteristic operator of the system to the non-autonomous stochastic differential equation via the pull-back of the discretised stochastic differential equation. We then use the completeness of the measurable function space which we construct skillfully and the Banach fixed point theorem to prove the existence of a fixed point of the gain operator. And we prove that the image for the input-to-state characteristic operator at this fixed point is a random periodic solution for the forward stochastic flow generated by the non-autonomous stochastic differential equation. Finally, we present some examples.

math.DS↗
arXiv2020

Spatial and temporal dynamics of an almost periodic reaction-diffusion system for West Nile virus

In current paper, we put forward a reaction-diffusion system for West Nile virus in spatial heterogeneous and time almost periodic environment with free boundaries to investigate the influences of the habitat differences and seasonal variations on the propagation of West Nile virus. The existence, uniqueness and regularity estimates of the global solution for this disease model are given. Focused on the effects of spatial heterogeneity and time almost periodicity, we apply the principal Lyapunov exponent $λ(t)$ with time $t$ to get the initial infected domain threshold $L^*$ to analyze the long-time dynamical behaviors of the solution for this almost periodic West Nile virus model and give the spreading-vanishing dichotomy regimes of the disease. Especially, we prove that the solution for this West Nile virus model converges to a time almost periodic function locally uniformly for $x$ in $\mathbb R$ when the spreading occurs, which is driven by spatial differences and seasonal recurrence. Moreover, the initial disease infected domain and the front expanding rate have momentous impacts on the permanence and extinction of the epidemic disease. Eventually, numerical simulations identify our theoretical results.

math.AP↗
arXiv2020

Exact non-Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice

We analytically determine the non-Hermitian mobility edges of a one-dimensional quasiperiodic lattice model with exponential decaying hopping and complex potentials as well as its dual model, which is just a non-Hermitian generalization of the Ganeshan-Pixley-Das Sarma model with nonreciprocal nearest-neighboring hopping. The presence of non-Hermitian term destroys the self-duality symmetry and thus prevents us exploring the localization-delocalization point through looking for self-dual points. Nevertheless, by applying Avila's global theory, the Lyapunov exponent of the Ganeshan-Pixley-Das Sarma model can be exactly derived, which enables us to get an analytical expression of mobility edge of the non-Hermitian dual model. Consequently, the mobility edge of the original model is obtained by using the dual transformation, which creates exact mappings between the spectra and wavefunctions of these two models.

cond-mat.dis-nn↗
arXiv2020

Coexistence of zero Lyapunov exponent and positive Lyapunov exponent for new quasi-periodic Schr$\ddot{o}$dinger operator

In this paper we solve a problem about the Schr$\ddot{o}$dinger operator with potential $v(θ)=2λcos2πθ/(1-αcos2πθ),\ (|α|<1)$ in physics. With the help of the formula of Lyapunov exponent in the spectrum, the coexistence of zero Lyapunov exponent and positive Lyapunov exponent for some parameters is first proved, and there exists a curve that separates them. The spectrum in the region of positive Lyapunov exponent is purely pure point spectrum with exponentially decaying eigenfunctions for almost every frequency and almost every phase. From the research, we realize that the infinite potential $v(θ)=2λtan^2(πθ)$ has zero Lyapunov exponent for some energies if $0<|λ|<1$.

math.DS↗
arXiv2020

A new metric for statistical properties of long time behaviors

Let $(X,T)$ be a topological dynamical system with metric $d$. We define a new function $\overline{F}(x,y)=\limsup\limits_{n \to +\infty} \inf\limits_{σ\in S_n} \frac 1n \sum\limits_{k=1}^n d(T^k x,T^{σ(k)} y)$ by using permutation group $S_n$. It's shown $F(x,y)=\lim\limits_{n \to +\infty} \inf\limits_{σ\in S_n} \frac 1n \sum\limits_{k=1}^n d(T^k x,T^{σ(k)} y)$ exists when $x,y \in X$ are generic points. Applying this function, we prove $(X,T)$ is uniquely ergodic if and only if $\overline{F}(x,y)=0$ for any $x,y \in X$. The characterizations of ergodic measures and physical measures by $\overline{F}(x,y)$ are given. We introduce the notion of weak mean equicontinuity and prove that $(X,T)$ is weak mean equicontinuous if and only if the time averages $f^{*}(x)=\lim\limits_{n \to +\infty}\frac 1n \sum\limits_{k=1}^n f(T^k x)$ exist and are continuous for all $f \in C(X)$.

math.DS↗
arXiv2020

Dynamics and Spreading Speed of a Reaction-Diffusion System with Advection Modeling West Nile Virus

This paper aims to explore the temporal-spatial spreading and asymptotic behaviors of West Nile virus by a reaction-advection-diffusion system with free boundaries, especially considering the impact of advection term on the extinction and persistence of West Nile virus. We define the spatial-temporal risk index $R^{F}_{0}(t)$ with the advection rate and the general basic disease reproduction number $R^D_0$ to get the vanishing-spreading dichotomy regimes of West Nile virus. We show that there exists a threshold value $μ^{*}$ of the advection rate, and obtain the threshold results of it. When the spreading occurs, we investigate the asymptotic dynamical behaviors of the solution in the long run and first give a sharper estimate that the asymptotic spreading speed of the leftward front is less than the rightward front for $0<μ<μ^*$. At last, we give some numerical simulations to identify the significant effects of the advection.

math.AP↗
arXiv2018

Some Spectrum Property of Periodic Coupling AMO Operator

We study spectrum of the periodic coupling AMO model. Meantime there establish the continuity of Lyapunov exponent about the the periodic coupling of AMO model. Through the dynamical method can find a interval the AMO model only have absolutely continuous spectrum. At the same time, some condition make the periodic coupling of AMO model is singular continuous.

math.DS↗
arXiv2017

On the convergence of the time average for skew-product structure and multiple ergodic system

In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation and observation function are neglected in some measure-theoretical sense. The theorems extend the classical results which have been established for continuous dynamical systems or continuous observation functions. Meanwhile, on the torus $\mathbb{T}^{d}$ with special rotation, we prove the pointwise convergence of multiple ergodic average $\disp \f 1 N \sum_{n=0}^{N-1} f_{1}(R_α^{n}x)f_{2}(R_α^{2n}x)$ in $\mathbb{T}^{d}$.

math.DS↗
arXiv2017

Ergodic behaviour of nonconventional ergodic averages for commuting transformations

Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to the space average, of diagonal measures is obtained and we give different result according to the classification of transformations. Additionally, on the torus with special rotation. we can not only get the convergence in T.Tao's paper for every point in Td, but also get a beautiful result for ergodic behaviour.

math.DS↗