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Discrete & Continuous Dynamical Systems - B

September 2011 , Volume 16 , Issue 2

A special issue
Dedicated to Qishao Lu on the occasion of his 70th birthday

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Preface
Zhaosheng Feng and Jinzhi Lei
2011, 16(2): i-iv doi: 10.3934/dcdsb.2011.16.2i +[Abstract](655) +[PDF](4833.4KB)
Abstract:
This issue of Discrete and Continuous Dynamical Systems–Series B, is dedicated to our professor and friend, Qishao Lu, on the occasion of his 70th birthday and in honor of his important and fundamental contributions to the fields of applied mathematics, theoretical mechanics and computational neurodynamics. His pleasant personality and ready helpfulness have won our hearts as his admirers, students, and friends.

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Lyapunov stability for conservative systems with lower degrees of freedom
Jifeng Chu, Jinzhi Lei and Meirong Zhang
2011, 16(2): 423-443 doi: 10.3934/dcdsb.2011.16.423 +[Abstract](815) +[PDF](450.8KB)
Abstract:
It is a central theme to study the Lyapunov stability of periodic solutions of nonlinear differential equations or systems. For dissipative systems, the Lyapunov direct method is an important tool to study the stability. However, this method is not applicable to conservative systems such as Lagrangian equations and Hamiltonian systems. In the last decade, a method that is now known as the 'third order approximation' has been developed by Ortega, and has been applied to particular types of conservative systems including time periodic scalar Lagrangian equations (Ortega, J. Differential Equations, 128(1996), 491-518). This method is based on Moser's twist theorem, a prototype of the KAM theory. Latter, the twist coefficients were re-explained by Zhang in 2003 through the unique positive periodic solutions of the Ermakov-Pinney equation that is associated to the first order approximation (Zhang, J. London Math. Soc., 67(2003), 137-148). After that, Zhang and his collaborators have obtained some important twist criteria and applied the results to some interesting examples of time periodic scalar Lagrangian equations and planar Hamiltonian systems. In this survey, we will introduce the fundamental ideas in these works and will review recent progresses in this field, including applications to examples such as swing, the (relativistic) pendulum and singular equations. Some unsolved problems will be imposed for future study.
Bursting and two-parameter bifurcation in the Chay neuronal model
Lixia Duan, Zhuoqin Yang, Shenquan Liu and Dunwei Gong
2011, 16(2): 445-456 doi: 10.3934/dcdsb.2011.16.445 +[Abstract](715) +[PDF](409.3KB)
Abstract:
In this paper, we study and classify the firing patterns in the Chay neuronal model by the fast/slow decomposition and the two-parameter bifurcations analysis. We show that the Chay neuronal model can display complex bursting oscillations, including the "fold/fold" bursting, the "Hopf/Hopf" bursting and the "Hopf/homoclinic" bursting. Furthermore, dynamical properties of different firing activities of a neuron are closely related to the bifurcation structures of the fast subsystem. Our results indicate that the codimension-2 bifurcation points and the related codimension-1 bifurcation curves of the fast-subsystem can provide crucial information to predict the existence and types of bursting with changes of parameters.
Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme
Fang Han, Bin Zhen, Ying Du, Yanhong Zheng and Marian Wiercigroch
2011, 16(2): 457-474 doi: 10.3934/dcdsb.2011.16.457 +[Abstract](769) +[PDF](190.6KB)
Abstract:
Global Hopf bifurcation analysis is carried out on a six-dimensional FitzHugh-Nagumo (FHN) neural network with a time delay. First, the existence of local Hopf bifurcations of the system is investigated and the explicit formulae which can determine the direction of the bifurcations and the stability of the periodic solutions are derived using the normal form method and the center manifold theory. Then the sufficient conditions for the system to have multiple periodic solutions when the delay is far away from the critical values of Hopf bifurcations are obtained by using the Wu's global Hopf bifurcation theory and the Bendixson's criterion. Especially, a synchronized scheme is used during the analysis to reduce the dimension of the system. Finally, example numerical simulations are given to support the theoretical analysis.
Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system
Cuncai Hua, Guanrong Chen, Qunhong Li and Juhong Ge
2011, 16(2): 475-488 doi: 10.3934/dcdsb.2011.16.475 +[Abstract](601) +[PDF](353.2KB)
Abstract:
A problem of reducing a general three-dimensional (3-D) autonomous quadratic system to a Lorenz-type system is studied. Firstly, under some necessary conditions for preserving the basic qualitative properties of the Lorenz system, the general 3-D autonomous quadratic system is converted to an extended Lorenz-type system (ELTS) which contains a large class of existing chaotic dynamical systems. Secondly, some different canonical forms of the ELTS are obtained with the aid of various nonsingular linear transformations and normalization techniques. Thirdly, the conjugate systems of the ELTS are defined and discussed. Finally, a sufficient condition for the nonexistence of chaos in such ELTS is derived.
Consensus of discrete-time linear multi-agent systems with observer-type protocols
Zhongkui Li, Zhisheng Duan and Guanrong Chen
2011, 16(2): 489-505 doi: 10.3934/dcdsb.2011.16.489 +[Abstract](882) +[PDF](588.2KB)
Abstract:
This paper concerns the consensus of discrete-time multi-agent systems with linear or linearized dynamics. An observer-type protocol based on the relative outputs of neighboring agents is proposed. The consensus of such a multi-agent system with a directed communication topology can be cast into the stability of a set of matrices with the same low dimension as that of a single agent. The notion of discrete-time consensus region is then introduced and analyzed. For neurally stable agents, it is shown that there exists an observer-type protocol having a bounded consensus region in the form of an open unit disk, provided that each agent is stabilizable and detectable. An algorithm is further presented to construct a protocol to achieve consensus with respect to all the communication topologies containing a spanning tree. Moreover, for the case where the agents have no poles outside the unit circle, an algorithm is proposed to construct a protocol having an origin-centered disk of radius
$\delta$ ($0<\delta<1$) as its consensus region. Finally, the consensus algorithms are applied to solve formation control problems of multi-agent systems.
Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation
Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian and Meisheng Li
2011, 16(2): 507-527 doi: 10.3934/dcdsb.2011.16.507 +[Abstract](751) +[PDF](361.8KB)
Abstract:
In this paper, a three dimensional Ginburg-Landau type equation is considered. Firstly, two families of new traveling wave solutions in term of explicit functions are presented by using the homogeneous balance method, in which one consists of variable-amplitude solutions and the other constant-amplitude solutions (namely, plane wave solutions). Moreover, the stability of plane wave solutions is analyzed by using the regular phase plane techniques.
Firing control of ink gland motor cells in Aplysia californica
Xiangying Meng, Quanbao Ji and John Rinzel
2011, 16(2): 529-545 doi: 10.3934/dcdsb.2011.16.529 +[Abstract](603) +[PDF](3710.6KB)
Abstract:
The release of ink in Aplysia californica occurs selectively to long-lasting stimuli. There is a good correspondence between features of the behavior and the firing pattern of the ink gland motor neurons. Indeed, the neurons do not fire for brief inputs and there is a delayed firing for long duration inputs. The biophysical mechanisms for the long delay before firing is due to a transient potassium current which activates rapidly but inactivates more slowly. Based on voltage-clamp experiments, a nine-variable Hodgkin-Huxley-like model for the ink gland motor neurons was developed by Byrne. Here, fast-slow analysis and two-parameter dynamical analysis are used to investigate the contribution of different currents and to predict various firing patterns, including the long latency before firing.
Border-collision bifurcations in a generalized piecewise linear-power map
Zhiying Qin, Jichen Yang, Soumitro Banerjee and Guirong Jiang
2011, 16(2): 547-567 doi: 10.3934/dcdsb.2011.16.547 +[Abstract](741) +[PDF](4999.1KB)
Abstract:
In this paper a class of generalized piecewise smooth maps is studied, which is linear at one side and nonlinear with power dependence at the other side. According to the value of the power in the term $x^z$, the bifurcations occurring in this map are classified into five types: $z>1$, $z=1$, $0<z<1$, $z=0$, and $z<0$. We derive the occurrence conditions of border collision bifurcation and smooth fold and flip bifurcation, especially the codimension-2 bifurcation points describing the interaction between border collision bifurcation and smooth bifurcation. The general results are then applied to the specific cases of the power $z$, and different bifurcation scenarios are shown for individual cases, from which the period-adding scenario is found to be general for any power.
A reliability study of square wave bursting $\beta$-cells with noise
Jiaoyan Wang, Jianzhong Su, Humberto Perez Gonzalez and Jonathan Rubin
2011, 16(2): 569-588 doi: 10.3934/dcdsb.2011.16.569 +[Abstract](622) +[PDF](1143.0KB)
Abstract:
Reliability of spike timing has been a hot topic recently. However reliability has not been considered for bursting behavior, as commonly observed in a variety of nerve and endocrine cells, including $\beta$-cells in intact pancreatic islets. In this paper, reliability of $\beta$-cells with noise is considered. A method to numerically study reliability of bursting cells is presented. Reliability of a single cell will decrease as noise level becomes larger. The reliability of networks of $\beta$-cells coupled by gap junctions or synaptic excitation is investigated. Simulations of the network of $\beta$-cells reveal that increasing noise level decreases the reliability. But the reliability of the network is higher than that of single cell. The effect of coupling strength on reliability is also investigated. Reliability will decrease when coupling strength is small and increase when coupling strength is large.
A constraint-stabilized method for multibody dynamics with friction-affected translational joints based on HLCP
Qi Wang, Huilian Peng and Fangfang Zhuang
2011, 16(2): 589-605 doi: 10.3934/dcdsb.2011.16.589 +[Abstract](594) +[PDF](1070.6KB)
Abstract:
In this paper, a constraint-stabilized numerical method is presented for the planar rigid multibody system with friction-affected translational joints, in which the sliders and the guides are treated as particles and bilateral constraints, respectively. The dynamical equations of the non-smooth system are obtained by using the first kind of Lagrange's equations and Baumgarte stabilization method. The normal forces of bilateral constraints are expressed by the Lagrange multipliers and described by complementarity condition, while frictional forces are characterized by a set-valued force law of the type of Coulomb's law for dry friction. Using event-driven scheme, the state transition problem of stick-slip and normal forces of bilateral constraints is formulated and solved as a horizontal linear complementarity problem (HLCP). Finally, the planar rigid multibody system with two translational joints is considered as a illustrative application example. The results obtained also show that the drift of constraints of the system remains bounded.
Delay-induced synchronization transition in small-world Hodgkin-Huxley neuronal networks with channel blocking
Qingyun Wang, Xia Shi and Guanrong Chen
2011, 16(2): 607-621 doi: 10.3934/dcdsb.2011.16.607 +[Abstract](796) +[PDF](2048.7KB)
Abstract:
We study the evolution of spatiotemporal dynamics and synchronization transition on small-world Hodgkin-Huxley (HH) neuronal networks that are characterized with channel noises, ion channel blocking and information transmission delays. In particular, we examine the effects of delay on spatiotemporal dynamics over neuronal networks when channel blocking of potassium or sodium is involved. We show that small delays can detriment synchronization in the network due to a dynamic clustering anti-phase synchronization transition. We also show that regions of irregular and regular wave propagations related to synchronization transitions appear intermittently as the delay increases, and the delay-induced synchronization transitions manifest as well-expressed minima in the measure for spatial synchrony. In addition, we show that the fraction of sodium or potassium channels can play a key role in dynamics of neuronal networks. Furthermore, We found that the fraction of sodium and potassium channels has different impacts on spatiotemporal dynamics of neuronal networks, respectively. Our results thus provide insights that could facilitate the understanding of the joint impact of ion channel blocking and information transmission delays on the dynamical behaviors of realistic neuronal networks.
Positive solutions of $p$-Laplacian equations with nonlinear boundary condition
Zuodong Yang, Jing Mo and Subei Li
2011, 16(2): 623-636 doi: 10.3934/dcdsb.2011.16.623 +[Abstract](860) +[PDF](390.9KB)
Abstract:
In this paper we study the following problem $$-\triangle_{p}u+|u|^{p-2}u=f(x,u) $$ in a bounded smooth domain $\Omega \subset {\bf R}^{N}$ with a nonlinear boundary value condition $|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}=g(x,u)$. Results on the existence of positive solutions are obtained by the sub-supersolution method and the Mountain Pass Lemma.
Bifurcation analysis of bursting solutions of two Hindmarsh-Rose neurons with joint electrical and synaptic coupling
Feng Zhang, Wei Zhang, Pan Meng and Jianzhong Su
2011, 16(2): 637-651 doi: 10.3934/dcdsb.2011.16.637 +[Abstract](612) +[PDF](1260.6KB)
Abstract:
In this paper, we investigate the dynamic behavior of a system of two coupled Hindmarsh-Rose (HR) neurons, based on bifurcation analysis of its fast subsystem. The individual HR neuron has chaotic behavior, but they can become regularized when coupled through synaptic coupling or joint electrical-synaptic coupling. Through numerical methods we first investigate the bifurcation structure of its fast subsystem. We show that the emerging of periodic patterns of neurons is related to topological changes of its underlying bifurcations. The Lyaponov exponent calculations further reveal the pathway from chaotic bursting behavior to regular bursting of HR neurons. Finally, we include both electrical and synaptic coupling in the system, and numerically calculate the time dynamics. Even though electrical couplings (or gap junctions) usually does not regularize chaotic trajectories, but joint coupling has been more effective than synaptic coupling alone in producing stable rhythms. The main contribution of this paper is that we provide a mathematical description for transitions of neuron dynamics from chaotic trajectories to regular bursting when synaptic and electrical-synaptic coupling strengthens, using bifurcation analysis.
Time-varying delayed feedback control for an internet congestion control model
Shu Zhang and Jian Xu
2011, 16(2): 653-668 doi: 10.3934/dcdsb.2011.16.653 +[Abstract](766) +[PDF](722.2KB)
Abstract:
A proportionally-fair controller with time delay is considered to control Internet congestion. The time delay is chosen to be a controllable parameter. To represent the relation between the delay and congestion analytically, the method of multiple scales is employed to obtain the periodic solution arising from the Hopf bifurcation in the congestion control model. A new control method is proposed by perturbing the delay periodically. The strength of the perturbation is predicted analytically in order that the oscillation may disappear gradually. It implies that the proved control scheme may decrease the possibility of the congestion derived from the oscillation. The proposed control scheme is verified by the numerical simulation.
Optimal regularity for $A$-harmonic type equations under the natural growth
Shenzhou Zheng, Xueliang Zheng and Zhaosheng Feng
2011, 16(2): 669-685 doi: 10.3934/dcdsb.2011.16.669 +[Abstract](685) +[PDF](431.1KB)
Abstract:
In this paper we are concerned with a class of nonlinear degenerate elliptic equations under the natural growth. We show that each bounded weak solution of $A$-harmonic type equations under the natural growth belongs to local Hölder continuity based on a density lemma and the Moser-Nash's argument. Then we show that its weak solution is of optimal regularity with the Hölder exponent for any $\gamma$: $0\le \gamma<\kappa$, where $\kappa$ is the same as the Hölder's index for homogeneous $A$-harmonic equations.

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