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Category Archives: ODE
More on Riccati equations and fractional linear transformations
In the last post, we saw that solutions of the nonautonomous Riccati equation are given by fractional linear transformations — that is, if is the map taking the initial condition to the solution at time , then for some functions … Continue reading
Riccati equations and fractional linear transformations
In a typical course on ordinary differential equations, the Picard–Lindelöf theorem on existence and uniqueness of solutions is followed at some point by an example illustrating that such solutions may not be defined for all time, but may go to … Continue reading
The stable manifold theorem
The stable manifold theorem is one of the most important in the theory of nonlinear ODEs and dynamical systems. Unfortunately, some of the standard introductory texts (Hirsch–Smale, Perko) either do not give a proof, or do not motivate the proof, … Continue reading