Construction of minimal cocycles arising from specific differential equations

Blending methods of Topological Dynamics and Control Theory, we develop a new technique to construct compact-Lie-group-valued minimal cocycles arising as fundamental matrix solutions of linear differential equations with recurrent coefficients subject to a given constraint. The precise requirement on the coefficients is that they belong to a specified closed convex subset S of the Lie algebra L of the Lie group. Our result is proved for a very thin class of cocycles, since the dimension of S is allowed to be much smaller than that of L, and the only assumption on S is that L₀(S) = L, where L₀(S) is the ideal of L(S) generated by the difference set S - S, and L(S) is the Lie subalgebra of L generated by S. This covers a number of differential equations arising in Mathematical Physics, and applies in particular to the widely studied example of the Rabi oscillator.