# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

January 1998 , Volume 4 , Issue 1

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1998, 4(1): 1-32 doi: 10.3934/dcds.1998.4.1 +[Abstract](2085) +[PDF](760.7KB)
Abstract:
In this paper we study the evolutions of the interfaces between gases and the vacuum for both inviscid and viscous one dimensional isentropic gas motions. The local (in time) existence of solutions for both inviscid and viscous models with initial data containing vacuum states is proved and some singular properties on the free surfaces separating the gas and the vacuum are obtained. It is found that the Euler equations are better behaved near the vacuum than the compressible Navier-Stokes equations. The Navier-Stokes equations with viscosity depending on density are introduced, which is shown to be well-posed (at least locally) and yield the desired solutions near vacuum.
1998, 4(1): 33-42 doi: 10.3934/dcds.1998.4.33 +[Abstract](1453) +[PDF](208.4KB)
Abstract:
We prove that for a dense $G_{\delta}$ of shift-invariant measures on $A^{\ZZ^d}$, all $d$ shifts have purely singular continuous spectrum and give a new proof that in the weak topology of measure preserving $\ZZ^d$ transformations, a dense $G_{\delta}$ is generated by transformations with purely singular continuous spectrum. We also give new examples of smooth unitary cocycles over an irrational rotation which have purely singular continuous spectrum. Quantitative weak mixing properties are related by results of Strichartz and Last to spectral properties of the unitary Koopman operators.
1998, 4(1): 43-54 doi: 10.3934/dcds.1998.4.43 +[Abstract](1275) +[PDF](188.9KB)
Abstract:
We define controlled dynamical systems and give a few of characterizations of ergodic attractors of controlled dynamical systems.
1998, 4(1): 55-72 doi: 10.3934/dcds.1998.4.55 +[Abstract](1236) +[PDF](222.3KB)
Abstract:
The quasistatic evolution of an elastic-viscoplastic body in bilateral contact with a rigid foundation is considered. The contact involves viscous friction of Tresca type. Two variational formulations of the problem are proposed, followed by existence and uniqueness results. Some properties involving the equivalence between the previous variational formulations, the continuous dependence of the solution with respect to the data as well as a convergence result with respect to the friction yield limit are also obtained.
1998, 4(1): 73-90 doi: 10.3934/dcds.1998.4.73 +[Abstract](1524) +[PDF](2982.1KB)
Abstract:
We consider positive perturbations $A = B+ C$ of resolvent positive operators $B$ by positive operators $C: D(A) \to X$ and in particular study their spectral properties. We characterize the spectral bound of $A$, $s(A)$, in terms of the resolvent outputs $F(\lambda) = C (\lambda - B)^{-1}$ and derive conditions for $s(A)$ to be an eigenvalue of $A$ and a (first order) pole of the resolvent of $A$. On our way we show that the spectral radii of a completely monotonic operator family form a superconvex function. Our results will be used in forthcoming publications to study the spectral and large-time properties of positive operator semigroups.
1998, 4(1): 91-98 doi: 10.3934/dcds.1998.4.91 +[Abstract](1487) +[PDF](161.6KB)
Abstract:
We prove the existence of bounded solutions to second order differential equations of Liénard type under asymptotic conditions generalizing recent results of Ahmad and Ortega.
1998, 4(1): 99-130 doi: 10.3934/dcds.1998.4.99 +[Abstract](2074) +[PDF](338.9KB)
Abstract:
This paper is concerned with attractors of randomly perturbed dynamical systems, called random attractors. The framework used is provided by the theory of random dynamical systems. We first define, analyze, and prove existence of random attractors. The main result is a technique, similar to Lyapunov's direct method, to ensure existence of random attractors for random differential equations. This method is formulated as a generally applicable procedure. As an illustration we shall apply it to the random Duffing-van der Pol equation. We then show, by the same example, that random attractors provide an important tool to analyze the bifurcation behavior of stochastically perturbed dynamical systems. We introduce new methods and techniques, and we investigate the Hopf bifurcation behavior of the random Duffing-van der Pol equation in detail. In addition, the relationship of random attractors to invariant measures and unstable sets is studied.
1998, 4(1): 131-140 doi: 10.3934/dcds.1998.4.131 +[Abstract](1200) +[PDF](199.5KB)
Abstract:
Suppose $(X,d)$ is a metric space and $f:X\to X$ a continuous map. Let $\sum = X^{\N}$ denote the set of all sequences of elements of $X$. $E_f: X\to\sum$ is given by $E_f(x) = (x,f(x),f^2(x),\ldots).$ $E_f(x)$ is called the trajectory or time evolution of $f$ at $x$. Let $\mathcal T$ be a toplogy on $\sum$. We define $f$ to be $\mathcal T$-stable ($\mathcal T$-sensitive) at $x$ if $E_f$ is $\mathcal T$-continuous at $x$ (if $E_f$ is $\mathcal T$-discontinuous at $x$). We construct topologies on $\sum$ by using a generalised notion of a metric on $\sum$ which we call a sensitivity function. We show that the classical notions of stability due to Liapunov, Birkhoff, Lefschetz and Poisson can be expressed in terms of suitably chosen sensitivity functions. This approach unifies old ideas and suggests new notions of sensitivity all stronger than that due to Liapunov. The mutual implications of these various notions are discussed in detail. Throughout, the analysis is in terms of elementary topology.
1998, 4(1): 141-158 doi: 10.3934/dcds.1998.4.141 +[Abstract](1176) +[PDF](230.7KB)
Abstract:
We study $C^2$-structural stability of interval maps with negative Schwarzian. It turns out that for a dense set of maps critical points either have trajectories attracted to attracting periodic orbits or are persistently recurrent. It follows that for any structurally stable unimodal map the $\omega$-limit set of the critical point is minimal.
1998, 4(1): 159-191 doi: 10.3934/dcds.1998.4.159 +[Abstract](1135) +[PDF](384.2KB)
Abstract:
In this part, we prove that the solitary wave solutions investigated in part I are extended as analytic functions in the complex plane, except for at most countably many branch points and branch lines. We describe in detail how the limiting behavior of the complex singularities allows the creation of non-analytic solutions with corners and/or compact support.

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