Sensitivities in complex-time flows: phase transitions, Hamiltonian structure and differential geometry

Reminiscent of physical phase transitions separatrices divide the phase space of dynamical systems with multiple equilibria into regions of distinct flow behavior and asymptotics. We introduce complex time in order to study corresponding Riemann surface solutions of holomorphic and meromorphic flows, explicitly solve their sensitivity differential equation and identify a related Hamiltonian structure and an associated geometry in order to study separatrix properties. As an application we analyze complex-time Newton flow of Riemann's $\xi$-function on the basis of a compactly convergent polynomial approximation of its Riemann surface solution defined as zero set of polynomials, e.g. algebraic curves over $\mathbb{C}$ (in the complex projective plane respectively), that is closely related to a complex-valued Hamiltonian system. Its geometric properties might contain information on the global separatrix structure and the root location of $\xi$ and $\xi'$.