Computer Algebra Recipes

320 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS
8.1.1 A Rattler Signals Chaos
Humor is emotional chaos remembered in tranquility.
James Thurber, American writer, humorist, and cartoonist (1894{1961)
As an illustrative example of how a Poincare section is produced and how it
changes character as a control parameter is varied, let's consider Du±ng's ODE
describing the force oscillations of a nonlinear spring system,
Äx +2g _x + ®x+ ¯x 3 = F cos(!t): (8.1)
Here x is the displacement, g the damping coe±cient, ® and ¯ real parameters,
F the force amplitude, and ! the driving frequency.
The solutions of Du±ng's equation were examined in some detail in Section
1.2.1 as the control parameter F was varied, all other parameters being
held ¯xed. In this recipe, we shall use exactly the same coe±cient values and
initial conditions as in the earlier treatment, including the same F values. This
will allow us to compare the results of the Poincare treatmentwiththeconclusions
reached previously on the response of the forced Du±ng oscillator.
The number of F values is taken to be N1 = 4, and the maximum number
of multiples of the driving period T0 =2¼=! considered is N2 =250.
> restart: with(plots): N1:=4: N2:=250:
The coe±cient values are entered and the driving period calculated.
> g:=0.25: alpha:=-1: beta:=1: omega:=1: T[0]:=2*Pi/omega;
T0 := 2 ¼
The force amplitudes are F1 =0:325, F2 =0:35, F3 =0:356, and F4 =0:42.
> F[1]:=0.325: F[2]:=0.35: F[3]:=0.356: F[4]:=0.42:
Du±ng's equation can be rewritten as a coupled set of ¯rst-order ODEs,
_x = y; _y = ¡2 gy¡ ®x¡ ¯x 3 + Fi cos(!t);
which will be numerically solved with the initial conditions x(0)=0:09, y(0)=0.
> ic:=x(0)=0.09,y(0)=0:
The coupled ¯rst-order di®erential equations are entered, an operator being
used for the second one, the subscript i of the force amplitude Fi to be given.
> de1:=diff(x(t),t)=y(t):
> de2:=i->diff(y(t),t)=-2*g*y(t)-alpha*x(t)-beta*x(t)^3
+F[i]*cos(omega*t):
An operator sol is introduced to numerically solve de1 and de2 (i), subject to
the initial conditions, for a given value of i. The option maxfun=0 overrules any
limit on the maximum number of function evaluations in Maple's numerical
algorithm. The output is given as a listprocedure.
> sol:=i->dsolve(fde1,de2(i),icg,fx(t),y(t)g,type=numeric,
maxfun=0,output=listprocedure):
Operators X and Y are formed to evaluate x(t) andy(t)
> X:=i->eval(x(t),sol(i)): Y:=i->eval(y(t),sol(i)):