Runge-Kutta discretization commutes with every affine map that relates two vector fields, so affine invariant submanifolds such as synchrony polydiagonals are preserved in the discrete dynamics.
Dynamics on Networks of Manifolds
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We propose a precise definition of a continuous time dynamical system made up of interacting open subsystems. The interconnections of subsystems are coded by directed graphs. We prove that the appropriate maps of graphs called graph fibrations give rise to maps of dynamical systems. Consequently surjective graph fibrations give rise to invariant subsystems and injective graph fibrations give rise to projections of dynamical systems.
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Runge-Kutta and Networks
Runge-Kutta discretization commutes with every affine map that relates two vector fields, so affine invariant submanifolds such as synchrony polydiagonals are preserved in the discrete dynamics.