DCDS
Robustly transitive singular sets via approach of an extended linear Poincaré flow
Ming Li Shaobo Gan Lan Wen
Morales, Pacifico and Pujals proved recently that every robustly transitive singular set for a 3-dimensional flow must be partially hyperbolic. In this paper we generalize the result to higher dimensions. By definition, an isolated invariant set $\Lambda$ of a $C^1$ vector field $S$ is called robustly transitive if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that for every $X\in\mathcal U$, the maximal invariant set of $X$ in $U$ is non-trivially transitive. Such a set $\Lambda$ is called singular if it contains a singularity. The set $\Lambda$ is called strongly homogeneous of index $i$, if there exist an isolating neighborhood $U$ of $\Lambda$ in $M$ and a $C^1$ neighborhood $\mathcal U$ of $S$ in $\mathcal X^1(M)$ such that all periodic orbits of all $X\in\mathcal U$ contained in $U$ have the same index $i$. We prove in this paper that any robustly transitive singular set that is strongly homogeneous must be partially hyperbolic, as long as the indices of singularities and periodic orbits fit in certain way. As corollaries we obtain that every robust singular attractor (or repeller) that is strongly homogeneous must be partially hyperbolic and, if dim$M\le 4$, every robustly transitive singular set that is strongly homogeneous must be partially hyperbolic. The main novelty of the proofs in this paper is an extension of the usual linear Poincaré flow "to singularities".
keywords: star flow. Robustly transitive set extended linear poincaré flow partial hyperbolicity
DCDS
Stability of parameterized Morse-Smale gradient-like flows
Ming-Chia Li
In this paper, we are concerned with the structural stability of a Morse-Smale gradient-like flow $\varphi^t$ and show that if $\{\varphi_\epsilon^t\}$ is a smooth one-parameter family of $C^3$ flows with $\varphi_0^t=\varphi^t$, and {$\psi_\epsilon^t$} is another one-parameter family of $C^3$ flows such that $\psi_\epsilon^t$ is $C^0$ $O(\epsilon^3)$-close and $C^1$ $O(\epsilon^2)$-close to $\varphi_\epsilon^t$, then for all small $|\epsilon|$, there is a homeomorphism $h_\epsilon$, which is $C^0$ $O(\epsilon^2)$-near the identity map, such that $h_\epsilon$ takes the trajectories of $\varphi^t_\epsilon$ to the ones of $\psi^t_\epsilon$.
keywords: Morse-Smale gradient flow averaging method. structural stability topological conjugacy
NACO
Stability analysis of parametric variational systems
Shengji Li Chunmei Liao Minghua Li
In this paper, Robinson's metric regularity of a positive order around/at some point of parametric variational systems is discussed. Under some suitable conditions, the relationships among H$\ddot{o}$lder-likeness, H$\ddot{o}$lder calmness, metric regularity of a positive order and Robinson's metric regularity of a positive order are discussed for the parametric variational systems. Then, some applications to the stabilities of the optimal value map and the solution map are studied for a parametric vector optimization problem, respectively.
keywords: H$\ddot{o}$lder-likeness Parametric variational systems metric regularity of a positive order. Robinson's metric regularity of a positive order
DCDS
Positive topological entropy for multidimensional perturbations of topologically crossing homoclinicity
Ming-Chia Li Ming-Jiea Lyu
In this paper, we consider a one-parameter family $F_{\lambda }$ of continuous maps on $\mathbb{R}^{m}$ or $\mathbb{R}^{m}\times \mathbb{R}^{k}$ with the singular map $F_{0}$ having one of the forms (i) $F_{0}(x)=f(x),$ (ii) $F_{0}(x,y)=(f(x),g(x))$, where $g:\mathbb{R}^{m}\rightarrow \mathbb{R} ^{k}$ is continuous, and (iii) $F_{0}(x,y)=(f(x),g(x,y))$, where $g:\mathbb{R}^{m}\times \mathbb{R}^{k}\rightarrow \mathbb{R}^{k}$ is continuous and locally trapping along the second variable $y$. We show that if $f:\mathbb{R}^{m}\rightarrow \mathbb{R}^{m}$ is a $C^{1}$ diffeomorphism having a topologically crossing homoclinic point, then $F_{\lambda }$ has positive topological entropy for all $\lambda $ close enough to $0$.
keywords: topological crossing homoclinicity. topological entropy Multidimensional perturbation
IPI
Near-field imaging of sound-soft obstacles in periodic waveguides
Ming Li Ruming Zhang

In this paper, we introduce a direct method for the inverse scattering problems in a periodic waveguide from near-field scattered data. The direct scattering problem is to simulate the point sources scattered by a sound-soft obstacle embedded in the periodic waveguide, and the aim of the inverse problem is to reconstruct the obstacle from the near-field data measured on line segments outside the obstacle. Firstly, we will approximate the scattered field by some solutions of a series of Dirichlet exterior problems, and then the shape of the obstacle can be deduced directly from the Dirichlet boundary condition. We will also show that the approximation procedure is reasonable as the solutions of the Dirichlet exterior problems are dense in the set of scattered fields. Finally, we will give several examples to show that this method works well for different periodic waveguides.

keywords: Near-field imaging direct method periodic waveguide least square method limiting absorption principle
DCDS
Topological conjugacy for Lipschitz perturbations of non-autonomous systems
Ming-Chia Li Ming-Jiea Lyu
In this paper, topological conjugacy for two-sided non-hyperbolic and non-autonomous discrete dynamical systems is studied. It is shown that if the system has covering relations with weak Lyapunov condition determined by a transition matrix, there exists a sequence of compact invariant sets restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by the transition matrix. Moreover, if the systems have covering relations with exponential dichotomy and small Lipschitz perturbations, then there is a constructive verification proof of the weak Lyapunov condition, and so topological dynamics of these systems are fully understood by symbolic representations. In addition, the tolerance of Lipschitz perturbation can be characterised by the dichotomy tuple . Here, the weak Lyapunov condition is adapted from [12,24,15] and the exponential dichotomy is from [2].
keywords: nonuniformly hyperbolic systems. exponential dichotomy Lipschitz perturbation Topological conjugacy non-autonomous systems
MBE
A mathematical model for the seasonal transmission of schistosomiasis in the lake and marshland regions of China
Yingke Li Zhidong Teng Shigui Ruan Mingtao Li Xiaomei Feng

Schistosomiasis, a parasitic disease caused by Schistosoma Japonicum, is still one of the most serious parasitic diseases in China and remains endemic in seven provinces, including Hubei, Anhui, Hunan, Jiangsu, Jiangxi, Sichuan, and Yunnan. The monthly data of human schistosomiasis cases in Hubei, Hunan, and Anhui provinces (lake and marshland regions) released by the Chinese Center for Disease Control and Prevention (China CDC) display a periodic pattern with more cases in late summer and early autumn. Based on this observation, we construct a deterministic model with periodic transmission rates to study the seasonal transmission dynamics of schistosomiasis in these lake and marshland regions in China. We calculate the basic reproduction number $R_{0}$, discuss the dynamical behavior of solutions to the model, and use the model to fit the monthly data of human schistosomiasis cases in Hubei. We also perform some sensitivity analysis of the basic reproduction number $R_{0}$ in terms of model parameters. Our results indicate that treatment of at-risk population groups, improving sanitation, hygiene education, and snail control are effective measures in controlling human schistosomiasis in these lakes and marshland regions.

keywords: Seasonal schistosomiasis model basic reproduction number extinction uniform persistence
DCDS
Periodic shadowing of vector fields
Jifeng Chu Zhaosheng Feng Ming Li
A vector field has the periodic shadowing property if for any $\varepsilon>0$ there is $d>0$ such that, for any periodic $d$-pseudo orbit $g$ there exists a periodic orbit or a singularity in which $g$ is $\varepsilon$-shadowed. In this paper, we show that a vector field is in the $C^1$ interior of the set of vector fields satisfying the periodic shadowing property if and only if it is $\Omega$-stable. More precisely, we prove that the $C^1$ interior of the set of vector fields satisfying the orbital periodic shadowing property is a subset of the set of $\Omega$-stable vector fields.
keywords: homoclinic connection diffeomorphism $\Omega$-stability. periodic shadowing property Vector fields hyperbolic singularity
MBE
Transmission dynamics and control for a brucellosis model in Hinggan League of Inner Mongolia, China
Mingtao Li Guiquan Sun Juan Zhang Zhen Jin Xiangdong Sun Youming Wang Baoxu Huang Yaohui Zheng
Brucellosis is one of the major infectious and contagious bacterial diseases in Hinggan League of Inner Mongolia, China. The number of newly infected human brucellosis data in this area has increased dramatically in the last 10 years. In this study, in order to explore effective control and prevention measures we propose a deterministic model to investigate the transmission dynamics of brucellosis in Hinggan League. The model describes the spread of brucellosis among sheep and from sheep to humans. The model simulations agree with newly infected human brucellosis data from 2001 to 2011, and the trend of newly infected human brucellosis cases is given. We estimate that the control reproduction number $\mathcal{R}_{c}$ is about $1.9789$ for the brucellosis transmission in Hinggan League and compare the effect of existing mixed cross infection between basic ewes and other sheep or not for newly infected human brucellosis cases. Our study demonstrates that combination of prohibiting mixed feeding between basic ewes and other sheep, vaccination, detection and elimination are useful strategies in controlling human brucellosis in Hinggan League.
keywords: Brucellosis vaccination and detection control strategy. basic reproduction number
DCDS
Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss
Sze-Bi Hsu Ming-Chia Li Weishi Liu Mikhail Malkin
This paper is concerned with the classical Nicholson-Bailey model [15] defined by $f_\lambda(x,y)=(y(1-e^{-x}), \lambda y e^{-x})$. We show that for $\lambda=1$ a heteroclinic foliation exists and for $\lambda>1$ global strict oscillations take place. The important phenomenon of delay of stability loss is established for a general class of discrete dynamical systems, and it is applied to the study of nonexistence of periodic orbits for the Nicholson-Bailey model.
keywords: singular perturbation. global oscillation Nicholson-Bailey model heteroclinic foliation delay of stability loss

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