AI 中文总结
该研究为封闭动力系统构建范畴框架，将其定义为从抽象演化形状\(S\)到系数范畴\(C\)的函子，涵盖多种系统。通过函子术语引入相关概念，结合余筛与邻域滤波器表述收敛，建立范畴李雅普诺夫原理，得出抽象稳定性和收敛标准。
AI 中文摘要
我们为封闭动力系统开发了一个范畴框架，其中可允许演化的抽象模式与其具体实现相分离。一个封闭动力系统被表述为一个从一个小范畴\(S\)（视为抽象演化形状）到一个系数范畴\(C\)的函子\(X\colon S\to C\)。通过改变\(S\)和\(C\)，这一定义涵盖了许多重要例子。在此框架内，我们用函子术语引入不变子系统、平衡点和轨道。然后通过将基于余筛的演化形状上的最终性内在概念与不变子系统的邻域滤波器相结合来表述收敛性。最后，基于范畴子水平邻域建立了一个范畴李雅普诺夫原理。这产生了抽象稳定性和收敛标准，在标准例子中恢复了经典李雅普诺夫方法。
英文摘要
We develop a categorical framework for closed dynamical systems in which the abstract pattern of admissible evolutions is separated from its concrete realisation. A closed dynamical system is formulated as a functor $X\colon S\to C$ from a small category $S$, viewed as an abstract evolution shape, to a coefficient category $C$. By varying $S$ and $C$, this single definition encompasses many important examples including autonomous, non-autonomous, switched, hybrid, and stochastic systems. Within this framework, we introduce invariant subsystems, equilibria, and orbits in functorial terms. We then formulate convergence by combining a cosieve-based intrinsic notion of eventuality on the evolution shape with neighbourhood filters of invariant subsystems. Finally, we establish a categorical Lyapunov principle based on categorical sublevel neighbourhoods. This yields abstract stability and convergence criteria that recover the classical Lyapunov method in standard examples.
Comments32 pages
