Exponential monotonicity of quadratic forms in ODEs and preserving methods

In this paper we consider ODEs whose solutions satisfy exponential monotonic quadratic forms. We show that the quadratic preserving Gauss-Legendre-Runge-Kutta methods do not preserve this qualitative feature, while certain Lie group preserving methods, as Crouch-Grossman methods and methods based on projection techniques are suitable integrators for such a kind of differential problems. We also show that these differential problems may be solved via associated differential systems with quadratic invariants. Numerical tests are reported in order to confirm our theoretical results.