Transient spectral theory, stable and unstable cones and Gershgorinʼs theorem for finite-time differential equations

Dynamical behaviour on a compact (finite-time) interval is called monotone-hyperbolic or M-hyperbolic if there exists an invariant splitting consisting of solutions with monotonically decreasing and increasing norms, respectively. This finite-time hyperbolicity notion depends on the norm. For arbitrary norms we prove a spectral theorem based on M-hyperbolicity and extend Gershgorinʼs circle theorem to this type of finite-time spectrum. Similarly to stable and unstable manifolds, we characterize M-hyperbolicity by means of existence of stable and unstable cones. These cones can be explicitly computed for D-hyperbolic systems with norms induced by symmetric positive definite matrices and also for row diagonally dominant systems with the sup-norm, thus providing sufficient and computable conditions for M-hyperbolicity.