# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

August 2009 , Volume 24 , Issue 3

A special issue

Dedicated to Peter W. Bates on the occasion of his 60th birthday

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2009, 24(3): i-ii doi: 10.3934/dcds.2009.24.3i +[Abstract](1331) +[PDF](40.3KB)
Abstract:
This special issue of Discrete and Continuous Dynamical Systems-A is dedicated to Peter W. Bates on the occasion of his 60th birthday, and in recognition of his outstanding contributions to infinite dimensional dynamical systems and the mathematical theory of phase transitions.
Peter Bates was born in Manchester, England on December 27, 1947. He graduated from the University of London in mathematics in 1969 after which he moved to United States with his family. Later, he attended the University of Utah and received his Ph.D. in 1976. Following his graduation, Peter moved to Texas and taught at University of Texas at Pan American and Texas A&M University. He returned to Utah in 1984 and taught at Brigham Young University until 2004. He is currently a professor of mathematics at Michigan State University.

2009, 24(3): 653-658 doi: 10.3934/dcds.2009.24.653 +[Abstract](1594) +[PDF](114.7KB)
Abstract:
We prove that a boundary value problem for a semilinear wave equation with smooth nonlinearity, smooth forcing, and no resonance cannot have continuous solutions. Our proof shows that this is due to the non-monotonicity of the nonlinearity.
2009, 24(3): 659-673 doi: 10.3934/dcds.2009.24.659 +[Abstract](1711) +[PDF](215.7KB)
Abstract:
The uniqueness and stability of traveling wave solutions for system of nonlocal evolution equations with bistable nonlinearity are established. It is also proved that traveling waves are monotone and exponentially asymptotically stable, up to translation.
2009, 24(3): 675-697 doi: 10.3934/dcds.2009.24.675 +[Abstract](1779) +[PDF](586.5KB)
Abstract:
The purpose of this paper is to introduce the model reference control method (MRC) in system biology. We review the main framework of MRC based on neural networks and some research issues. The model reference control for some model biological systems plant is considered.
2009, 24(3): 699-729 doi: 10.3934/dcds.2009.24.699 +[Abstract](1431) +[PDF](386.9KB)
Abstract:
Quadratic perturbations of a one-parameter family of quadratic reversible systems with two centers (without other singularities in finite plane) are studied. The exact upper bound of the number of limit cycles, the configurations of limit cycles, and the bifurcation diagrams for different range of the parameter are given.
2009, 24(3): 731-761 doi: 10.3934/dcds.2009.24.731 +[Abstract](1632) +[PDF](291.7KB)
Abstract:
We study a system of elliptic equations arising from biology with a chemotaxis term. This system is non-variational. Using a reduction argument, we show that the system has solutions with peaks near the boundary and inside the domain.
2009, 24(3): 763-780 doi: 10.3934/dcds.2009.24.763 +[Abstract](1694) +[PDF](251.0KB)
Abstract:
In this paper, we study a model of insect and animal dispersal where both density-dependent diffusion and nonlinear rate of growth are present. We analyze the existence of bounded traveling wave solution under certain parametric conditions by using the qualitative theory of dynamical systems. An explicit traveling wave solution is obtained by means of the first integral method. Traveling wave solutions in parametric forms for three particular cases are established by the Lie symmetry method.
2009, 24(3): 781-807 doi: 10.3934/dcds.2009.24.781 +[Abstract](1414) +[PDF](327.7KB)
Abstract:
The problem of discerning key features of steady turbulent flow adjacent to a wall has drawn the attention of some of the most noted fluid dynamicists of all time. Standard examples of such features are found in the mean velocity profiles of turbulent flow in channels, pipes or boundary layers. The aim of this article is to explain and further develop the recent concept of scaling patch for the time-averaged equations of motion of incompressible flow made highly turbulent by friction at a fixed boundary (introduced in recent papers by Wei et al, Fife et al, and Klewicki et al.) Besides outlining ways to identify the patches, which provide the scaling structure of mean profiles, a critical comparison will be made between that approach and more traditional ones.
Our emphasis will be on the question of how and how well these arguments supply insight into the structure of the mean flow profiles. Although empirical results may initiate the search for explanations, they will be viewed simply as means to that end.
2009, 24(3): 809-826 doi: 10.3934/dcds.2009.24.809 +[Abstract](1278) +[PDF](245.0KB)
Abstract:
We consider wavefronts that arise in a mathematical model for high Lewis number combustion processes. An efficient method for the proof of the existence and uniqueness of combustion fronts is provided by geometric singular perturbation theory. The fronts supported by the model with very large Lewis numbers are small perturbations of the front supported by the model with infinite Lewis number. The question of stability for the fronts is more complicated. Besides discrete spectrum, the system possesses essential spectrum up to the imaginary axis. We show how a geometric approach which involves construction of the Stability Index Bundles can be used to relate the spectral stability of wavefronts with high Lewis numbers to the spectral stability of the front in the case of infinite Lewis number. We discuss the implication for nonlinear stability of fronts with high Lewis numbers. This work builds on the ideas developed by Gardner and Jones [12] and generalized in the papers by Bates, Fife, Gardner and Jones [3, 4].
2009, 24(3): 827-840 doi: 10.3934/dcds.2009.24.827 +[Abstract](1265) +[PDF](160.1KB)
Abstract:
For the near-Hamiltonian system $\dot{x}=y+\varepsilon P(x,y),\dot{y}=x-x^2+\varepsilon Q(x,y)$, where $P$ and $Q$ are polynomials of $x,y$ having degree 3 with varying coefficients we obtain 5 limit cycles.
2009, 24(3): 841-854 doi: 10.3934/dcds.2009.24.841 +[Abstract](1690) +[PDF](263.2KB)
Abstract:
In this paper we investigate critical periods for a planar cubic differential system with a periodic annulus linking to equilibria at infinity. The monotonicity of the period function is decided by the sign of the second order derivative of a Abelian integral. We derive a Picard-Fuchs equation from a system of Abelian integrals and further give an induced Riccati equation for a ratio of derivatives of Abelian integrals. The number of critical points of the period function for periodic annulus is determined by discussing an planar autonomous system, the orbits of which describe solutions of the Riccati equation.
2009, 24(3): 855-882 doi: 10.3934/dcds.2009.24.855 +[Abstract](1910) +[PDF](331.7KB)
Abstract:
The current paper is devoted to the study of pullback attractors for general nonautonomous and random parabolic equations on non-smooth domains $D$. Mild solutions are considered for such equations. We first extend various fundamental properties for solutions of smooth parabolic equations on smooth domains to solutions of general parabolic equations on non-smooth domains, including continuous dependence on parameters, monotonicity, and compactness, which are of great importance in their own. Under certain dissipative conditions on the nonlinear terms, we prove that mild solutions with initial conditions in $L_q(D)$ exist globally for $q$ » $1$. We then show that pullback attractors for nonautonomous and random parabolic equations on non-smooth domains exist in $L_q(D)$ for $1$ « $q$ < $\infty$.
2009, 24(3): 883-896 doi: 10.3934/dcds.2009.24.883 +[Abstract](1595) +[PDF](163.6KB)
Abstract:
Consider a reaction-diffusion model for a microbial flow reactor with two competing species. Suppose that the amount of nutrient is input in a constant velocity at one end of the flow reactor and is washed out at the other end of the reactor. We study the dynamical behavior of population growth of these two species. In particular we are interested in the problem on the coexistence of traveling waves that best describes the long time dynamical behavior. By developing shooting method and continuation argument with the aid of an appropriately Liapunov function, we obtain the sufficient conditions for the coexistence of traveling waves as well as the minimum wave speed.
2009, 24(3): 897-907 doi: 10.3934/dcds.2009.24.897 +[Abstract](1210) +[PDF](1087.2KB)
Abstract:
This paper gives a family of nonlinear wave equations, which can yield so called loop solution, cusp wave solution and solitary wave solution depending on the values of parameter $A$. For two third order systems, the dynamical behavior of these solutions are considered. The exact explicit parametric representations of solitary wave solutions and periodic wave solutions are given. It concerns with the properties of singular traveling wave systems.
2009, 24(3): 909-932 doi: 10.3934/dcds.2009.24.909 +[Abstract](1539) +[PDF](4867.3KB)
Abstract:
The purpose of this paper is to analyze the asymptotic properties of collision orbits of Newtonian $N$-body problems. We construct new coordinates and time transformation that regularize the singularities of simultaneous binary collisions in the collinear four-body problem. The motion in the new coordinates and time scale across simultaneous binary collisions at least $C^2$. The explicit formulae are given in detail for the transformations and the extension of solutions. Furthermore, we study the behaviors of the motion approaching, across and after the simultaneous binary collision. Numerical simulations have been conducted for the special case in which the bodies are distributed symmetrically about the center of mass.
2009, 24(3): 933-978 doi: 10.3934/dcds.2009.24.933 +[Abstract](1461) +[PDF](388.1KB)
Abstract:
This paper concerns the lowest eigenvalue $\mu(b\N^Q)$ of the Schrödinger operator in three-dimensions with a magnetic potential $b\N^Q$, where the vector field $\N^Q$ depends on a matrix $Q$ varying in $SO(3)$ and $b$ is a real parameter. The eigenvalue variation problem is to minimize the lowest eigenvalue among all $Q$ in $SO(3)$. This problem arises in the phase transitions of smectic liquid crystals. We give an estimate of the minimum value inf${\mu(b\N^Q):~Q\in SO(3)\}$ for large $b$, and examine its dependence on geometry of the domain surface.
2009, 24(3): 979-1003 doi: 10.3934/dcds.2009.24.979 +[Abstract](1477) +[PDF](300.3KB)
Abstract:
A shell like structure is sought as a solution of a free boundary problem derived from the Ohta-Kawasaki theory of diblock copolymers. The boundary of the shell satisfies an equation that involves its mean curvature and the location of the entire shell. A variant of Lyapunov-Schmidt reduction process is performed that rigorously reduces the free boundary problem to a finite dimensional problem. The finite dimensional problem is solved numerically. The problem has two parameters: $a$ and $\gamma$. When $a$ is small, there are a lower bound and a sequence such that if $\gamma$ is greater than the lower bound and stays away from the sequence, there is a shell like solution.
2009, 24(3): 1005-1023 doi: 10.3934/dcds.2009.24.1005 +[Abstract](2379) +[PDF](455.4KB)
Abstract:
In this paper, an effective existence theorem for periodic Markov process is first established. Using the theorem, we consider a class of periodic $It\hat{o}$ stochastic functional differential equations, and some sufficient conditions for the existence of periodic solution of the equations are given. To overcome the difficulties created by the special features possessed by the periodic stochastic differential equations with delays, as one will see, several lemmas are introduced. These existence theorems are rather general and therefore have great power in applications. Especially, our results are natural generalization of some classical periodic theorems on the model without stochastic perturbation. An example is worked out to demonstrate the advantages of our results.
2009, 24(3): 1025-1045 doi: 10.3934/dcds.2009.24.1025 +[Abstract](1520) +[PDF](293.9KB)
Abstract:
We study the existence and uniqueness, as well as various qualitative properties of periodic traveling waves for a reaction-diffusion equation in infinite cylinders. We also investigate the spectrum of the operator obtained by linearizing with respect to such a traveling wave. A detailed description of the spectrum is obtained.

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