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,


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


In view of the KBM, we assume the ansatz form


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



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



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.