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Discrete & Continuous Dynamical Systems - S

June 2017 , Volume 10 , Issue 3

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Preface
Zhaosheng Feng
2017, 10(3): i-i doi: 10.3934/dcdss.201703i +[Abstract](156) +[HTML](82) +[PDF](71.7KB)
Abstract:
Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term
Lianzhang Bao and Zhengfang Zhou
2017, 10(3): 395-412 doi: 10.3934/dcdss.2017019 +[Abstract](167) +[HTML](3) +[PDF](424.3KB)
Abstract:

This work is concerned with the properties of the traveling wave solutions of a one dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion with net birth term

where \begin{document}$ D(ρ) $\end{document} may take positive or negative values with different population density \begin{document}$ ρ $\end{document} and adhesion coefficient \begin{document}$ α ∈ [0,1] $\end{document}, and the negative one will lead to the ill-posedness of the equation. In all these cases we prove the existence of infinitely many traveling wave solutions, where these solutions are parameterized by their wave speed and monotonically connect the stationary states \begin{document}$ ρ\equiv0 $\end{document} and \begin{document}$ ρ\equiv 1 $\end{document}.

State transformations of time-varying delay systems and their applications to state observer design
Dinh Cong Huong and Mai Viet Thuan
2017, 10(3): 413-444 doi: 10.3934/dcdss.2017020 +[Abstract](291) +[HTML](4) +[PDF](630.0KB)
Abstract:

In this paper, we derive new state transformations of linear systems with a time-varying delay in the state vector. We first provide a new algebraic and systematic method for computing forward state transformations to transform time-delay systems into a novel form where time-varying delay appears in the input and output vectors, but not in the state vector. In the new coordinate system, a Luenberger-type state observer with a guaranteed \begin{document}$ β $\end{document}-exponential stability margin can be designed. Then, a backward state transformation problem which allows us to reconstruct the original state vector of the system is investigated. By using both the forward and the backward state transformations, state observers for time-varying delay systems can be systematically designed. Conditions for ensuring the existence of the forward and backward state transformations and an effective algorithm for computing them are given in this paper. We illustrate our results by three examples and simulation results.

A periodic and diffusive predator-prey model with disease in the prey
Xiaoling Li, Guangping Hu, Zhaosheng Feng and Dongliang Li
2017, 10(3): 445-461 doi: 10.3934/dcdss.2017021 +[Abstract](245) +[HTML](1) +[PDF](433.7KB)
Abstract:

In this paper, we are concerned with a time periodic and diffusivepredator-prey model with disease transmission in the prey. Firstwe consider a \begin{document}$ SI $\end{document} model when the predator species is absent. Byintroducing the basic reproduction number for the \begin{document}$ SI $\end{document} model, weshow the sufficient conditions for the persistence and extinctionof the disease. When the presence of the predator is taken intoaccount, a number of sufficient conditions for the co-existence ofthe prey and predator species, the global extinction of predatorspecies and the global extinction of both the prey and predatorspecies are given.

Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales
Yongkun Li and Pan Wang
2017, 10(3): 463-473 doi: 10.3934/dcdss.2017022 +[Abstract](351) +[HTML](1) +[PDF](366.3KB)
Abstract:

We first propose a concept of almost periodic functions in the sense of Stepanov on time scales. Then, we consider a class of neutral functional dynamic equations with Stepanov-almost periodic terms on time scales in a Banach space. By means of the contraction mapping principle, we establish some criteria for the existence and uniqueness of almost periodic solutions for this class of dynamic equations on time scales. Finally, we give an example to illustrate the effectiveness of our results.

Condensing operators and periodic solutions of infinite delay impulsive evolution equations
Jin Liang, James H. Liu and Ti-Jun Xiao
2017, 10(3): 475-485 doi: 10.3934/dcdss.2017023 +[Abstract](202) +[HTML](2) +[PDF](367.3KB)
Abstract:

By showing the existence of the fixed point of the condensing operators in the phasespace \begin{document}$ C_μ $\end{document} for the Cauchy problem for impulsive evolution equations with infinite delay in a Banach space \begin{document}$ X $\end{document}:

where \begin{document}$ \mathfrak{A}(t) $\end{document} is \begin{document}$ \varpi $\end{document}-periodic, the operator \begin{document}$ \mathfrak{A}(t) $\end{document} is unbounded for each \begin{document}$ t>0 $\end{document}, \begin{document}$ x_t (s)=x(t+s),\; s≤0$\end{document}, \begin{document}$ Δ x(t_i)= x(t_i ^+)-x(t_i ^- ) $\end{document}, \begin{document}$ \mathfrak{F} $\end{document}, \begin{document}$ φ $\end{document} and \begin{document}$ \mathfrak{I}_i\ (i=1,···,n) $\end{document} are given functions, we derive periodic solutions from bounded solutions. The new periodic solution existence results obtained here extend earlier results in this area for evolution equations without impulsive conditions or without infinite delay.

The mixed-mode oscillations in Av-Ron-Parnas-Segel model
Bo Lu, Shenquan Liu, Xiaofang Jiang, Jing Wang and Xiaohui Wang
2017, 10(3): 487-504 doi: 10.3934/dcdss.2017024 +[Abstract](173) +[HTML](1) +[PDF](4335.6KB)
Abstract:

Mixed-mode oscillations (MMOs) as complex firing patterns with both relaxation oscillations and sub-threshold oscillations have been found in many neural models such as the stellate neuron model, HH model, and so on. Based on the work, we discuss mixed-mode oscillations in the Av-Ron-Parnas-Segel model which can govern the behavior of the neuron in the lobster cardiac ganglion. By using the geometric singular perturbation theory we first explain why the MMOs exist in the reduced Av-Ron-Parnas-Segel model. Then the mixed-mode oscillatory phenomenon and aperiodic mixed-mode behaviors in the model have been analyzed numerically. Finally, we illustrate the influence of certain parameters on the model.

Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative
Chun Wang and Tian-Zhou Xu
2017, 10(3): 505-521 doi: 10.3934/dcdss.2017025 +[Abstract](235) +[HTML](2) +[PDF](403.2KB)
Abstract:

In this paper, using the weighted space method and a fixed point theorem, we investigate the Hyers-Ulam-Rassias stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville derivative on the continuous function space. We obtain some sufficient conditions in order that the nonlinear fractional differential equations are stable on the continuous function space. The results improve and extend some recent results. Finally, we construct some examples to illustrate the theoretical results.

Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay
Runxia Wang, Haihong Liu, Fang Yan and Xiaohui Wang
2017, 10(3): 523-542 doi: 10.3934/dcdss.2017026 +[Abstract](224) +[HTML](10) +[PDF](8151.2KB)
Abstract:

In this paper, we study a coupled FitzHugh-Nagumo (FHN) neurons model with time delay. The existence conditions on Hopf-pitchfork singularity are given. By selecting the coupling strength and time delay as the bifurcation parameters, and by means of the center manifold reduction and normal form theory, the normal form for this singularity is found to analyze the behaviors of the system. We perform the bifurcation analysis and numerical simulations, and present the bifurcation diagrams. Some interesting phenomena are observed, such as the existence of a stable fixed point, a stable periodic solution, a pair of stable fixed points, and the coexistence of a pair of stable fixed points and a stable periodic solution near the Hopf-pitchfork critical point.

Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term
Xiaoming Wang
2017, 10(3): 543-556 doi: 10.3934/dcdss.2017027 +[Abstract](161) +[HTML](1) +[PDF](403.0KB)
Abstract:

In this paper we consider the existence of Aubry-Mather sets and quasi-periodic solutions for a class of second order differential equation with a nonlinear damping term

where \begin{document} $q, f∈ C^1(\mathbb{R}),$ \end{document} \begin{document} $g(t,x)∈ C^{0,1}(\mathbf{S}^1× \mathbb{R})$ \end{document} and \begin{document} $p(t)∈ C^0(\mathbf{S}^1)$ \end{document}, \begin{document} $\mathbf{S}^1= \mathbb{R}/2π\mathbb{Z}$ \end{document}, \begin{document} $α$ \end{document} and \begin{document} $β $ \end{document} are two positive constants satisfying

with \begin{document} $ω∈ \mathbb{R}^+ $ \end{document}. Under some assumptions on the parities of \begin{document} $f,$ \end{document} \begin{document} $g$ \end{document} and \begin{document} $p$ \end{document}, we obtain the existence of infinitely many generalized quasi-periodic solutions via a result of Chow and Pei from the Aubry-Mather theory of reversible mapping. In particular, an advantage of our approach is that it does not require any high smoothness assumptions on the functions \begin{document} $q, f, g$ \end{document} and \begin{document} $p$ \end{document}.

Exponential stability of 1-d wave equation with the boundary time delay based on the interior control
Yaru Xie and Genqi Xu
2017, 10(3): 557-579 doi: 10.3934/dcdss.2017028 +[Abstract](207) +[HTML](7) +[PDF](425.1KB)
Abstract:

In this paper, the stability problem of 1-d wave equation with the boundary delay and the interior control is considered. The well-posedness of the closed-loop system is investigated by the linear operator. Based on the idea of Lyapunov functional technology, we give the condition on the relationship between the control parameter α and the delay parameter k to guarantee the exponential stability of the system.

Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay
Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian and Zhaosheng Feng
2017, 10(3): 581-603 doi: 10.3934/dcdss.2017029 +[Abstract](255) +[HTML](4) +[PDF](453.9KB)
Abstract:

This paper is concerned with traveling waves for temporally delayed, spatially discrete reaction-diffusion equations without quasi-monotonicity. We first establish the existence of non-critical traveling waves (waves with speeds c>c*, where c* is minimal speed). Then by using the weighted energy method with a suitably selected weight function, we prove that all noncritical traveling waves Φ(x+ct) (monotone or nonmonotone) are time-asymptotically stable, when the initial perturbations around the wavefronts in a certain weighted Sobolev space are small.

On a hyperbolic-parabolic mixed type equation
Huashui Zhan
2017, 10(3): 605-624 doi: 10.3934/dcdss.2017030 +[Abstract](142) +[HTML](1) +[PDF](443.6KB)
Abstract:

In this paper, the hyperbolic-parabolic mixed type equation

with the homogeneous boundary condition is considered. We find that only a part of the boundary condition is able to ensure the posedness of the solutions. By introducing a new kind of entropy solution matching the part boundary condition in a special way, we obtain the existence of the solution by the $BV$ estimate method, and establish the stability of the solutions by the Kruzkov bi-variables method.

Bifurcation analysis of the three-dimensional Hénon map
Ming Zhao, Cuiping Li, Jinliang Wang and Zhaosheng Feng
2017, 10(3): 625-645 doi: 10.3934/dcdss.2017031 +[Abstract](528) +[HTML](4) +[PDF](366.3KB)
Abstract:

In this paper, we consider the dynamics of a generalized three-dimensional Hénon map. Necessary and sufficient conditions on the existence and stability of the fixed points of this system are established. By applying the center manifold theorem and bifurcation theory, we show that the system has the fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation under certain conditions. Numerical simulations are presented to not only show the consistence between examples and our theoretical analysis, but also exhibit complexity and interesting dynamical behaviors, including period-10, -13, -14, -16, -17, -20, and -34 orbits, quasi-periodic orbits, chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors. These results demonstrate relatively rich dynamical behaviors of the three-dimensional Hénon map.

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