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 <
b or I <
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 =
LC; where p is a constant, the result is
dt2 + (3bp2x2 a)
+ x = 0
With p =
aI(3b) and = a
C=L this gives the standard form
x00 + (x2 1)x0 + x = 0
of van der Pol's equation.
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
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.
This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.
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....