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Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica

Solutions of non-linear second order differential equations formed for
r = Sech[th],Tanh[th] after two differentiations  are repeated  at
about th=8.0 which value should actually be Infinity. After this value
they do not tend to the expected asymptotic value but become  periodic
functions !.

The NDSolve Runge-Kutta numerical algorithm appears to involve a small
positive error at large argument values that allows build-up of
function back to its full value as a reflected periodic function.This
is seen more clearly in a polar plot of r = Sech[th] and Tanh[th] where
periodic lobes reappear, simulating  periodic functions ( 0.02 added in
plots just to distinguish the curves).

BTW, I would be much enthused if one can identify the ( Jacobi Elliptic
like ??) functions labeled as Reborn, that result by this Mathematica
command/program.  Or please indicate whether there is a command to
convert asymptotic functions with introduced second order errors to
their periodic counterparts where  the time period can be numerically
defined /specified.  It appears as an unacceptable error for Sech or
Tanh because the hyperbolic monotonic nature of the non-linear second
order differential equation gets  numerically depicted  or compromised
as an elliptic periodic one.

Regards,  Narasimham

"-- Sech --"
eq = { r''[th] == r[th]*(1 - 2 r[th]^2), r[0]==1, r'[0]==0};
NDSolve[ eq, r,{th,0,15}] ;
"-- Tanh --"
eq = { r''[th] == 2*(r[th]^3 - r[th]), r[0]==0, r'[0]==1};
NDSolve[ eq, r,{th,0,38}] ;

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