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Hypercomplex analysis and integration of systems of ordinary differential equations. (English) Zbl 1351.34042

Summary: We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati-type systems, we obtain invariants of Abel-type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
30G35 Functions of hypercomplex variables and generalized variables
34A05 Explicit solutions, first integrals of ordinary differential equations
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