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Discrete & Continuous Dynamical Systems - A

2000 , Volume 6 , Issue 2

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The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps
Qiudong Wang
2000, 6(2): 255-274 doi: 10.3934/dcds.2000.6.255 +[Abstract](29) +[PDF](5343.8KB)
We improve Mather's proof on the existence of the connecting orbit around rotation number zero (Proposition 8.1 in [7]) in this paper. Our new proof not only assures the existences of the connecting orbit, but also gives a quantitative estimation on the diffusion time.
Symbolic dynamics in nondifferentiable system originating in R-L-Diode driven circuit
Safya Belghith
2000, 6(2): 275-292 doi: 10.3934/dcds.2000.6.275 +[Abstract](26) +[PDF](252.6KB)
In this paper, we will study a simple, piecewise linear model which mimics the transformation in a chaotic electrical circuit: R-L-Diode driven by a sinusoïdal voltage source. This map leads to a complicated chaotic structure, with infinitly many distinct, prime homoclinic points. We prove here that there are infinitly many distinct homoclinic points. Their dynamical classification is not completely understood. They are derived through a nonlinear ($-+$) map, built with piecewise linear pieces. To different two sequences, should correspond two distinct prime homoclinic points. We have derived, we believe, the basic phenomena leading to the complicated dynamics.
Qualitative analysis of periodic oscillations of an earth satellite with magnetic attitude stabilization
Alexandr A. Zevin and  Mark A. Pinsky
2000, 6(2): 293-297 doi: 10.3934/dcds.2000.6.293 +[Abstract](23) +[PDF](127.0KB)
The equation of motion of a magnetically stabilized satellite in the plane of a circular polar orbit is studied through qualitative methods. Sufficient uniqueness conditions and bilateral bounds for odd periodic solutions are found. A solution with the largest amplitude is indicated and a criterion for its stability is obtained.
Symbol sequences and entropy for piecewise monotone transformations with discontinuities
C. Kopf
2000, 6(2): 299-304 doi: 10.3934/dcds.2000.6.299 +[Abstract](30) +[PDF](153.7KB)
After introducing a coding definition for the entropy of a piecewise monotone transformation with discontinuities, an algorithm is presented by which the entropy can be determined by the symbol sequences of the separation points.
Finite speed of propagation for the porous media equation with lower order terms
S. Bonafede , G. R. Cirmi and  A.F. Tedeev
2000, 6(2): 305-314 doi: 10.3934/dcds.2000.6.305 +[Abstract](29) +[PDF](196.4KB)
We study the finite speed of propagation of the Cauchy-Dirichlet problem for the porous media equation with absorption or convection terms in the strip $\mathfrak R_k^N\times (0,T)$, where $\mathfrak R_k^N=\mathfrak R^N\cap\{x_1, ..., x_k>0\}$, $1\leq k\leq N $ and we find new upper bounds of the free boundary.
Finally, we consider the case of higher order parabolic equations.
Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles
Ana Maria Bertone and  J.V. Goncalves
2000, 6(2): 315-328 doi: 10.3934/dcds.2000.6.315 +[Abstract](35) +[PDF](208.3KB)
This paper deals with existence and regularity of positive solutions of sublinear equations of the form $-\Delta u + b(x)u =\lambda f(u)$ in $\Omega$ where either $\Omega\in R^N$ is a bounded smooth domain in which case we consider the Dirichlet problem or $\Omega =R^N$, where we look for positive solutions, $b$ is not necessarily coercive or continuous and $f$ is a real function with sublinear growth which may have certain discontinuities. We explore the method of lower and upper solutions associated with some subdifferential calculus.
On the shift differentiability of the flow generated by a hyperbolic system of conservation laws
Stefano Bianchini
2000, 6(2): 329-350 doi: 10.3934/dcds.2000.6.329 +[Abstract](52) +[PDF](274.6KB)
We consider the notion of shift tangent vector introduced in [7] for real valued BV functions and introduced in [9] for vector valued BV functions. These tangent vectors act on a function $u\in L^1$ shifting horizontally the points of its graph at different rates, generating in such a way a continuous path in $L^1$. The main result of [7] is that if the semigroup $\mathcal S$ generated by a scalar strictly convex conservation law is shift differentiable, i.e. paths generated by shift tangent vectors at $u_0$ are mapped in paths generated by shift tangent vectors at $\mathcal S_t u_0$ for almost every $t\geq 0$. This leads to the introduction of a sort of differential, the "shift differential", of the map $u_0 \to \mathcal S_t u_0$.
In this paper, using a simple decomposition of $u\in $BV in terms of its derivative, we extend the results of [9] and we give a unified definition of shift tangent vector, valid both in the scalar and vector case. This extension allows us to study the shift differentiability of the flow generated by a hyperbolic system of conservation laws.
Examples of topologically transitive skew-products
Viorel Nitica
2000, 6(2): 351-360 doi: 10.3934/dcds.2000.6.351 +[Abstract](25) +[PDF](201.8KB)
Let $\sigma:\Sigma\to\Sigma$ be a topologically mixing shift of finite type. If $G$ is a group, and $\beta:\Sigma\to G$ is a continuous function, denote by $\sigma_\beta$ the skew-product of $\sigma$ by $\beta$. If $G$ is $\mathbb R^n$, we show examples of continuous multiparameters families of functions $\beta$ for which the skew-products $\sigma_\beta$ are topologically transitive for sets of parameters of full measure. If $G$ is a connected semisimple matrix Lie group, we show examples of functions $\beta$ for which the skew-products $\sigma_beta$ are topologically transitive.
Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations
Martino Bardi , Shigeaki Koike and  Pierpaolo Soravia
2000, 6(2): 361-380 doi: 10.3934/dcds.2000.6.361 +[Abstract](43) +[PDF](272.2KB)
In this paper we study the boundary value problem for the Hamilton-Jacobi-Isaacs equation of pursuit-evasion differential games with state constraints. We prove existence of a continuous viscosity solution and a comparison theorem that we apply to establish uniqueness of such a solution and its uniform approximation by solutions of discretized equations.
Transition tori near an elliptic-fixed point
Antonio Garcia
2000, 6(2): 381-392 doi: 10.3934/dcds.2000.6.381 +[Abstract](44) +[PDF](218.3KB)
Let $F:(M,\omega) \mapsto (M,\omega)$ be a smooth symplectic diffeomorphism with a fixed point a and a heteroclinic orbit in the sense of been in the intersection of the central stable and the central unstable manifolds of the fixed point. It is studied the case when the tangent space of a point in the heteroclinic orbit is the direct sum of three subspaces. The first one is the characteristic bundle of the central stable manifold of $\mathbf a$ the second one is the characteristic bundle of the central unstable manifold of $\mathbf a$, and the third one is tangent to the intersection of the central stable and unstable manifolds.
In this situation, the homoclinic map $\Lambda$ is a smooth and symplectic diffeomorphism of open subsets of the central manifold of $\mathbf a$.
Moreover, if an invariant circle intersects the domain of definition of $\Lambda$ and its image intersects other circle, there are orbits that wander from one circle to the other. This phenomenon is similar to the Arnold diffusion.
The Melnikov Method gives sufficient conditions for the existence of homoclinic maps, and non identity homoclinic maps in a perturbation of a Hamiltonian system.
Uniform inertial sets for damped wave equations
P. Fabrie , C. Galusinski and  A. Miranville
2000, 6(2): 393-418 doi: 10.3934/dcds.2000.6.393 +[Abstract](29) +[PDF](294.7KB)
In this paper, we establish the existence of inertial sets for a class of wave equations in which the coefficient of the second order time derivative is $\varepsilon$. We show that the fractal dimension of these inertial sets does not depend on $\varepsilon$ for $\varepsilon$ small enough. We then compare the asymptotic behavior of the problem (as $\varepsilon\to 0$) through a continuity like property of the inertial sets. The autonomous case and nonautonomous case are studied.
On the 2-D Riemann problem for the compressible Euler equations II. Interaction of contact discontinuities
Tung Chang , Gui-Qiang Chen and  Shuli Yang
2000, 6(2): 419-430 doi: 10.3934/dcds.2000.6.419 +[Abstract](125) +[PDF](695.3KB)
We are concerned with the Riemann problem for the two-dimensional compressible Euler equations in gas dynamics. This paper is a continuation of our program (see [CY1,CY2]) in studying the interaction of nonlinear waves in the Riemann problem. The central point in this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that connect two neighboring constant initial states in the quadrants. In this paper we focus mainly on the interaction of contact discontinuities, which consists of two genuinely different cases. For each case, the structure of the Riemann solution is analyzed by using the method of characteristics, and the corresponding numerical solution is illustrated via contour plots by using the upwind averaging scheme that is second-order in the smooth region of the solution developed in [CY1]. For one case, the four contact discontinuities role up and generate a vortex, and the density monotonically decreases to zero at the center of the vortex along the stream curves. For the other, two shock waves are formed and, in the subsonic region between two shock waves, a new kind of nonlinear hyperbolic waves (called smoothed Delta-shock waves) is observed.
Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints
J.-P. Raymond and  F. Tröltzsch
2000, 6(2): 431-450 doi: 10.3934/dcds.2000.6.431 +[Abstract](106) +[PDF](280.9KB)
In this paper, optimal control problems for semilinear parabolic equations with distributed and boundary controls are considered. Pointwise constraints on the control and on the state are given. Main emphasis is laid on the discussion of second order sufficient optimality conditions. Sufficiency for local optimality is verified under different assumptions imposed on the dimension of the domain and on the smoothness of the given data.
Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions
Jiu Ding and  Aihui Zhou
2000, 6(2): 451-458 doi: 10.3934/dcds.2000.6.451 +[Abstract](30) +[PDF](179.2KB)
In this paper, by using a trace theorem in the theory of functions of bounded variation, we prove the existence of absolutely continuous invariant measures for a class of piecewise expanding mappings of general bounded domains in any dimension.
Bound sets for floquet boundary value problems: The nonsmooth case
Valentina Taddei
2000, 6(2): 459-473 doi: 10.3934/dcds.2000.6.459 +[Abstract](21) +[PDF](200.7KB)
We give a definition of bound set for a very general boundary value problem that generalizes those already known in literature. We then find sufficient conditions for the intersection of the sublevelsets of a family of scalar functions to be a bound set for the Floquet boundary value problem. Indeed, we distinguish the two cases of locally Lipschitz continuous and only continuous scalar functions.
Positively homogeneous equations in the plane
Alessandro Fonda and  Rafael Ortega
2000, 6(2): 475-482 doi: 10.3934/dcds.2000.6.475 +[Abstract](27) +[PDF](182.1KB)
We prove the multiplicity of periodic solutions to second order ordinary differential equations in $\mathbb R^2$ with nonlinearities crossing the two first eigenvalues of the differential operator.
Global well-posedness for the Kadomtsev-Petviashvili II equation
Hideo Takaoka
2000, 6(2): 483-499 doi: 10.3934/dcds.2000.6.483 +[Abstract](31) +[PDF](243.3KB)
We study the global well-posedness of the Cauchy problem for the KP II equation. We prove the global well-posedness in the inhomogeneous-homogeneous anisotropic Sobolev spaces $H_{x,y}^{-1/78+\epsilon,0}\cap H_{x,y}^{-17/144,0}$. Though we require the use of the homogeneous Sobolev space of negative index, we obtain the global well-posedness below $L^2$.

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