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1 Flows vs. Maps

A system of ordinary differential equations can often be written in the form [1,2]

where t is the independent variable (normally time) and

Equation (1) consists of n first-order differential equations. If the original set of equations is of higher-order, they can often be transformed to Eq. (1) by redefining variables and expanding their number. (It may be recalled that in classical mechanics the m second-order differential equations arising from a Lagrangian formulation are equivalent to the n=2m first-order equations from the Hamiltonian formulation.)