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(a) Chau’s electric circuit is a simple electronic circuit thatcan exhibit chaotic behaviour. The voltages x(t) and y(t), andcurrent z(t), across components in the circuit can be investigatedusing the Matlab command

[t,xyz] = ode45(@ChuaFunction,[-10 100],[0.7 0.2 0.3]);

and the function:

1 function dxyzdt = ChuaFunction(~,xyz)

2 % xyz(1) = X, xyz(2) = Y, xyz(3) = Z

3 4 dxdt = 15.6*(xyz(2) - xyz(1) + 2*tanh(xyz(1)));

5 dydt = xyz(1) - xyz(2) + xyz(3);

6 dzdt = -28*xyz(2); 7

8 dxyzdt = [dxdt dydt dzdt]’;

9 end

(i) What is the differential equation involving xË™(t)? Here adot represents differentiation with respect to time t.

(ii) What is the initial condition for the variable y(t)?

(iii) What does the apostrophe after the square brackets in line8 of the function signify and why is the apostrophe neededhere?

(b) For a given function u(t), explain how the derivative ofu(t) with respect to t can be approximated on a uniform grid withgrid spacing ∆t, using the one-sided forward differenceapproximation

du/dt ≈ Ui+1 − Ui/ ∆t ,

where ui = u(ti). You should include a suitable diagramexplaining your answer

(c) Using the one-sided forward difference approximation frompart (b) and Euler’s method, calculate the approximate solution tothe initial value problem

du/dt + t cos(u) = 0, subject to u(0) = −0.2,

at t = 0.4, on a uniform grid with spacing ∆t = 0.1.