The Van Der Pol differential equation
is solved using perturbation with first order approximation. Two different solutions are obtained. The first solution restricted the initial conditions to be which resulted in the forcing function that caused resonance to be eliminated. This gave a stable solution but with initial conditions restricted to be near the origin of the phase plane space. The second solution allowed arbitrary initial conditions any where in the phase plane but the the resulting forcing function that caused resonance resulting in a solution which became unstable after some time. Phase plane plots are used to compare the two solutions.
The Van Der Pol equation is
And assuming initial conditions are and .
If , then the equation becomes which has an exact solution. We perturb the exact solution for to approximate the solution of the nonlinear equation in terms of perturbation parameter Let the solution of the Van Der Pol equation be and the solution to the exact equation be , then
First order approximation results in
Now we need to determine and . Substituting (2) into (1) gives
Collecting terms with same power of gives
Setting terms which multiply by higher power to zero and since it is assumed that is very small therefore
For the LHS to be zero implies that
Equation (3) is solved for and (4) is solved for in order to find the solution . Solution to (3) is
Assuming initial conditions for are zero, then initial conditions for can be taken to be those given for . Solving for gives
Substituting the solution just found (and its derivative) into (4) gives
Using and the above can be simplified to
Using the above can be simplified further to
The above is the equation we will now solve for , which has four forcing functions, hence four particular solutions. two of these particular solutions will cause a complete solution blows up as time increases (the first two in the RHS shown above). If we however restrict the initial conditions such that
Then those forcing function will vanish. The above two equation result in the same restriction, which can be found to be
By restricting to be exactly , then the solution we obtain for from (5) will not blow up. Hence (4) can now be rewritten without the two forcing functions which caused resonance resulting in
But and , hence the above becomes after some simplification
The homogeneous solution to the above is
And we guess two particular solutions and . By substituting each of these particular solutions into (6) one at a time, and comparing coefficients, we can determine . This results in
Hence the solution to (6) becomes
Now applying initial conditions, which are and , to (7) and determining and results in
Therefore we now have the final perturbation solution, which is
Where and and the above solution is valid under the restriction that
We notice that the above restriction implies that both and have to be less than to avoid getting a quantity which is complex. This implies that his solution is valid near the center of the phase plane only. In addition, it is valid only for small . To illustrate this solution, We show the phase plot which results from the above solution, and also plot the solution . Selecting and
The same initial steps are repeated as in the first solution, up the equation to solve for
There exists four particular solutions relating to the above four forcing functions
For each of the above, we guess a particular solution, and determine the solution parameters. We guess , , , and . By substituting each of the above into (8) one at a time and comparing coefficients, we arrive at the following solutions
Now applying initial conditions which are and and determining and results in
We can now write the final solution for as
To illustrate this solution, we show the phase plot which results from the above solution, and also plot the solution . We select and . We note that the same initial conditions are used as in the earlier solution to compare both solutions.
We see clearly the effect of including resonance particular solutions. The limit cycle boundary is increasing with time.
The effect becomes more clear if we plot the solution itself, and compare it to the solution earlier with no resonance. Below, We show from both solutions. We clearly see the effect of including the resonance terms. This is the solution with no resonance terms
This is the solution with resonance terms
In the second solution (with resonance), there is no restriction on initial conditions. Hence we can for example look at the solution starting from any and . This is the phase plot for and . This would not be possible using the first solution, due to the restriction on having to be equal to
Van Der Pol was solved using perturbation for first order approximation. It was solved using two methods. In the first method, the solution was restricted to not include forcing functions which leads to resonance.
In the second method, no such restriction was made. In both methods, this problem was solved for general initial conditions (i.e. both and can both be nonzero.
In the first method, it was found that the we must restrict the initial conditions to be such that . This is the condition which resulted in the resonance terms vanishing from the solution. This leads to a solution which generated a limit cycle which did not blow up as the solution is run for longer time. In other words, once the solution enters a limit cycle, it remains in the limit cycle.
In the second method, no restriction on the initial conditions was made. This allowed the solution to start from any state. However, the limit cycle would grow with time, and the solution will suffer from fluttering due to the presence of the resonance terms. However, even though the solution was not stable in the long term, this second approach allowed one to examine the solution for a shorter time but with the flexibility of choosing any initial conditions.
It was also observed that increasing the value of the perturbation parameter gradually resulted in an inaccurate solution as would be expected, as the solutions derived here all assumed a very small1 value of was used in generating the solutions and plots.
For future research, it would be interesting to consider the effect of higher order approximation on the solutions and compare with accurate numerical solutions.