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Discrete & Continuous Dynamical Systems - A

1997 , Volume 3 , Issue 1

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The global attractor of semilinear parabolic equations on $S^1$
Hiroshi Matano and Ken-Ichi Nakamura
1997, 3(1): 1-24 doi: 10.3934/dcds.1997.3.1 +[Abstract](577) +[PDF](3712.6KB)
Abstract:
We study the global attractor of semilinear parabolic equations of the form

$u_t=u_{x x}+f(u,u_x),\ x\in\mathbb{R}$/$\mathbb{Z}, \ t>0.$

Under suitable conditions on $f$, the equation generates a global semiflow on a suitable function space. The general theory of inertial manifolds does not apply to this equation due to lack of the so-called spectral gap condition. Using a totally different method, we show that the global attractor is the graph of a continuous mapping of finite dimension. We also show that this dimension is equal to $2[N$/$2]+1$, where $N$ is the maximal value of the generalized Morse index of equilibria and periodic solutions. Note that we do not make any assumption regarding the hyperbolicity of those solutions. We further prove that there exists no homoclinic orbit nor heteroclinic cycle.

Multiple periodic solutions of second order equations with asymmetric nonlinearities
C. Rebelo
1997, 3(1): 25-34 doi: 10.3934/dcds.1997.3.25 +[Abstract](449) +[PDF](1750.1KB)
Abstract:
The problem of the existence and multiplicity of periodic (harmonic and subharmonic) solutions to the parameter-dependent second order equation $x' '+g(x)=s+w(t)$ is investigated for $|s|$ large under suitable "jumping" conditions on $g'(\pm\infty)$. The results which are obtained complete and complement some recent theorems on the periodic Ambrosetti-Prodi problem.
Shift-differentiabilitiy of the flow generated by a conservation law
Alberto Bressan and Graziano Guerra
1997, 3(1): 35-58 doi: 10.3934/dcds.1997.3.35 +[Abstract](526) +[PDF](3038.5KB)
Abstract:
The paper introduces a notion of "shift-differentials" for maps with values in the space BV. These differentials describe first order variations of a given function $u$, obtained by horizontal shifts of the points of its graph. The flow generated by a scalar conservation law is proved to be generically shift-differentiable, according to the new definition.
Exact controllability of a beam in an incompressible inviscid fluid
Scott W. Hansen and Andrei A. Lyashenko
1997, 3(1): 59-78 doi: 10.3934/dcds.1997.3.59 +[Abstract](541) +[PDF](2748.4KB)
Abstract:
It is well known that an Euler-Bernoulli beam may be exactly controlled with a single control acting on an end of the beam. In this article we show that for certain boundary conditions, the same result holds for a beam that is surrounded by an incompressible, inviscid fluid with a sufficiently small density. The proof involves reducing the control problem to a moment problem and using compactness properties of the Neumann to Dirichlet map for the Laplacian operator to obtain the needed estimates.
Generic 1-parameter families of binary differential equations of Morse type
J.W. Bruce and F. Tari
1997, 3(1): 79-90 doi: 10.3934/dcds.1997.3.79 +[Abstract](497) +[PDF](2231.1KB)
Abstract:
In a previous paper [2] we made a classification of generic binary differential equations (BDE's)

$a(x,y)dy^2+2b(x,y)dxdy+c(x,y)dx^2=0$

near points at which the discriminant function $b^2-ac$ has a Morse singularity. Such points occur naturally in families of BDE's and here we describe the manner in which the configuration of solution curves change in their natural 1-parameter versal deformations.
The results in this paper can be used to describe, for instance, the changes in the structure of the asymptotic curves on a 1-parameter family of smooth surfaces acquiring a flat umbilic and on integral curves determined by eigenvectors of 1-parameter families of $2\times 2$ matrices. It also sheds light on the structure of the rarefraction curves associated to a $2\times 2$ system of conservation laws in 1 space variable.

On the global solvability of symmetric hyperbolic systems of Kirchhoff type
Renato Manfrin
1997, 3(1): 91-106 doi: 10.3934/dcds.1997.3.91 +[Abstract](413) +[PDF](1994.2KB)
Abstract:
We shall prove here the global solvability for small initial data for symmetric hyperbolic systems with integro-differential coefficients. In this way, we will extend some results obtained in [5], [6], [8], [11] for the classic Kirchhoff equation and in [3] for regularly hyperbolic systems.
A billiard in the hyperbolic plane with decay of correlation of type $n^{-2}$
M. Bauer and A. Lopes
1997, 3(1): 107-116 doi: 10.3934/dcds.1997.3.107 +[Abstract](497) +[PDF](1797.8KB)
Abstract:
We consider billiard trajectories on the geodesic triangle $D$ in the hyperbolic half-plane with internal angles $0,0$ and $\pi /2$ at the vertices $\infty,1$, and $i$. The billiard map of $D$ sends a given trajectory $\gamma$ with the boundary $\delta D$ of $D$ to the next intersection point of $\gamma$ with $\delta D$.
By choosing an appropriate cross section for the billiard map, we show that the decay of correlation of the first return map is slower than $n^-2$. As a by-product, we enumerate the billiard trajectories in terms of their cutting sequences and relate the boundary expansion of the billiard map to continued fractions with even partial quotients.
Exact spiral solutions of the two-dimensional Euler equations
Tong Zhang and Yuxi Zheng
1997, 3(1): 117-133 doi: 10.3934/dcds.1997.3.117 +[Abstract](695) +[PDF](2590.2KB)
Abstract:
We construct a two-parameter family of self-similar solutions to both the compressible and incompressible two-dimensional Euler equations with axisymmetry. The equations can be reduced under the situation to two systems of ordinary differential equations. In the compressible and polytropic case, the system in autonomous form consists of four ordinary differential equations with a two-dimensional set of stationary points, one of which is degenerate up to order four. Through asymptotic analysis and computations of numerical solutions, we are fortunate to be able to recognize a one-parameter family of exact solutions in explicit form. All the solutions (exact or numerical) are globally bounded and continuous, have finite local energy and vorticity, and have well-defined initial and boundary values at time zero and spatial infinity respectively. Particle trajectories of some of these solutions are spiral-like. In the incompressible case, we also find explicit self-similar axisymmetric spiral solutions, which are, however, somewhat less physical due to unbounded pressures or infinite local energy near their swirling centers.
Aubry-Mather theory for functions on lattices
Hans Koch, Rafael De La Llave and Charles Radin
1997, 3(1): 135-151 doi: 10.3934/dcds.1997.3.135 +[Abstract](526) +[PDF](3162.0KB)
Abstract:
We generalize the Aubry-Mather theorem on the existence of quasi-periodic solutions of one dimensional difference equations to situations in which the independent variable ranges over more complicated lattices. This is a natural generalization of Frenkel-Kontorova models to physical situations in a higher dimensional space. We also consider generalizations in which the interactions among the particles are not just nearest neighbor, and indeed do not have finite range.

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