Analysis of almost periodic attractors for nonlinear evolution equations

近年来，在种群动力学或反应扩散模型的研究中，人们对复杂性或混沌行为进行了大量的研究。在这项研究中，我们通过估计半线性偏微分方程的分数维来研究其概周期吸引子。特别是在[1]和[2]（of 11. REF.）本文研究了一类反应扩散方程，假定扩散系数具有周期性，反应函数为非线性，作为扩散率具有季节性波动的种群动力学模型。利用同时丢番图近似，我们可以证明几乎周期吸引子的维数由这些周期函数上的保持器条件的指数控制。在文献[3]中，我们进一步研究了具有关于空间变量的拟周期扰动的线性Schrodinger方程的拟周期解。我们可以估计解的范围的分形维数，通过迭代方法构造ε-几乎周期，也就是ε-同步点，这依赖于对频率的无理数的同时逼近。计算吸引子的维数是为了衡量它们的复杂性和随机性。另一方面，众所周知，周期或概周期态在通向混沌的各种途径中占据着重要的地位。在以下论文中（11. REF.）我们的主要研究者和合作者已经得到了一些基本的结果，这些结果对于研究非线性动力学模型的混沌行为具有重要的意义。

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.