# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2000 , Volume 6 , Issue 4

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2000, 6(4): 751-772 doi: 10.3934/dcds.2000.6.751 +[Abstract](241) +[PDF](248.7KB)
Abstract:
We study the Vlasov-Poisson-Fokker-Planck system. For arbitrary data we prove the global well-posedness and gain of regularity of solutions under improved assumptions. We also prove that if the initial data are sufficiently small, the solutions satisfy optimal rates of asymptotic decay.
2000, 6(4): 773-796 doi: 10.3934/dcds.2000.6.773 +[Abstract](153) +[PDF](253.7KB)
Abstract:
If $(X,T)$ is a rank one system and $g$ a positive concave funtion on $(0,\infty)$ such that $g(x)^2 / x^3$ is integrable, then limsup $_{n\to\infty}$ $H(\alpha_0^{n-1})$/$g(log_2 n) =\infty$, for all partitions $\alpha$ of $X$ into two sets with $\lim_{n\to\infty} \max\{\mu(A)|A\in\alpha_0^{n-1}\}=0$.
2000, 6(4): 797-802 doi: 10.3934/dcds.2000.6.797 +[Abstract](153) +[PDF](159.5KB)
Abstract:
We investigate multiplicity of solutions $u(x,t)$ for a piecewise linear perturbation of the one-dimensional wave operator $u_{t t} - u_{x x}$ under Dirichlet boundary condition on the interval $(-\pi/2, \pi/2)$ and periodic condition on the variable $t$. Our concern is to investigate a relation between multiplicity of solutions and source terms of (1.4) when the nonlinearity $-(bu^+ -au^-)$ crosses two eigenvalues and the source term $f$ is generated by two eigenfunctions $\phi_{0 0}$, $\phi_{10}$.
2000, 6(4): 803-808 doi: 10.3934/dcds.2000.6.803 +[Abstract](362) +[PDF](156.4KB)
Abstract:
A class of linear systems subject to sudden jumps in parameter values is considered. To solve this class of stochastic control problem, we try to find the best feedback control law depending both on the measurable output as well as the mode of the system. A gradient flow based algorithm is derived for this problem. It is shown that an optimal solution can be successfully computed by finding the limiting solution of an ordinary differential equation which is given in terms of the gradient flow associated with the cost function. Several important properties are obtained. A numerical example is solved.
2000, 6(4): 809-828 doi: 10.3934/dcds.2000.6.809 +[Abstract](147) +[PDF](270.4KB)
Abstract:
Recently, in [9] we characterized the set of planar homogeneous vector fields that are structurally stable and we obtained the exact number of the topological equivalence classes. Furthermore, we gave a first extension of the Hartman-Grobman Theorem for planar vector fields. In this paper we study the structural stability in the set $H_{m,n}$ of planar semi-homogeneous vector fields $X = (P_m,Q_n)$, where $P_m$ and $Q_n$ are homogeneous polynomial of degree $m$ and $n$ respectively, and $0 < m < n$. Unlike the planar homogeneous vector fields, the semi-homogeneous ones can have limit cycles, which prevents to characterize completely those planar semi-homogeneous vector fields that are structurally stable. Thus, in the first part of this paper we will study the local structural stability at the origin and at infinity for the vector fields in $H_{m,n}$. As a consequence of these local results, we will complete the extension of the Hartman-Grobman Theorem to the nonlinear planar vector fields. In the second half of this paper we define a subset $\Delta_{m,n}$ that is dense in $H_{m,n}$ and whose elements are structurally stable. We prove that there exist vector fields in $\Delta_{m,n}$ that have at least $(m+n)/2$ hyperbolic limit cycles.
2000, 6(4): 829-840 doi: 10.3934/dcds.2000.6.829 +[Abstract](161) +[PDF](1668.6KB)
Abstract:
We consider the system of parabolic equations with distributed delay. The existence of Inertial Manifolds with Delay is proved. We prove that the system has finite number of determining modes and can be reproduced by a finite-dimensional system with concerntrated delays.
2000, 6(4): 841-860 doi: 10.3934/dcds.2000.6.841 +[Abstract](225) +[PDF](2891.7KB)
Abstract:
The convergence characteristics of an isolated Hopfield-type neuron in time varying environments are considered in particular when the neuronal parameters are assumed to be almost periodic. This study includes the investigations of neurons having periodic parameters but the periods are not integrally dependent. Both continuous-time-continuous-state and discrete-time-continuous-state models are discussed. Sufficient conditions are established for associative stimulus. It is shown that when the nreronal gain is dominated by the neuronal dissipation on average, associative recall of the encoded temporal pattern is guaranteed and this is achieved by the global stability of the encoded pattern.
2000, 6(4): 861-874 doi: 10.3934/dcds.2000.6.861 +[Abstract](181) +[PDF](212.2KB)
Abstract:
In this paper we study the existence of radial solutions to sublinear systems of elliptic equations.
We first give a multiplicity result on solutions with prescribed nodal properties; then, we show the existence of positive solutions. The proofs are based on topological degree arguments.
2000, 6(4): 875-892 doi: 10.3934/dcds.2000.6.875 +[Abstract](243) +[PDF](251.3KB)
Abstract:
We study the asymptotic behaviour of a reaction-diffusion equation, and prove that the addition of multiplicative white noise (in the sense of Itô) stabilizes the stationary solution $x\equiv 0$. We show in addition that this stochastic equation has a finite-dimensional random attractor, and from our results conjecture a possible bifurcation scenario.
2000, 6(4): 893-900 doi: 10.3934/dcds.2000.6.893 +[Abstract](150) +[PDF](199.9KB)
Abstract:
In this paper we give a characterization for the set of periods for a class of skew-products that we can see as deterministic systems driven by some stochastic process. This class coincides with a set of skew product maps from $\Sigma_N \times \mathbb S^1$ into itself, where $\Sigma_N$ is the space of the bi-infinite sequences on $N$ symbols and $\mathbb S^1$ is the unit circle.
2000, 6(4): 901-914 doi: 10.3934/dcds.2000.6.901 +[Abstract](154) +[PDF](243.7KB)
Abstract:
The spectra of Poincaré recurrences for two classes of dynamical systems are obtained in the framework of the Carathéodory construction. One class contains systems which are topologically conjugate to subshifts with the specification property, the other consists of minimal multipermutative symbolic systems. The spectra are shown to be solutions of a non-homogeneous Bowen equation, and their relationship with multifractal spectra of Lyapunov exponents is exposed.
2000, 6(4): 915-934 doi: 10.3934/dcds.2000.6.915 +[Abstract](157) +[PDF](226.3KB)
Abstract:
Dynamical systems generated by expanding piecewise linear transformations are considered. Decay rates of higher order mixing properties are characterized by the spectra of the Perron-Frobenius operator.
2000, 6(4): 935-946 doi: 10.3934/dcds.2000.6.935 +[Abstract](134) +[PDF](2177.8KB)
Abstract:
A twisted cocyle with values in a Lie group $G$ is a cocyle that incorporates an automorphism of $G$. Suppose that the underlying transformation is hyperbolic. We prove that if two Hölder continuous twisted cocycles with a sufficiently high Hölder exponent assign equal 'weights' to the periodic orbits of $\phi$, then they are Hölder cohomologous. This generalises a well-known theorem due to Livšic in the untwisted case. Having determined conditions for there to be a solution to the twisted cocycle equation, we consider how many other solution there may be. When $G$ is a toius, we determine conditions for there to be only finitely many solutions to the twisted cocycle equation.
2000, 6(4): 947-973 doi: 10.3934/dcds.2000.6.947 +[Abstract](180) +[PDF](344.4KB)
Abstract:
The long-time dynamical properties of an arbitrary positive solution $u(t)$, $t\ge 0$, to autonomous gradient-like systems are investigated. These evolutionary systems are generated by semilinear parabolic Dirichlet problems where coefficients and nonlinearities are allowed to be unbounded near the boundary $\partial \Omega$­ of the underlying bounded domain ­$\Omega\subset \mathbb R^N$. Analyticity of the potential is used to show that every positive solution of the system asymptotically approaches a (single) steady-state solution. A key tool in the proof is a Lojasiewicz-Simon-type inequality. Weighted Lebesgue and Sobolev spaces are employed. Important applications include the nonlinear heat and porous medium equations that contain nonlinearities which are not necessarily analytic on the boundary of the domain.
2000, 6(4): 975-997 doi: 10.3934/dcds.2000.6.975 +[Abstract](172) +[PDF](258.3KB)
Abstract:
In this paper a semiconcavity result is obtained for the value function of an optimal exit time problem. The related state equation is of general form

$\dot y(t)=f(y(t),u(t))$,  $y(t)\in\mathbb R^n$, $u(t)\in U\subset \mathbb R^m$.

However, suitable assumptions are needed relating $f$ with the running and exit costs.
The semiconcavity property is then applied to obtain necessary optimality conditions, through the formulation of a suitable version of the Maximum Principle, and to study the singular set of the value function.

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