Abstract
In this paper we show the way to apply some and analytical and numerical techniques in order to solve the forced Van der Pol oscillator. We illustrate the obtained results with examples. A comparison with Runge–Kutta numerical method is made in order to see the accuracy of the approximated analytical solution.
1. Introduction
We consider the following forced Van der Pol oscillator:In the case when and we obtain the well known Van der Pol oscillator [1].
Oscillator (2) may be solved using perturbation methods as described.
2. The Lindstedt–Poincaré Method
Let us consider the i.v.p.
The Lindstedt–Poincaré method assumes the solution in the ansatz formwhere
Using this method gives the following solution up to :
This solution is periodic with period .
3. He’s Homotopy Perturbation Method
The homotopy perturbation method has been shown to solve a large class of nonlinear differential problems effectively, easily, and accurately; generally one iteration is enough for engineering applications with acceptable accuracy, making the method accessible to nonmathematicians. However, in this method secularity terms appears. We perform iterations until such terms appear. We construct the following homotopy [2].where
After some algebra one finds that
The constants and are determined from nthe initial conditions.
4. The Krylov–Bogoliubov–Mitropolsky Method (KBM)
The Krylov–Bogoliubov–Mitropolsky method (KBM) [2], [3] is a technique to give approximate analytical solution to the weakly nonlinear second-order equation
When the solution of (10) may be expressed aswhere and are constants. For the case when is small, Krylov and Bogoliubov (1947) assumed that the solution is still given by (11) but with time-varying and , and subject to the condition
In the general case, the solution is assumed in the ansatz formwhere each is a periodic function of with a period , and and are assumed to vary with time according to
In order to uniquely determine and , we require that no contains . Let . Then,
Here,
Following is the solution obtained using KBM to accuracy [2]:where and obey the oddsso that
The constants and are determined from the initial conditions and . For an illustration, see Figure 1 and Figure 2.
4.1. A More Accurate Analytical Solution
Let
In view of the KBM, we assume the ansatz form
Hence,
Making use of (15) and (16), we will havewhere
Equating to zero the coefficients of and in (25) gives
From the condition , we get the second order linear ode
The general solution to the ode (29) reads
Define
Then,
Replacing the expressions (31) and (32) into (26), we obtain
Equating to zero the coefficients of and in (33), we get
From the condition we obtain the following linear second order ode
The general solution to the ode (35) is given by
Define
Then,
Finally, we must have . Proceeding in a similar manner as before, we get
We thus have that
Also, from (23),
Ode (41) is hard to solve in closed form. We use the approximation
Solving the odeone gets
The expression for is obtained from the ode :
Example 1. LetThe i.v.p to be solved readsFrom (43),We haveSee Figure 3.
5. Analytical Solution to a Forced Van der Pol Oscillator
Let us consider the i v p.given that
We will assume the ansatz
Here, the function is the solution to the i v p
We now will determine suitable values for the constants and . We have
We choose the values of and so that
From (55), it follows that
We choose the least in magnitude real root to (56) and the least in magnitude real root to (57).
Example 2. Let , , , , and . Consider the i.v.p.Our calculations giveThe [4] expression for is [5] obtained from (40). In Figure 4, we compare[6] the approximate [7] analytical solution (dashed curve) with the Runge–Kutta numerical solution. The error equals 0.0756086.
Data Availability
No data were used to support this paper
Conflicts of Interest
The authors declare that they have no conflicts of interest.