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The solution space of a nonlinear ordinary differential equation (ODE) often includes a separatrix that is unstable in the sense that nearby solutions depart exponentially from it. The nonlinear Bessel's equation, for instance,

f''[x] + f'[x]/x +f[x](1 – f[x]^2) == 0

has this behavior, with a separatrix satisfying f[0] = 0, f[∞] = 1. What algorithms work well for computing an unstable separatrix from zero to a large value of the independent variable in a nonlinear ODE?

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There are a few and they most typically rely on setting up a boundary value problem. You can find solutions to these in the AUTO continuation package, and the XPP package that provides a (partial) front end to it. However, you'll still have to understand how to set it up because it's not an automated process. There are tutorials for those packages that include that application, I believe.

Alternatively, my own package, PyDSTool, is meant for dynamical systems analysis and comes with a prototype (and somewhat naive) tool to compute separatrices -- in fact, any geometrically simple sub-manifold around a fixed point. It is described in graphic detail here. It uses the shooting method. A similar computation was achieved using my package's PyCont sub-package (uses AUTO internally), which is described here.

The shooting method will, however, struggle more if you need to cover a very large domain of the independent variable unless its geometry remains relatively "simple". E.g. it doesn't get tangled or squeezed close to other sub-manifolds. It should be a good starting point for further analysis, though.

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  • $\begingroup$ This is helpful and I shall study it in a few days. Thanks. I don't suppose there is a Mathematica version. $\endgroup$ – bbgodfrey Nov 4 '15 at 2:04

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