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Reducibility of zero curvature equations. (English) Zbl 1058.34049

A pair \(( \Theta _{1},\Theta _{2}) \) of smooth complex \(n\times n\)-matrix functions in the variables \(( s,t) \in \mathbb{R}^{2}\) is called a compatible pair if there exists a smooth complex function \(G( s,t) \) satisfying \[ \frac{\partial G}{\partial s}=\Theta _{1}G, \quad \frac{\partial G}{\partial t}=\Theta _{2}G, \] with the initial condition \(G( 0,0) =I_{n}\). The commutativity of the second mixed derivatives \(\frac{\partial ^{2}G}{\partial s\partial t}=\frac{\partial ^{2}G}{\partial t\partial s}\) implies \(\frac{\partial \Theta _{1}}{\partial t}-\frac{\partial \Theta _{2}}{\partial s}+[ \Theta _{1},\Theta _{2}] =0\) which is called the zero-curvature equation, because the pair \(( \Theta _{1},\Theta _{2}) \) yields a linear connection \(\mathbb{R}^{2}\times \mathbb{C}^{n}\to \mathbb{R}^{2}\) with the connection 1-form \(\Omega =-( \Theta _{1}ds+\Theta _{2}dt) \) having the curvature \(d\Omega +\Omega \wedge \Omega =0\).
For this type of systems, a natural concept of reducibility is introduced as the existence of a global “gauge” transformation reducing the initial system to another one with constant coefficients. The main result is a Floquet representation for such equations. As applications, the reducibility problems for quasiperiodic 2-dimensional linear systems and for fiberwise linear dynamical systems on trivial vector bundles are discussed.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34A26 Geometric methods in ordinary differential equations
34A30 Linear ordinary differential equations and systems
37C55 Periodic and quasi-periodic flows and diffeomorphisms
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