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];