frc : A Periodically Forced System.

This demo illustrates the computation of periodic solutions to a periodically forced system. In AUTO this can be done by adding a nonlinear oscillator with the desired periodic forcing as one of the solution components. An example of such an oscillator is

$$\begin{array}{cl} x'&=x + \beta y - x (x^{2} + y^{2}), \\ y'&=-\beta x + y - y (x^{2} + y^{2}), \\ \end{array}$$ (14.5)

which has the asymptotically stable solution $x=sin (\beta t)$, $y=cos (\beta t)$. We couple this oscillator to the Fitzhugh-Nagumo equations :

$$\begin{array}{cl} v'&=\bigl( F(v) - w \bigr) / \epsilon , \\ w'&=v - dw - \bigl( b + r \sin(\beta t) \bigr) , \end{array}$$ (14.6)

by replacing $\sin(\beta t)$ by $x$. Above, $F(v) = v (v-a) (1-v)$ and $a,b,\epsilon$ and $d$ are fixed. The first run is a homotopy from $r=0$, where a solution is known analytically, to $r=0.2$. Part of the solution family with $r=0.2$ and varying $\beta$ is computed in the second run. For detailed results see AlDoOt:90 AlDoOt:90.

Table 14.12: Commands for running demo frc.

AUTO -COMMAND	ACTION
! mkdir frc	create an empty work directory
cd frc	change directory
demo('frc')	copy the demo files to the work directory
ld('frc')	load the problem definition
run(c='frc.1')	homotopy to $r=0.2$
sv('0')	save output-files as b.0, s.0, d.0
run(c='frc.2',s='0')	compute solution family; restart from s.0. Constants changed : IRS, ICP(1), NTST, NMX, DS, DSMAX
sv('frc')	save output-files as b.frc, s.frc, d.frc