A note on topological methods for a class of Differential-Algebraic Equations

2009/02/21 by Marco Spadini, Spadini, Marco
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #Numerical methods for differential equations #math.CA #math.DS #msc:34C25 #msc:34C40

paper · pdf · doi:10.48550/arxiv.0902.3745

16 pages Changes in new version: A few typos across the paper; Wrong cross references at page one; Statement and proof of Theorem 5.1 revised; $°(F,U)=1$ in Example 5.2; $p_n\to p_0$ in the proof of Lemma 5.5; inequality instead of equality in line 18, page 15; Assumption $A$ bounded added in Corollary 5.3

Abstract

We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine an easy-to-use straightforward formula for the computation of the degree of the associated tangent vector field that does not require any explicit knowledge of the manifold. We use this formula to study the set of harmonic solutions to periodic perturbations of our equations. Two different classes of applications are provided.

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