# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

July 2010 , Volume 14 , Issue 1

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2010, 14(1): 1-16 doi: 10.3934/dcdsb.2010.14.1 +[Abstract](2162) +[PDF](231.7KB)
Abstract:
The purpose of this paper is to study the relations between different concepts of dispersive solution for the Vlasov-Poisson system in the gravitational case. Moreover we give necessary conditions for the existence of partially and totally dispersive solutions and a sufficient condition for the occurence of statistical dispersion. These conditions take the form of inequalities involving the energy, the mass and the momentum of the solution. Examples of dispersive and non-dispersive solutions-steady states, periodic solutions and virialized solutions-are also considered.
2010, 14(1): 17-40 doi: 10.3934/dcdsb.2010.14.17 +[Abstract](3043) +[PDF](430.7KB)
Abstract:
This paper introduces a simplified dynamical systems framework for the study of the mechanisms behind the growth of cooperative learning in large communities. We begin from the simplifying assumption that individual-based learning focuses on increasing the individual's "fitness" while collaborative learning may result in the increase of the group's fitness. It is not the objective of this paper to decide which form of learning is more effective but rather to identify what types of social communities of learners can be constructed via collaborative learning. The potential value of our simplified framework is inspired by the tension observed between the theories of intellectual development (individual to collective or vice versa) identified with the views of Piaget and Vygotsky. Here they are mediated by concepts and ideas from the fields of epidemiology and evolutionary biology. The community is generated from sequences of successful "contacts'' between various types of individuals, which generate multiple nonlinearities in the corresponding differential equations that form the model. A bifurcation analysis of the model provides an explanation for the impact of individual learning on community intellectual development, and for the resilience of communities constructed via multilevel epidemiological contact processes, which can survive even under conditions that would not allow them to arise. This simple cooperative framework thus addresses the generalized belief that sharp community thresholds characterize separate learning cultures. Finally, we provide an example of an application of the model. The example is autobiographical as we are members of the population in this "experiment".
2010, 14(1): 41-74 doi: 10.3934/dcdsb.2010.14.41 +[Abstract](3024) +[PDF](3771.9KB)
Abstract:
The purpose of this paper is to develop a numerical procedure for the determination of frequencies and amplitudes of a quasi--periodic function, starting from equally-spaced samples of it on a finite time interval. It is based on a collocation method in frequency domain. Strategies for the choice of the collocation harmonics are discussed, in order to ensure good conditioning of the resulting system of equations. The accuracy and robustness of the procedure is checked with several examples. The paper is ended with two applications of its use as a dynamical indicator. The theoretical support for the method presented here is given in a companion paper [21].
2010, 14(1): 75-109 doi: 10.3934/dcdsb.2010.14.75 +[Abstract](2505) +[PDF](566.9KB)
Abstract:
In a previous paper [6], a numerical procedure for the Fourier analysis of quasi-periodic functions was developed, allowing for an accurate determination of frequencies and amplitudes from equally-spaced samples of the input function on a finite time interval. This paper is devoted to a complete error analysis of that procedure, from which computable bounds are deduced. These bounds are verified and further discussed in examples.
2010, 14(1): 111-128 doi: 10.3934/dcdsb.2010.14.111 +[Abstract](2308) +[PDF](248.0KB)
Abstract:
In this work we consider the existence of traveling plane wave solutions of systems of delayed lattice differential equations in competitive Lotka-Volterra type. Employing iterative method coupled with the explicit construction of upper and lower solutions in the theory of weak quasi-monotone dynamical systems, we obtain a speed, c *, and show the existence of traveling plane wave solutions connecting two different equilibria when the wave speeds are large than c *.
2010, 14(1): 129-141 doi: 10.3934/dcdsb.2010.14.129 +[Abstract](2551) +[PDF](196.7KB)
Abstract:
Stability and dynamic bifurcation in the ac-driven complex Ginzburg-Landau (GL) equation with periodic boundary conditions and even constraint are investigated using central manifold reduction procedure and attractor bifurcation theory. The results show that the bifurcation into an attractor near a small-amplitude limit cycle takes place on a two dimensional central manifold, as bifurcation parameter crosses a critical value. Furthermore, the component of the bifurcated attractor is analytically described for the non-autonomous system.
2010, 14(1): 143-158 doi: 10.3934/dcdsb.2010.14.143 +[Abstract](2269) +[PDF](321.0KB)
Abstract:
The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In [18, 21] the method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study of the geometrical properties of the unit balls of some specific Barabanov norms. In this context the situation when one fails to find among current publications any detailed analysis of the geometrical properties of the unit balls of Barabanov norms looks a bit paradoxical. Partially this is explained by the fact that Barabanov norms are defined nonconstructively, by an implicit procedure. So, even in simplest cases it is very difficult to visualize the shape of their unit balls. The present work may be treated as the first step to make up this deficiency. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.
2010, 14(1): 159-176 doi: 10.3934/dcdsb.2010.14.159 +[Abstract](2773) +[PDF](343.6KB)
Abstract:
Recently a discrete-time prey-predator model with Holling type II was discussed for its bifurcations so as to show its complicated dynamical properties. Simulation illustrated the occurrence of invariant cycles. In this paper we first clarify the parametric conditions of non-hyperbolicity, correcting a known result. Then we apply the center manifold reduction and the method of normal forms to completely discuss bifurcations of codimension 1. We give bifurcation curves analytically for transcritical bifurcation, flip bifurcation and Neimark-Sacker bifurcation separately, showing bifurcation phenomena not indicated in the previous work for the system.
2010, 14(1): 177-197 doi: 10.3934/dcdsb.2010.14.177 +[Abstract](2229) +[PDF](248.4KB)
Abstract:
In this article, we study the stability of weak solutions to the stochastic two dimensional (2D) primitive equations (PEs) with multiplicative noise. In particular, we prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions.
2010, 14(1): 199-207 doi: 10.3934/dcdsb.2010.14.199 +[Abstract](2396) +[PDF](162.8KB)
Abstract:
In this work, we generalize the idea of Ginzburg-Landau approximation to study the existence and asymptotic behaviors of global weak solutions to the one dimensional periodical fractional Landau-Lifshitz equation modeling the soft micromagnetic materials. We apply the Galerkin method to get an approximate solution and, to get the convergence of the nonlinear terms we introduce the commutator structure and take advantage of special structures of the equation.
2010, 14(1): 209-231 doi: 10.3934/dcdsb.2010.14.209 +[Abstract](3000) +[PDF](342.8KB)
Abstract:
The paper presents an SEIQHRS model for evaluating the combined impact of quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms) on the spread of a communicable disease. Rigorous analysis of the model, which takes the form of a deterministic system of nonlinear differential equations with standard incidence, reveal that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. Further, the model has a unique endemic equilibrium when the threshold quantity exceeds unity. Using a Krasnoselskii sub-linearity trick, it is shown that the unique endemic equilibrium is locally-asymptotically stable for a special case. A nonlinear Lyapunov function of Volterra type is used, in conjunction with LaSalle Invariance Principle, to show that the endemic equilibrium is globally-asymptotically stable for a special case. Numerical simulations, using a reasonable set of parameter values (consistent with the SARS outbreaks of 2003), show that the level of transmission by individuals isolated in hospitals play an important role in determining the impact of the two control measures (the use of quarantine and isolation could offer a detrimental population-level impact if isolation-related transmission is high enough).
2010, 14(1): 233-250 doi: 10.3934/dcdsb.2010.14.233 +[Abstract](2864) +[PDF](356.0KB)
Abstract:
We study the interaction of saddle-node and transcritical bifurcations in a Lotka-Volterra model with a constant term representing harvesting or migration. Because some of the equilibria of the model lie on an invariant coordinate axis, both the saddle-node and the transcritical bifurcations are of codimension one. Their interaction can be associated with either a single or a double zero eigenvalue. We show that in the former case, the local bifurcation diagram is given by a nonversal unfolding of the cusp bifurcation whereas in the latter case it is a nonversal unfolding of a degenerate Bogdanov-Takens bifurcation. We present a simple model for each of the two cases to illustrate the possible unfoldings. We analyse the consequences of the generic phase portraits for the Lotka-Volterra system.
2010, 14(1): 251-274 doi: 10.3934/dcdsb.2010.14.251 +[Abstract](2377) +[PDF](231.2KB)
Abstract:
Backward stochastic Volterra integral equations (BSVIEs in short) are studied. We introduce the notion of adapted symmetrical solutions (S-solutions in short), which are different from the M-solutions introduced by Yong [16]. We also give some new results for them. At last a class of dynamic coherent risk measures were derived via certain BSVIEs.
Fuke Wu and
2010, 14(1): 275-288 doi: 10.3934/dcdsb.2010.14.275 +[Abstract](2179) +[PDF](195.5KB)
Abstract:
In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, this paper perturbs the Lotka--Volterra system

$\dot{x}(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))$

into the corresponding stochastic system

$dx(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))[(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))dt+\beta dw(t)].$

This paper obtains one condition under which the above stochastic system has a global almost surely positive solution and gives the asymptotic pathwise estimation of this solution. This paper also shows that when the noise is sufficiently large, the solution of this stochastic system will converge to zero with probability one. This reveals that the sufficiently large noise may make the population extinct.

2010, 14(1): 289-306 doi: 10.3934/dcdsb.2010.14.289 +[Abstract](3426) +[PDF](239.8KB)
Abstract:
The dynamics of Leslie-Gower predator-prey models with constant harvesting rates are investigated. The ranges of the parameters involved in the systems are given under which the equilibria of the systems are positive. The phase portraits near these positive equilibria are studied. It is proved that the positive equilibria on the $x$-axis are saddle-nodes, saddles or unstable nodes depending on the choices of the parameters involved while the interior positive equilibria in the first quadrant are saddles, stable or unstable nodes, foci, centers, saddle-nodes or cusps. It is shown that there are two saddle-node bifurcations and by computing the Liapunov numbers and determining its signs, the supercritical or subcritical Hopf bifurcations and limit cycles for the weak centers are obtained.

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