Re: Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg66003] Re: Periodic Rebirth of Hyperbolic Functions by ODE in Mathematica*From*: Maxim <m.r at inbox.ru>*Date*: Wed, 26 Apr 2006 04:38:17 -0400 (EDT)*References*: <e2i7qg$8tg$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On Mon, 24 Apr 2006 10:04:00 +0000 (UTC), Narasimham <mathma18 at hotmail.com> wrote: > 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]; > One way to obtain a more accurate solution is to figure out an invariant of the system: In[1]:= inv = Integrate[(r''[th] - r[th]*(1 - 2*r[th]^2))*r'[th], th] Out[1]= (-(1/2))*r[th]^2 + r[th]^4/2 + (1/2)*Derivative[1][r][th]^2 Hence this quantity should remain constant. Then sol = First@ NDSolve[ {r''[th] - r[th]*(1 - 2*r[th]^2) == 0, r[0] == 1, r'[0] == 0}, r, {th, 0, 100}, Method -> {Projection, Invariants -> inv}, MaxStepFraction -> .001] Plot[r[th] - Sech[th] /. sol, {th, 0, 100}, PlotRange -> All] shows that the error remains bounded (in fact, the accuracy even increases linearly). We could also get the exact solution from the system {inv == 0, r[0] == 1} but in general that may be problematic, as the solution to that system is not unique, there is another solution, namely (r -> (1&)). This method doesn't work very well for your second system of ODEs. In that case we can use (1 - r[th]^2)/r'[th] as an invariant but I don't know how to obtain this form without guessing. Maxim Rytin m.r at inbox.ru