同伦算法及其在微分方程周期解问题中的应用研究

The homotopy algorithm has received high concerns from many famous scholars since it was proposed, then, it was used to give constructive proofs of lots of important theorems and became a powerful toll for solving a large number of nonlinear problems. In this project, we want to investigate the homotopy algorithm and its applications in the periodicity problems of differential equations deeply, and make great efforts to obtain some meaningful and innovative research results. First, we modify the idea of aggregate functions and the homotopy algorithm simultaneously, construct new aggregate functions and new aggregate homotopy equations, and hence propose a new homotopy algorithm, i.e., an aggregate constraint homotopy algorithm for equality constraint cases, to solve the optimization problems with equality constraints; second, use the aggregate constraint homotopy algorithm for equality constraint cases, we give a constructive proof of the existence of the periodic solution of the dissipative system under certain non-convexity conditions, and also give a constructive proof of the existence of a homotopy path connecting the periodic solution of Liénard differential equation to a given point at several non-resonance cases, as a result, we can provide a new efficient globally convergent algorithm to solve the periodicity problems of the above two class of important differential equations; finally, when solving the periodic solutions of differential equations, we enlarge the scope of choice of initial points, and hence improve the computational efficiency of the algorithm.

同伦算法一经提出，便引起了众多著名学者的高度关注，并被用于给出一系列重要定理的构造性证明，而且也成为求解诸多非线性问题的强大工具。本项目拟对同伦算法及其在微分方程周期解问题中的应用进行系统深入的研究，力争取得一些有意义的创新性研究成果。首先，针对含等式约束的非线性优化问题，同时对凝聚函数思想和同伦算法进行改进，构造新的凝聚函数和凝聚同伦方程，进而提出求解该问题的新的同伦算法，即等式约束下的凝聚约束同伦算法；其次，利用等式约束下的凝聚约束同伦算法，在一定非凸条件下给出耗散系统周期解存在性的构造性证明，在几类常见的非共振条件下，给出连接Liénard微分方程周期解和给定初始点的同伦路径存在性的构造性证明，从而为求解这两类重要的微分方程周期解问题提供一种新的高效的全局收敛性算法；最后，在求解上述两类微分方程周期解时，扩大算法的初始点选择范围，大大提高算法的计算效率。

微分方程周期解问题的研究一直是微分方程定性理论的中心课题，人们在周期解的存在性证明方面已经取得了丰硕成果。目前，如何数值求解微分方程周期解问题是人们需要加以探讨的困难问题。本项目组对同伦算法及其在周期解问题数值计算方面的应用进行了系统深入的研究，并取得了如下研究成果：针对非线性优化问题，采用一定的技巧，使得由凝聚函数形成的约束集合始终具备边界正则性和弱法锥条件，进而提出了等式约束下的凝聚约束同伦算法；把耗散系统周期解问题进转化为等价的不动点问题，同时使用同伦技巧、内点法思想以及凝聚函数方法，提出了等式约束下的凝聚约束同伦算法处理了该类周期解问题；构造了连接Liénard方程和一个平凡方程的线性同伦，引入了新的凝聚函数和对应的牛顿同伦，进而给出了求解Liénard方程周期解问题的等式约束下的凝聚约束同伦算法；在求解微分方程周期解问题时，针对约束函数进行合理的摄动，进一步利用非内点法思想和同伦技巧，扩大了初始点的选择范围，大大提高了算法的计算效率；在处理不动点问题时，给出了更一般的非凸性条件和一类无界性条件；所有这些研究成果都是对微分方程理论及其数值计算领域有关方法的有益的补充和发展。