MSC: 93C10, 45G15
DOI: 10.21538/0134-4889-2021-27-3-271-285
Full text
We obtain an analogue of Wilcox–Snyder formula for flows of diffeomorphisms of $C^m$-smooth vector fields on infinite-dimensional Banach manifolds. For classical linear system this formula can be efficiently used, for example, to obtain Magnus expansion of solutions. The generalized Wilcox formula is obtained by using an extended Chronological Calculus for Banach manifold. We apply this formula to derive new structured differential equations which solutions approximate solutions of the original differential equation.
Keywords: flow of diffeomorphisms, Wilcox formula, chronological calculus, Magnus expansion.
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Received July 20, 2021
Revised August 11, 2021
Accepted August 23, 2021
Yuri S. Ledyaev, Dr. Phys.-Math. Sci., Prof., Member of Steklov Institute of Mathematics, Moscow, 117966 Russia; Department of Mathematics Western Michigan University, Kalamazoo, MI 49008, USA, e-mail: ledyaev@wmich.edu
Cite this article as: Yuri S. Ledyaev. Wilcox Formula for Vector Fields on Banach Manifolds, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 271–285.
Русский
Ю.С. Ледяев. Формула Уилкокса для векторных полей на банаховых многообразиях
Получен аналог формулы Уилкокса — Снайдера для потоков диффеоморфизмов $C^m$-гладких векторных полей на бесконечномерных банаховых многообразиях. Эта формула может эффективно использоваться, например, для получения разложения Магнуса решений классических линейных систем. Обобщенная формула Уилкокса получена с использованием расширения хронологического исчисления для банаховых многообразий. Эта формула применена для вывода новых стуктурированных дифференциальных уравнений, решения которых приближают решения исходного дифференциального уравнения.
Ключевые слова: поток диффеоморфизмов, формула Уилкокса, хронологическое исчисление, разложение Магнуса.
