# Algorithm to Compute Separatrix of Nonlinear ODE

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?