火成岩蠕变中分数阶本构模型的动力学及约化方法研究

In recent decades, fractional differential systems have attracted a great deal of attention, in which Hadamard type fractional differential systems have played an important role in rheology, ultraslow kinetics, material mechanics and other applied sciences. However, theoretical problems such as the dynamic behaviors and reductions of fractional differential systems in creep of igneous rock are still challenging. The main contents of this project can be summarized as follows: A generalized Lomnitz logarithmic creep law which be characterized by Caputo-Hadamard type fractional differential operators is established; The Lyapunov stability and equilibrium bifurcation and other dynamic problems of Caputo-Hadamard type fractional differential systems are also investigated; Based on stability theory and Fredholm operator theory, the center manifold reduction and Lyapunov-Schmidt reduction for Caputo-Hadamard type fractional differential systems will be implemented, respectively. Consequently, this project can enrich the related theory and methods for Caputo-Hadamard type fractional differential systems. In addition that, it may also provide the necessary theoretical guides for realistic models described by Caputo-Hadamard type fractional operators.

近几十年，分数阶微分系统受到了广泛的关注，其中Hadamard型分数阶微分系统已在流变学、超慢动力学、材料力学等应用科学中占据重要的地位。然而，火成岩蠕变中分数阶微分系统的动力学行为及其约化方法等理论问题仍具有挑战性。鉴于此，本项目拟开展如下研究：构建由Caputo-Hadamard型分数阶微分算子所刻画的广义Lomnitz对数蠕变定律；研究Caputo-Hadamard型分数阶微分系统的稳定性及分岔等动力学问题；基于稳定性以及Fredholm算子理论，分别建立Caputo-Hadamard型分数阶微分系统的中心流形约化方法与Lyapunov-Schmidt约化方法。从而丰富并发展Caputo-Hadamard型分数阶微分系统相关理论与方法，并为实际应用中的Caputo-Hadamard型分数阶模型提供必要的理论支撑。

本项目采用Katugampola型分数阶微分算子（兼容Caputo-Hadamard分数阶算子）建立火成岩蠕变的分数阶本构模型，进而获得广义的Lomnitz蠕变定律。其次，通过采用修正经典Laplace积分变换的定义方式，获得了适用于Hadamard型与Katugampola型分数阶微积分的Laplace积分变换技术；最后，还针对Hadamard型分数阶微分系统，论证了系统不存在非平凡周期解的重要事实，明确了爆破解存在的条件与影响因素，提出了有限时间稳定性的标准，给出了更为精准的分数阶Lyapunov指数的定义方式。本项目的主要结果为进一步建立Caputo-Hadamard型分数阶微分系统的分岔标准型、中心流形与Lyapunov-Schmidt约化方法奠定了理论基础，同时也为复杂环境下岩石蠕变建模提供可行性方案。