American Institute of Mathematical Sciences

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1078-0947

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1553-5231

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Discrete & Continuous Dynamical Systems - A

July 2004 , Volume 10 , Issue 3

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2004, 10(3): 581-587 doi: 10.3934/dcds.2004.10.581 +[Abstract](766) +[PDF](132.0KB)
Abstract:
Let $T$ be a transformation from $I=[0,1)$ onto itself and let $Q_n(x)$ be the subinterval $[i/2^n,(i+1)/2^n)$, $0 \leq i < 2^n$ containing $x$. Define $K_n (x) =$min{$j\geq 1:T^j (x)\in Q_n(x)$} and $K_n(x,y) =$min{$j\geq 1:T^{j-1} (y) \in Q_n(x)$}. For various transformations defined on $I$, we show that

$\lim_{n\to\infty}\frac{\log K_n(x)}{n}=1 \quad$and$\quad \lim_{n\to\infty}\frac{\log K_n(x,y)}{n}=1 \quad$a.e.

2004, 10(3): 589-616 doi: 10.3934/dcds.2004.10.589 +[Abstract](663) +[PDF](287.7KB)
Abstract:
We develop a new framework that allows to prove that the limiting distribution of return times for a large class of mixing dynamical systems are Poisson distributed. We demonstrate our technique in several settings and obtain more general results than previously has been proven. We also obtain error estimates. For $\phi$-mixing maps we obtain a close to exhausting description of return times. For $(\phi,f)$-mixing maps it is shown how the separation function affects error estimates for the limiting distribution. As examples of $(\phi,f)$-mixing we prove that for piecewise invertible maps and for rational maps return times are in the limit Poisson distributed.
2004, 10(3): 617-634 doi: 10.3934/dcds.2004.10.617 +[Abstract](692) +[PDF](195.2KB)
Abstract:
In this paper, existence and multiplicity of nontrivial solutions are obtained for some nonlinear elliptic boundary value problems with perturbation terms of arbitrary growth. Results are obtained via variational arguments.
2004, 10(3): 635-656 doi: 10.3934/dcds.2004.10.635 +[Abstract](790) +[PDF](256.6KB)
Abstract:
We provide a detailed study of the quantitative behavior of first return times of points to small neighborhoods of themselves. Let $K$ be a self-conformal set (satisfying a certain separation condition) and let $S:K\to K$ be the natural self-map induced by the shift. We study the quantitative behavior of the first return time,

$\tau_{B(x,r)}(x)=$inf{$1\le k \le n | S^k x \in B(x,r)},$

of a point $x$ to the ball $B(x,r)$ as $r$ tends to $0$. For a function $\varphi:(0,\infty)\to\mathbb R$, let A$(\varphi(r))$ denote the set of accumulation points of $\varphi(r)$ as $r\to 0$. We show that the first return time exponent, $\frac{\log\tau_{B(x,r)}(x)} {-\log r}$, has an extremely complicated and surprisingly intricate structure: for any compact subinterval $I$ of $(0,\infty)$, the set of points $x$ such that for each $t\in I$ there exists arbitrarily small $r>0$ for which the first return time $\tau_{B(x,r)}(x)$ of $x$ to the neighborhood $B(x,r)$ behaves like $1/r^t$, has full Hausdorff dimension on any open set, i. e.

dim$(G\cap${$x\in K|$A ($\frac {\log\tau_{B(x,r)}(x)}{-\log r}) =I$}) $=$dim $K$

for any open set $G$ with $G\cap K$≠$\emptyset$. As a consequence we deduce that the so-called multifractal formalism fails comprehensively for the first return time multifractal spectrum. Another application of our results concerns the construction of a certain class of Darboux functions.

2004, 10(3): 657-678 doi: 10.3934/dcds.2004.10.657 +[Abstract](766) +[PDF](1042.0KB)
Abstract:
Closed physical systems eventually come to rest, the reason being that due to friction of some kind they continuously lose energy. The mathematical extension of this principle is the concept of a Lyapunov function. A Lyapunov function for a dynamical system, of which the dynamics are modelled by an ordinary differential equation (ODE), is a function that is decreasing along any trajectory of the system and with exactly one local minimum. This implies that the system must eventually come to rest at this minimum. Although it has been known for over 50 years that the asymptotic stability of an ODE's equilibrium is equivalent to the existence of a Lyapunov function for the ODE, there has been no constructive method for non-local Lyapunov functions, except in special cases. Recently, a novel method to construct Lyapunov functions for ODEs via linear programming was presented [5], [6], which includes an algorithmic description of how to derive a linear program for a continuous autonomous ODE, such that a Lyapunov function can be constructed from any feasible solution of this linear program. We will show how to choose the free parameters of this linear program, dependent on the ODE in question, so that it will have a feasible solution if the equilibrium at the origin is exponentially stable. This leads to the first constructive converse Lyapunov theorem in the theory of dynamical systems/ODEs.
2004, 10(3): 679-685 doi: 10.3934/dcds.2004.10.679 +[Abstract](679) +[PDF](116.1KB)
Abstract:
For certain Newtonian $N$-body problems in $\mathbf R^3$, we proved the existence of new symmetrical noncollision periodic solutions.
2004, 10(3): 687-707 doi: 10.3934/dcds.2004.10.687 +[Abstract](711) +[PDF](243.7KB)
Abstract:
In the paper we prove that the Lagrangian system

$\ddot{q} = \alpha(\omega t) V'(q), \quad t \in \mathbb R, q \in \mathbb R^N,$ $\qquad\qquad (L_\omega)$

has, for some classes of functions $\alpha$, a chaotic behavior---more precisely the system has multi-bump solutions---for all $\omega$ large. These classes of functions include some quasi-periodic and some limit-periodic ones, but not any periodic function.
We prove the result using global variational methods.

2004, 10(3): 709-717 doi: 10.3934/dcds.2004.10.709 +[Abstract](662) +[PDF](125.9KB)
Abstract:
Solitary-wave solutions of a nonlinearly dispersive equation are considered. It is found that solitary waves are peaked or smooth waves, depending on the wave speed. The stability of the smooth solitary waves also depends on the wave speed. Orbital stability is proved for some wave speeds, while instability is proved for others.
2004, 10(3): 719-730 doi: 10.3934/dcds.2004.10.719 +[Abstract](691) +[PDF](161.4KB)
Abstract:
This paper is a continuation of [3] by the same authors to study the problem of global existence of strong solutions for the Shigesada-Kawasaki-Teramoto model. We shall prove global existence of strong solutions assuming that there are self- and cross-diffusions in the first species and there is no cross-diffusion in the second species. If self-diffusion is also present in the second species, then our result requires that the space dimension be less than 6.
2004, 10(3): 731-753 doi: 10.3934/dcds.2004.10.731 +[Abstract](792) +[PDF](239.8KB)
Abstract:
Considered herein is an initial-value problem for the Ostrovsky equation that arises in modelling the unidirectional propagation of long waves in a rotating homogeneous incompressible fluid. Nonlinearity and dispersion are taken into account, but dissipation is ignored. Local- and global-in-time solvability is investigated. For the case of positive dispersion a fundamental solution of the Cauchy problem for the linear equation is constructed, and its asymptotics is calculated as $t\rightarrow \infty, x/t=$const. For the nonlinear problem solutions are constructed in the form of a series and the analogous long-time asymptotics is obtained.
2004, 10(3): 755-768 doi: 10.3934/dcds.2004.10.755 +[Abstract](816) +[PDF](182.7KB)
Abstract:
In this paper we give, as far as we know, the first method to detect non-algebraic invariant curves for polynomial planar vector fields. This approach is based on the existence of a generalized cofactor for such curves. As an application of this algorithmic method we give some Lotka-Volterra systems with non-algebraic invariant curves.
2004, 10(3): 769-786 doi: 10.3934/dcds.2004.10.769 +[Abstract](652) +[PDF](200.9KB)
Abstract:
When $\alpha\le 2\beta$, we will prove the non-existence of solutions of the equation $\Delta v+\alpha e^v+\beta (x\cdot\nabla v)e^v=0$ in $R^2$ which satisfy $\gamma =\int_{R^2}e^vdx/(2\pi) <\infty$ and $|x|^2e^{v(x)}\le C_1$ in $R^2$ for some constant $C_1>0$. When $\alpha>2\beta$, we will prove that if $v$ is a solution of the above equation, then there exist constants $0<\tau\le 1$ and $a_1$ such that $v(x)=-\gamma \log |x|+a_1+O(|x|^{-\tau})$ as $|x|\to\infty$ where $\gamma=(\alpha -2\beta)\gamma$. We will also show that $\gamma$ satisfies $\gamma>2$ and $\gamma<\alpha$.
2004, 10(3): 787-804 doi: 10.3934/dcds.2004.10.787 +[Abstract](853) +[PDF](187.7KB)
Abstract:
We study the qualitative behavior of solutions of a wave equation with nonlinear damping and a source term. We give a characterization of blow-up of solutions, improving a previous result. When the dissipation dominates the source term, we show existence of unbounded global solutions. We use the stable (potential well) and unstable sets, introduced by Sattinger and Payne. We study all bounded global solutions, and we charaterize their convergence as $t \rightarrow \infty$. In particular, we prove that every solution, with energy larger or equal than the depth of the potential well, is global, bounded and converges to the set of nonzero equilibria.
2004, 10(3): 805-826 doi: 10.3934/dcds.2004.10.805 +[Abstract](772) +[PDF](506.5KB)
Abstract:
We are interested in the asymptotic behaviors of a discrete-time neural network. This network admits transiently chaotic behaviors which provide global searching ability in solving combinatorial optimization problems. As the system evolves, the variables corresponding to temperature in the annealing process decrease, and the chaotic behaviors vanish. We shall find sufficient conditions under which evolutions for the system converge to a fixed point of the system. Attracting sets and uniqueness of fixed point for the system are also addressed. Moreover, we extend the theory to the neural networks with cycle-symmetric coupling weights and other output functions. An application of this annealing process in solving travelling salesman problems is illustrated.
2004, 10(3): 827-833 doi: 10.3934/dcds.2004.10.827 +[Abstract](960) +[PDF](130.5KB)
Abstract:
About 5 years ago, Dai, Zhou and Geng proved the following result. If $X$ is a metric compact space and $f:X\to X$ a Lipschitz continuous map, then the Hausdorff dimension of $X$ is bounded from below by the topological entropy of $f$ divided by the logarithm of its Lipschitz constant. We show that this is a simple consequence of a 30 years old Bowen's definition of topological entropy for noncompact sets. Moreover, a modification of this definition provides a new insight into the entropy of subshifts of finite type.

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