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Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg65924] Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica
*From*: "Narasimham" <mathma18 at hotmail.com>
*Date*: Mon, 24 Apr 2006 06:01:15 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
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 --"
Clear[r,th,R];
eq = { r''[th] == r[th]*(1 - 2 r[th]^2), r[0]==1, r'[0]==0};
NDSolve[ eq, r,{th,0,15}] ;
R[th_]=r[th]/.First[%];
Reborn=Plot[%,{th,0,15}];
soln=Plot[Sech[th]+.02,{th,0,15}];
Show[Reborn,soln];
{x,y}={R[th]Cos[th],R[th]Sin[th]};
ParametricPlot[{x,y},{th,0,15},AspectRatio->Automatic];
"-- Tanh --"
Clear[r,th,R];
eq = { r''[th] == 2*(r[th]^3 - r[th]), r[0]==0, r'[0]==1};
NDSolve[ eq, r,{th,0,38}] ;
R[th_]=r[th]/.First[%];
Reborn=Plot[%,{th,0,38}];
soln=Plot[Tanh[th]+.02,{th,0,38}];
Show[Reborn,soln];
{x,y}={R[th]Cos[th],R[th]Sin[th]};
ParametricPlot[{x,y},{th,0,15},AspectRatio->Automatic];
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