# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

October 1999 , Volume 5 , Issue 4

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1999, 5(4): 697-728 doi: 10.3934/dcds.1999.5.697 +[Abstract](1106) +[PDF](4458.3KB)
Abstract:
The paper is devoted to investigation of a number of difference-deiiferential equations, among them the following one plays the central role:

$dF_n$/$dt=\varphi(F_n)(F_{n-1}-F_n)\quad\qquad\qquad (\star)$

where, for every $t, \{F_n(t), n=0,1,2,\ldots\}$ is a probability distribution function, and $\varphi$ is a positive function on $[0, 1]$. The equation $(\star)$ arose as a description of industrial economic development taking into accout processes of creation and propagation of new technologies. The paper contains a survey of the earlier received results including a multi-dimensional generalization and an application to the economic growth theory.
If $\varphi$ is decreasing then solutions of Cauchy problem for $(\star)$ approach to a family of wave-trains. We show that diffusion-wise asymptotic behavior takes place if $\varphi$ is increasing. For the nonmonotonic case a general hypothesis about asymtotic behavior is formulated and an analogue of a Weinberger's (1990) theorem is proved. It is argued that the equation can be considereded as an analogue of Burgers equation.

1999, 5(4): 729-740 doi: 10.3934/dcds.1999.5.729 +[Abstract](782) +[PDF](1445.4KB)
Abstract:
In this paper, we consider the initial value problem with periodic boundary condition for a class of general systems of the ferromagnetic chain

$z_t=-\alpha z\times (z\times z_{x x})+ z\times z_{x x}+z\times f(z), \qquad (\alpha \geq 0).$

The existence of unique smooth solutions is proved by using the technique of spatial difference and a priori estimates of higher-order derivatives in Sobolev spaces.

1999, 5(4): 741-752 doi: 10.3934/dcds.1999.5.741 +[Abstract](746) +[PDF](2114.5KB)
Abstract:
The general topic is the connection between a change of stability of an equilibrium point or invariant set $M$ of a (semi-) dynamical system depending on a parameter and a bifurcation of $M$ (generalizing the Hopf bifurcation). In particular, we address the case where $M$ is unstable (for instance a saddle) for a certain value $\lambda_0$ of a parameter $\lambda$, and stable for certain nearby values. Two kinds of bifurcations are considered: "extracritical", i.e. splitting of the set $M$ as $\lambda$ passes the value $\lambda_0$, and "critical" (also called "vertical"), a term which refers to the accumulation of closed invariant set at $M$ for $\lambda=\lambda_0$. Also, two kinds of change of stability are considered, corresponding to the presence or absence of a certain generalized equistability property for $\lambda\ne\lambda_0$. Connections are established between the type of change of stability and the types of bifurcation arising from them.
1999, 5(4): 753-763 doi: 10.3934/dcds.1999.5.753 +[Abstract](984) +[PDF](214.9KB)
Abstract:
We study the scattering theory for nonlinear Klein-Gordon equations $u_{t t} + (m^2-\Delta)u = f_1(u) + f_2(u)$. We show that the scattering operator carries a band in $H^s \times H^{s-1}$ into $H^s \times H^{s-1}$ for all $s\in [1/2,\ \infty)$ if $f_i(u)\ (i = 1,\ 2)$ have $H^s$-critical or $H^s$-subcritical powers.
1999, 5(4): 765-782 doi: 10.3934/dcds.1999.5.765 +[Abstract](962) +[PDF](395.3KB)
Abstract:
We present results on homoclinic and multibump solutions for perturbed second order systems. Using topological degree, we generalize results recently obtained by variational methods. We give Melnikov type conditions for the existence of one homoclinic solution and for the existence of infinitely many multibump solutions. We give also an example for which the set of zeros of the Poincaré-Melnikov function contains an interval and results requiring a simple zero of this function can not be applied. In the case of multibump solutions, when the perturbation is periodic, we prove the existence of approximate Bernoulli shift structures leading to some form of chaos.
1999, 5(4): 783-798 doi: 10.3934/dcds.1999.5.783 +[Abstract](887) +[PDF](246.1KB)
Abstract:
In this paper we give new properties of the dimension introduced by Afraimovich to characterize Poincaré recurrence and which we proposed to call Afraimovich-Pesin's (AP's) dimension. We will show in particular that AP's dimension is a topological invariant and that it often coincides with the asymptotic distribution of periodic points : deviations from this behavior could suggest that the AP's dimension is sensitive to some "non-typical" points.
1999, 5(4): 799-804 doi: 10.3934/dcds.1999.5.799 +[Abstract](941) +[PDF](109.2KB)
Abstract:
In this article, we study harmonic maps between two complete noncompact manifolds M and N by a heat flow method. We find some new sufficient conditions for the uniform convergence of the heat flow, and hence the existence of harmonic maps.
Our condition are: The Ricci curvature of M is bounded from below by a negative constant, M admits a positive Green’s function and

$\int_M G(x, y)|\tau(h(y))|dV_y$ is bounded on each compact subset. $\qquad$ (1)

Here $\tau(h(x))$ is the tension field of the initial data $h(x)$.
Condition (1) is somewhat sharp as is shown by examples in the paper.

1999, 5(4): 805-812 doi: 10.3934/dcds.1999.5.805 +[Abstract](1022) +[PDF](1365.7KB)
Abstract:
We give a uniform rate function for large deviations of the random occupational measures of an expanding random dynamical system.
1999, 5(4): 813-824 doi: 10.3934/dcds.1999.5.813 +[Abstract](884) +[PDF](185.0KB)
Abstract:
This article discusses the relationship between the inertial manifolds "with delay" introduced by Debussche & Temam, and the standard definition. In particular, the "multi-valued" manifold of the same paper is shown to arise naturally from the manifolds "with delay" when considering issues of convergence as the delay time tends to infinity. This leads to a new characterisation of the multi-valued manifold, which allows a fuller understanding of its structure.
1999, 5(4): 825-848 doi: 10.3934/dcds.1999.5.825 +[Abstract](896) +[PDF](442.7KB)
Abstract:
The complex Ginzburg-Landau equation (CGL, for short)

$\partial_t u = (1 + i\nu)\Delta u + Ru- (1 + i\mu) |u|^2 u; \quad 0\le t < \infty, x\in\Omega$,

is investigated in a bounded domain ­$\Omega\subset \mathbb R^n$ with suffciently smooth boundary. Standard boundary conditions are considered: Dirichlet, Neumann or periodic. Existence and uniqueness of global smooth solutions is established for all real parameter values $\mu$ and $\nu$ if $n\le 2$, and for certain parameter values $\mu$ and $\nu$ if $n\ge 3$. Furthermore, dynamical properties of the CGL equation, such as existence of determining nodes, are shown. The proof of existence of smooth solutions hinges on the following inequality using the $L^2(\Omega)$-duality,

$|\mathfrak Im$ $<\Delta u ,\ |u|^{p-2}u>\le (|p-2|)/(2\sqrt{p-1})\mathfrak Re$ $< -\Delta u ,\ |u|^{p-2}u >.$

1999, 5(4): 849-870 doi: 10.3934/dcds.1999.5.849 +[Abstract](1115) +[PDF](464.8KB)
Abstract:
We study here the blow-up set of the maximal classical solution of $u_t -\Delta u = g(u)$ on a ball of $\mathbb R^N$, $N \geq 2$ for a large class of nonlinearities $g$, with $u(x,0) = u_0(|x|)$. Numerical experiments show the interesting behaviour of the blow-up set in respect of $u_0$. As a theoretical background to the method used in this work, we prove an important monotonicity property, that is for a fixed positive radius $r_0$, when the solution gets large enough at a certain time $t_0$, then $u$ is monotone increasing at $r_0$ after $t_0$. Finally, a single radius blow-up property is proved for some large initial conditions.
1999, 5(4): 871-880 doi: 10.3934/dcds.1999.5.871 +[Abstract](965) +[PDF](284.4KB)
Abstract:
It is shown that the complex Ginzburg-Landau (CGL) equation on the real line admits nontrivial $2\pi$-periodic vortex solutions that have $2n$ simple zeros ("vortices") per period. The vortex solutions bifurcate from the trivial solution and inherit their zeros from the solution of the linearized equation. This result rules out the possibility that the vortices are determining nodes for vortex solutions of the CGL equation.
1999, 5(4): 881-896 doi: 10.3934/dcds.1999.5.881 +[Abstract](1026) +[PDF](222.7KB)
Abstract:
The wellposedness of the delay problem in a Banach space $X$

$u'(t) = Au(t) + \int_{-r}^0 k(s)A_1 u(s) ds + f(t),\quad t\ge 0;\quad u(t) = z(t), \quad t\in [-r,0]$

(where $A : D(A)\subset X \to X$ is a closed operator and $A_1 : D(A)\to X$ is continuous) is proved and applied to get a classical solution of the wave equation with memory effects

$w_{t t} (t,x) = w_{x x}(t, x) + \int_{-r}^0 k(s) w_{x x} (t + s, x)ds + f(t, x), \quad t\ge 0,\quad x\in [0,l]$

To include also the Dirichlet boundary conditions and to get $C^2$-solutions, $D(A)$ is not supposed to be dense hence A is only a Hille-Yosida operator. The methods used are based on a reduction of the inhomogeneous equation to a homogeneous system of the first order and then on an immersion of $X$ in its extrapolation space, where the regularity and perturbation results of the classical semigroup theory can be applied.

1999, 5(4): 897-904 doi: 10.3934/dcds.1999.5.897 +[Abstract](969) +[PDF](1262.5KB)
Abstract:
We consider hyperbolic flows on one dimensional basic sets. Any such flow is conjugate to a suspension of a shift of finite type. We consider compact Lie group skew-products of such symbolic flows and prove that they are stably ergodic and stably mixing, within certain naturally defined function spaces.
1999, 5(4): 905-928 doi: 10.3934/dcds.1999.5.905 +[Abstract](1074) +[PDF](3274.8KB)
Abstract:
The aim of this paper is to study the blow up behavior of a radially symmetric solution $u$ of the semilinear parabolic equation

$u_t - \Delta u = |u|^{p-1} u, \quad x\in\Omega,\quad t\in [0,T]$,

$u(t,x)=0, x\in\partial\Omega, \quad t\in [0,T]$,

$u(0,x) =u_0(x),\quad x\in\Omega$,

around a blow up point other than its centre of symmetry. We assume that $\Omega$ is a ball in $\mathbb R^N$ or $\Omega =\mathbb R^N$, and $p>1$. We show that $u$ behave as of a one-dimensional problem was concerned, that is, the possible asymptotic behaviors and final time profiles around an unfocused blow up point are the ones corresponding to the case of dimesion $N=1$.

1999, 5(4): 929-945 doi: 10.3934/dcds.1999.5.929 +[Abstract](789) +[PDF](190.4KB)
Abstract:
In this paper we establish several results concerning the asymptotic behavior of (random) infinite products of generic sequences of homogeneous order-preserving mappings on a cone in an ordered Banach space. In addition to weak ergodic theorems we also obtain convergence to an operator $f(\cdot)\eta$ where $f$ is a functional and $\eta$ is a common fixed point.

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