# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

January 2012 , Volume 32 , Issue 1

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2012, 32(1): 1-26 doi: 10.3934/dcds.2012.32.1 +[Abstract](1663) +[PDF](531.1KB)
Abstract:
Consider a closed subset $K \subset \mathbb{R}^n$ and $f:[0,T]\times \mathbb{R}^n\times U \to \mathbb{R}^n$, where $U$ is a complete separable metric space. We associate to these data the control system under a state constraint \begin{equation*}\label{dm400} \left \{ \begin{array}{lll} x'(t) &=&f(t,x(t),u(t)), \; \; u(t)\in U \quad\; \mbox{ a.e. in }\; [0,T] \\ x(t) & \in & K \quad\; \mbox{ for all }\; t \in [0,T]\\ x(0)& = &x_0 . \end{array} \right. \end{equation*} When the boundary of $K$ is smooth, then an inward pointing condition guarantees that under standard assumptions on $f$ (measurable in $t$, Lipschitz in $x$, continuous in $u$) the sets of solutions to the above system depend on the initial state $x_0$ in a Lipschitz way. This follows from the so-called Neighboring Feasible Trajectories (NFT) theorems. Some recent counterexamples imply that NFT theorems are not valid when $f$ is discontinuous in time and $K$ is a finite intersection of sets with smooth boundaries, that is in the presence of multiple state constraints.
In this paper we prove that for multiple state constraints the inward pointing condition yields local Lipschitz dependence of solution sets on the initial states from the interior of $K$. Furthermore we relax the usual inward pointing condition. The novelty of our approach lies in an application of a generalized inverse mapping theorem to investigate feasible solutions of control systems. Our results also imply a viability theorem without convexity of right-hand sides for initial states taken in the interior of $K$.
2012, 32(1): 27-40 doi: 10.3934/dcds.2012.32.27 +[Abstract](1175) +[PDF](378.0KB)
Abstract:
In this note, we consider a partially hyperbolic horseshoe and prove uniqueness of equilibrium states for a class of potentials. In particular we obtain that there exists a unique maximal entropy measure.
2012, 32(1): 41-56 doi: 10.3934/dcds.2012.32.41 +[Abstract](920) +[PDF](468.2KB)
Abstract:
In this paper NMS flows on $S^{3}$ with a round handle decomposition made up of connected sum of tori are considered. We describe these flows from the corresponding filtrations and obtain conditions for the existence of transversal intersections of invariant manifolds of saddle orbits.
Moreover, we build the phase portrait for each case in order to obtain a complete description of the flow and to visualize how invariant manifolds of saddles intersect.
2012, 32(1): 57-79 doi: 10.3934/dcds.2012.32.57 +[Abstract](1129) +[PDF](474.9KB)
Abstract:
This paper is concerned with the Cauchy problem of three-dimensional modified Navier-Stokes equations with fractional dissipation $\nu (-\Delta)^{\alpha} u$. The results are three-fold. We first prove the global existence of weak solutions for $0<\alpha\leq1$ and global smooth solution for $\frac{3}{4}<\alpha\leq1.$ Second, we obtain the optimal decay rates of both weak solutions and the higher-order derivative of the smooth solution. Finally, we investigate the asymptotic stability of the large solution to the system under large initial and external forcing perturbation.
2012, 32(1): 81-100 doi: 10.3934/dcds.2012.32.81 +[Abstract](1253) +[PDF](437.8KB)
Abstract:
We introduce a dual sequence condition (DSC) for a discrete dynamical system given by a continuous map $f:X\to X$ of some metric space $X$. It is defined in terms of the Lefschetz sequence and its dual sequence of the endomorphism of a graded vector space of finite type associated to the dynamical system $f$. We prove the arithmetical properties of the dual Lefschetz sequence and we show some of its dynamical consequences, mainly concerning the topological methods for detecting chaotic dynamics.
2012, 32(1): 101-124 doi: 10.3934/dcds.2012.32.101 +[Abstract](1711) +[PDF](412.9KB)
Abstract:
We study traveling wave solutions for a lattice dynamical system with convolution type nonlinearity. We consider the monostable case and discuss the asymptotic behaviors, monotonicity and uniqueness of traveling wave. First, we characterize the asymptotic behavior of wave profile at both wave tails. Next, we prove that any wave profile is strictly decreasing. Finally, we prove the uniqueness (up to translation) of wave profile for each given admissible wave speed.
2012, 32(1): 125-166 doi: 10.3934/dcds.2012.32.125 +[Abstract](1537) +[PDF](574.6KB)
Abstract:
We consider the spectrum associated with the linear operator obtained when a Cahn--Hilliard system on $\mathbb{R}$ is linearized about a transition wave solution. In many cases it's possible to show that the only non-negative eigenvalue is $\lambda = 0$, and so stability depends entirely on the nature of this neutral eigenvalue. In such cases, we identify a stability condition based on an appropriate Evans function, and we verify this condition under strong structural conditions on our equations. More generally, we discuss and implement a straightforward numerical check of our condition, valid under mild structural conditions.
2012, 32(1): 167-190 doi: 10.3934/dcds.2012.32.167 +[Abstract](1088) +[PDF](296.9KB)
Abstract:
This paper is devoted to the study of long-time behavior of the solutions to a one-dimensional full model for the first order phase transitions. Our system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature $\theta$, which is coupled with an evolution equation for the phase change parameter $f$ with a third-order nonlinearity $G_2'(f)$ in place of the customarily constant latent heat. The main novelty of this paper is that we perform an argument to establish Lemma 3.1 which enables us to obtain uniform estimates of the global solutions with respect to time. Asymptotic behavior of the solutions as time goes to infinity and the compactness of the orbit are obtained. Furthermore, we investigate the dynamics of the system and prove the existence of global attractors.
2012, 32(1): 191-221 doi: 10.3934/dcds.2012.32.191 +[Abstract](1468) +[PDF](576.3KB)
Abstract:
We consider the mass-critical generalized Korteweg--de Vries equation $$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$ for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed, then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear Schrödinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution.
2012, 32(1): 223-263 doi: 10.3934/dcds.2012.32.223 +[Abstract](1201) +[PDF](588.0KB)
Abstract:
We consider Hamilton Jacobi Bellman equations in an infinite dimensional Hilbert space, with quadratic (respectively superquadratic) Hamiltonian and with continuous (respectively lipschitz continuous) final condition. This allows to study stochastic optimal control problems for suitable controlled state equations with unbounded control processes. The results are applied to controlled heat equations.
2012, 32(1): 265-291 doi: 10.3934/dcds.2012.32.265 +[Abstract](1265) +[PDF](523.2KB)
Abstract:
In this article, we consider a non-autonomous three-dimensional primitive model of the ocean with a singularly oscillating external force depending on a small parameter $\epsilon.$ We prove the existence of the uniform global attractor $\mathcal{A}^{\epsilon}$ in $V,$ (i.e., with the $H^1-$regularity). Furthermore, using the method of [13] in the case of the two-dimensional Navier-Stokes systems, we study the convergence of $\mathcal{A}^{\epsilon}$ as $\epsilon$ goes to zero.
2012, 32(1): 293-301 doi: 10.3934/dcds.2012.32.293 +[Abstract](1745) +[PDF](323.3KB)
Abstract:
We study dynamical systems for which at most $n$ orbits can accompany a given arbitrary orbit. For simplicity we call them $n$-expansive (or positively $n$-expansive if positive orbits are considered instead). We prove that these systems can satisfy properties of expansive systems or not. For instance, unlike positively expansive maps [3], positively $n$-expansive homeomorphisms may exist on certain infinite compact metric spaces. We also prove that a map (resp. bijective map) is positively $n$-expansive (resp. $n$-expansive) if and only if it is so outside finitely many points. Finally, we prove that a homeomorphism on a compact metric space is $n$-expansive if and only if it is so outside finitely many orbits. These last resuls extends previous ones for expansive systems [2],[11],[12].
2012, 32(1): 303-329 doi: 10.3934/dcds.2012.32.303 +[Abstract](1193) +[PDF](483.8KB)
Abstract:
This paper deals with an unstirred chemostat model of competition between plasmid-bearing and plasmid-free organisms when the plasmid-bearing organism produces toxins. The toxins are lethal to the plasmid-free organism, which leads to the conservation principle cannot be applied, and the resulting dynamical system is described by three nonlinear partial differential equations and is not monotone. First, the existence and multiplicity of the positive steady-state solutions are determined by bifurcation theory and degree theory. Second, the effects of the toxins are considered by perturbation technique. The results show that if the parameter $r$, which measures the effect of the toxins, is sufficiently large, this model has at least two positive solutions provided that the maximal growth rate $a$ of $u$ lies in a certain range; and has only a unique asymptotically stable positive solution when $a$ belongs to another range.
2012, 32(1): 331-352 doi: 10.3934/dcds.2012.32.331 +[Abstract](1433) +[PDF](441.9KB)
Abstract:
In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The global unique solvability is proved based on the energy method. In particular, our main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameter $\beta$ goes to zero. We show that the boundary layer thickness is of the order $O\left(\beta^\gamma\right)$ with $0<\gamma<\frac{1}{2}$.

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