chu : A Non-Smooth System (Chua's Circuit).

Chua's circuit is one of the simplest electronic devices to exhibit complex behavior. For related calculations see KhRoCh:93 KhRoCh:93. The equations modeling the circuit are

$$\begin{array}{cl} u_1' &= \alpha \bigl[~ u_2 - h(u_1) ~\bigr]... ...u_2' &= u_1 - u_2 + u_3~, \\ u_3' &= - \beta~ u_2~, \end{array}$$ (14.13)

where

$$h(x) = a_1 x + \frac{1}{2}~ (a_0 - a_1) ~ \bigl\{ \mid x+1\mid - \mid x-1\mid \bigr\}~,$$

and where we take $\beta = 14.3$, $a_0 = - 1/7$, $a_1 = 2/7$.

Note that $h(x)$ is not a smooth function, and hence the solution to the equations may have non-smooth derivatives. However, for the orthogonal collocation method to attain its optimal accuracy, it is necessary that the solution be sufficiently smooth. Moreover, the adaptive mesh selection strategy will fail if the solution or one of its lower order derivatives has discontinuities. For these reasons we use the smooth approximation

$$\mid x\mid ~\approx~ \frac{2 x}{\pi } ~ {\rm arctan}(Kx),$$

which get better as $K$ increases. In the numerical calculations below we use $K = 10$. The free parameter is $\alpha$.

Table 14.20: Commands for running demo chu.

AUTO -COMMAND	ACTION
! mkdir chu	create an empty work directory
cd chu	change directory
demo('chu')	copy the demo files to the work directory
ld('chu')	load the problem definition
run(c='chu.1')	1st run; stationary solutions
sv('chu')	save output-files as b.chu, s.chu, d.chu
run(c='chu.2',s='chu')	2nd run; periodic solutions, with detection of period-doubling. constants changed : IPS, IRS, ICP, ICP
ap('chu')	append the output-files to b.chu, s.chu, d.chu