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In the 1920s E.V. Appleton and B. van der Pol conducted experimental research on the oscillation of the electric current and voltage of early radio sets. The circuits they consid- ered were LRC loops, but with the resistors satisfying Ohm's law replaced by a vacuum tube. (Nowadays a semiconductor would be used instead of a vacuum tube.) The dif- ference between a semiconductor and a resistor is this: a resistor dissipates energy at all levels, but a semiconductor pumps energy into a circuit at low levels and absorbs energy at high levels. Here is a typical van der Pol circuit. Suppose that a power supply is attached to this circuit in order to energize it, and then the power supply is removed. Let assume that the voltage drop across the semiconductor is given by the nonlinear function Vs = bI3 􀀀 aI = f(I) of the current I, where b and a are positive constants. The voltage drop across a usual (passive) resistor is given by a linear function, VR = IR, and always has the same sign as I. In contrast, the nonlinear function Vs is negative when 0 < I < pa b or I < 􀀀 pa b . If we substitute f(I) for RI in the familiar RLC-equation LI00 +RI0 +I=C = 0 we obtain 2-nd order equation LI00 + f0(I)I0 + I=C = 0 For f(I) = bI3 􀀀 aI equation (1) becomes LI00 + (3bI2 􀀀 a)I0 + I=C = 0 If we denote  the time variable in equation (1) and make the substitutions I = px; t = = p LC; where p is a constant, the result is d2x dt2 + (3bp2x2 􀀀 a) r C L dx dt + x = 0 With p = p aI(3b) and  = a p C=L this gives the standard form x00 + (x2 􀀀 1)x0 + x = 0 of van der Pol's equation. 1 Assignment: 1. Rewrite the van der Pol's equation as a system of two 1-st order equations. 2. For the system from question 1 write the Euler's method for solving the system numerically. 3. Convince yourself in the fact that for any   0 the solution of the van der Pol's equation with the initial conditions x(0) = 2, x'(0)=0 is periodic. For this translate the IVP into the corresponding IVP for the system from 1 and solve it by Euler's method with the step-size h=0.05 for  = 0; 1:5; and 3. Sketch the corresponding trajectories on the phase-plane. Supply the table of the rst 20 values of xt)and x'(¢) of your numerical solutions in each case  = 0; 1:5 and 3. Estimate the period of the periodic solutions for z = 0; 1:5 and 3. 4. For  = 1:5 and the initial conditions x(0) = 0:a and x(0) = 0:b, where a and b are before the last and the last digits of your student ID (replace zeros by 9), respectively, use the Euler's method to convince yourself that the trajectory is attracted to the closed orbit from question 3 for  = 1:5 from inside. Supply the table of the rst 20 values of x(t) and x0(t) of your numerical solution. Sketch on the same phase-plane both trajectories: the trajectory you obtained with the initial condition x(0) = 0:a and x0(0) = 0:b and the closed trajectory with the initial conditions x(0) = 2; x0(0) = 0. 5. The same as question 4 with the initial conditions x(0) = 3:a and x(0) = 1:b. 2

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Part 1

The Vander Pol equation written as a system:

Part 2: Euler Method

Let h be given.
Given initial conditons x(0) and y(0)=x'(0) determine recursively for k >= 0
y(k+1)=y(k)+h*(-x(k) - mu(x(k)^2-1)y(k))
Here x(k) is shorthand for x(k*h), y(k) is shorthand for y(k*h)

Part 3: mu = 0

We choose h=0.05 in the remainder of this assignment.
The initial conditions are x(0)=2, x’(0)=y(0)=0;
In this case the solution is x(t)=2cos(t) which is periodic of period 2 π.
All solutions are periodic, but the Euler method does not produce periodic solutions in this case: solutions produced by this method spiral out....
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