## 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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