动力学中的连续层：层上同调与分歧
Continuation Sheaves in Dynamics: Sheaf Cohomology and Bifurcation
摘要：
连续的代数结构在家庭的动力系统描述使用范畴论，层，和格代数。动力学中的著名概念，如吸引子或不变集，被公式化为动力系统映射到格、偏序集、环或阿贝尔群的适当范畴上的函子。层是由这样的函子构造而成的，函子在系统参数变化时对结构的连续性进行编码。类似地，所讨论的层的态射来自自然变换。这个框架适用于各种格代数和环结构相关的动力系统，其代数性质结转到各自的层。此外，这些层的上同调是代数不变式，它包含了参数化系统的分叉信息。
英文摘要：
Continuation of algebraic structures in families of dynamical systems is described using category theory, sheaves, and lattice algebras. Well-known concepts in dynamics, such as attractors or invariant sets, are formulated as functors on appropriate categories of dynamical systems mapping to categories of lattices, posets, rings or abelian groups. Sheaves are constructed from such functors, which encode data about the continuation of structure as system parameters vary. Similarly, morphisms for the sheaves in question arise from natural transformations. This framework is applied to a variety of lattice algebras and ring structures associated to dynamical systems, whose algebraic properties carry over to their respective sheaves. Furthermore, the cohomology of these sheaves are algebraic invariants which contain information about bifurcations of the parametrized systems.
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