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## Discrete & Continuous Dynamical Systems - A

1999 , Volume 5 , Issue 1

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1999, 5(1): 1-34
doi: 10.3934/dcds.1999.5.1

*+*[Abstract](52)*+*[PDF](344.7KB)**Abstract:**

This paper deals with various applications of two basic theorems in order- preserving systems under a group action -- monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equa- tions and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudo- traveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.

1999, 5(1): 35-42
doi: 10.3934/dcds.1999.5.35

*+*[Abstract](59)*+*[PDF](187.5KB)**Abstract:**

We establish the existence of solutions to an anti-periodic non-monotone boundary value problem. Our approach relies on a combination of monotonicity and compactness methods.

1999, 5(1): 43-60
doi: 10.3934/dcds.1999.5.43

*+*[Abstract](170)*+*[PDF](236.0KB)**Abstract:**

This paper is a study of the global structure of the attractors of a dynamical system. The dynamical system is associated with an oriented graph called a Symbolic Image of the system. The symbolic image can be considered as a finite discrete approximation of the dynamical system flow. Investigation of the symbolic image provides an opportunity to localize the attractors of the system and to estimate their domains of attraction. A special sequence of symbolic images is considered in order to obtain precise knowledge about the global structure of the attractors and to get filtrations of the system.

Stability of symmetric periodic solutions with small amplitude of $\dot x(t)=\alpha f(x(t), x(t-1))$

1999, 5(1): 61-82
doi: 10.3934/dcds.1999.5.61

*+*[Abstract](89)*+*[PDF](267.2KB)**Abstract:**

We study special symmetric periodic solutions of the equation

$\dot x(t) =\alphaf(x(t), x(t-1))$

where $\alpha$ is a positive parameter and the nonlinearity $f$ satisfies the symmetry conditions $f(-u, v) = -f(u,-v) = f(u, v).$ We establish the existence and stability properties for such periodic solutions with small amplitude.

1999, 5(1): 83-92
doi: 10.3934/dcds.1999.5.83

*+*[Abstract](100)*+*[PDF](297.7KB)**Abstract:**

Topological transitivity, weak mixing and non-wandering are definitions used in topological dynamics to describe the ways in which open sets feed into each other under iteration. Using finite directed graphs, these definitions are generalized to obtain topological mapping properties. The extent to which these mapping properties are logically distinct is examined. There are three distinct properties which entail "interesting" dynamics. Two of these, transitivity and weak mixing, are already well known. The third does notappear in the literature but turns out to be close to weak mixing in a sense to be discussed. The remaining properties comprise a countably infinite collection of distinct properties entailing somewhat less interesting dynamics and including non-wandering.

1999, 5(1): 93-106
doi: 10.3934/dcds.1999.5.93

*+*[Abstract](80)*+*[PDF](224.9KB)**Abstract:**

We study the Cauchy problem for a nonlinear Schrödinger equation which is the generalization of a one arising in plasma physics. We focus on the so called subcritical case and prove that when the initial datum is "small", the solution exists globally in time and decays in time just like in the linear case. For a certain range of the exponent in the nonlinear term, we prove that the solution is asymptotic to a "final state" and the nonexistence of asymptotically free solutions. The method used in this paper is based on some gauge transformation and on a certain phase function.

1999, 5(1): 107-116
doi: 10.3934/dcds.1999.5.107

*+*[Abstract](85)*+*[PDF](184.2KB)**Abstract:**

The rich diversity of patterns and concepts intrinsic to the Julia and the Mandelbrot sets of the quadratic map in the complex plane invite a search for higher dimensional generalisations. Quaternions provide a natural framework for such an endeavour. The objective of this investigation is to provide explicit formulae for the domain of stability of multiple cycles of classes of quaternionic maps $F(Q)+C$ or $CF(Q)$ where $C$ is a quaternion and $F(Q)$ is an integral function of $Q$. We introduce the concept of quaternionic differentials and employ this in the linear stability analysis of multiple cycles.

1999, 5(1): 117-136
doi: 10.3934/dcds.1999.5.117

*+*[Abstract](80)*+*[PDF](269.1KB)**Abstract:**

Nonlinear stability and some other dynamical properties for a KS type equation in space dimension two are studied in this article. We consider here a variation of the KS equation where the derivatives in the nonlinear and the antidissipative linear terms are in one single direction. We prove the nonlinear stability for all positive times and study the corresponding attractor.

1999, 5(1): 137-156
doi: 10.3934/dcds.1999.5.137

*+*[Abstract](78)*+*[PDF](259.7KB)**Abstract:**

Given a control system (formulated as a nonconvex and unbounded differential inclusion) we study the problem of reaching a closed target with trajectories of the system. A controllability condition around the target allows us to construct a path that steers each point nearby into it in finite time and using a finite amount of energy. In applications to minimization problems, limits of such trajectories could be discontinuous. We extend the inclusion so that all the trajectories of the extension can be approached by (graphs of) solutions of the original system. In the extended setting the value function of an exit time problem with Lagrangian affine in the unbounded control can be shown to coincide with the value function of the original problem, to be continuous and to be the unique (viscosity) solution of a Hamilton-Jacobi equation with suitable boundary conditions.

1999, 5(1): 157-184
doi: 10.3934/dcds.1999.5.157

*+*[Abstract](127)*+*[PDF](324.2KB)**Abstract:**

We study the regularity of the composition operator $((f, g)\to g \circ f)$ in spaces of Hölder differentiable functions. Depending on the smooth norms used to topologize $f, g$ and their composition, the operator has different differentiability properties. We give complete and sharp results for the classical Hölder spaces of functions defined on geometrically well behaved open sets in Banach spaces. We also provide examples that show that the regularity conclusions are sharp and also that if the geometric conditions fail, even in finite dimensions, many elements of the theory of functions (smoothing, interpolation, extensions) can have somewhat unexpected properties.

1999, 5(1): 185-196
doi: 10.3934/dcds.1999.5.185

*+*[Abstract](162)*+*[PDF](236.3KB)**Abstract:**

In this paper, we give some existence results for equilibrium problems by proceeding to a perturbation of the initial problem and using techniques of recession analysis. We develop and describe thoroughly recession condition which ensure existence of at least one solution for hemivariational inequalities introduced by Panagiotopoulos. Then we give two applications to resolution of concrete variational inequalities. We shall examine two examples. The first one concerns the unilateral boundary condition. In the second, we shall consider the contact problem with given friction on part of the boundary.

1999, 5(1): 197-214
doi: 10.3934/dcds.1999.5.197

*+*[Abstract](78)*+*[PDF](254.7KB)**Abstract:**

In this paper we consider the notion of determining projections for two classes of stochastic dissipative equations: a reaction-diffusion equation and a 2-dimensional Navier-Stokes equation.

We define certain finite dimensional objects that can capture the asymptotic behavior of the related dynamical system. They are projections on a space of polynomial functions, generalizing the classical (but not very much studied in a stochastic context) concepts of determining modes, nodes and volumes.

1999, 5(1): 215-231
doi: 10.3934/dcds.1999.5.215

*+*[Abstract](68)*+*[PDF](243.5KB)**Abstract:**

We show the local in time solvability of the Cauchy problem for nonlinear wave equations in the Sobolev space of critical order with nonlinear term of exponential type.

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