Analysis of almost periodic attractors for nonlinear evolution equations

近年来，在群体动力学或反应扩散模型的研究中，对复杂性或混沌行为进行了大量的研究。本文通过估计半线性偏微分方程的分形维数，研究了半线性偏微分方程的概周期吸引子。特别是在[1]和[2]（11.REF.）中，我们研究了一个假设扩散系数周期性和非线性反应函数的反应-扩散方程，作为具有扩散速率季节性波动的种群动态模型。通过同时使用Diophantine近似，我们可以证明概周期吸引子的维数被这些周期函数上的Holder条件的指数所占据。在[3]中，我们进一步研究了关于空间变量具有拟周期扰动的线性薛定谔方程的拟周期解。我们可以估计解范围的分形维数，构造近似周期，换句话说，就是同步点，通过迭代方法，它依赖于对频率无理数的同时逼近。计算吸引子的维度是为了衡量它们的复杂性和随机性。另一方面，众所周知，在各种混沌路径中，周期或准周期状态作为主要门户占据着重要地位。在接下来的论文（11.REF.）中，我们已经展示了各种基本结果，这些结果将对研究非线性动力学模型的混沌行为起重要和必不可少的作用。

In recent years great efforts have been made to analyze complexity or chaotic behaviors in the study of population dynamics or reaction-diffusion models. In this research we investigate almost periodic attractors for the semilinear partial differential equations by estimating their fractal dimensions. Especially, in [1] and [2] (of 11.REF.) we study a reaction-diffusion equation, assuming the periodicity of the diffusion coefficient and the nonlinear reaction function, as a model of population dynamics which has seasonal fluctuations in the diffusion rates. By using simultaneous Diophantine approximation, we can show that the dimension of the almost periodic attractor is majorized by the exponents of Holder's conditions on these periodic functions. Furthermire, in [3] we investigate a quasi-periodic solution of a linear Schrodinger equation with a quasi periodic perturbation with respect to the space variables. We can estimate the fractal dimension of the range of the solution, constructing the epsilon-almost periods, the epsilon-synchronous points in other words, by the iterative method, which depends on the simultaneous approximation for the irrational numbers of the frequencies.Calculating the dimensions of the attractors is to measure their level of complexity and randomness. On the other hand, it is well known that periodic or almost periodic states occupy the important positions as main gateways in various routes to chaos. In the following papers (of 11.REF.) by the head and co-investigators we have shown various fundamental results, which will play important and essential roles for investigating chaotic behaviors of nonlinear dynamical models.