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Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method
Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method Turgut Öziş ∗ , Ahmet Yıldırım Department of Mathematics, Faculty of Science, Ege University, Campus, 35100, Bornova-İzmir, Turkey
article
info
Article history: Received 30 December 2007 Accepted 5 March 2008 Keywords: Nonlinear systems Forcing nonlinear oscillators van der Pol’s equation
a b s t r a c t In this paper, the iteration perturbation method proposed by He [J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation. de-Verlag im Internet GmbH, 2006; J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26 (2005) 827–833] is used to generate periodic solutions of van der Pol oscillator with a forcing term, forcing oscillator with quadratic type damping and van der Pol oscillator with excitation term. The comparison of the obtained results verifies its convenience and effectiveness. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we will consider the second-order differential equation x¨ + x + εf (x, x˙ , t) = 0
(1)
where ε need not be small, and f (x, x˙ , t) is nonlinear analytic function of the displacement x, the velocity x˙ , and the time t. Equations of this kind arise directly in various engineering applications. One of the most studied equations within this class is the forcing van der Pol’s equation. In the present work, we study how to obtain the amplitude and period for a system described by Eq. (1) where f will be defined later. There are many approaches for approximating solutions to nonlinear oscillatory systems. The most widely studied approximation methods are the perturbation methods [1]. In the past few decades, new perturbation methods and nonperturbative methods are proposed. A review of the recently developed analytical methods is given in review article and the comprehensive book by He [2,3]. There also exist considerable number of works dealing with the problem of approximate determination of limit cycles for strongly nonlinear oscillators by using different methodologies. For example, the energy method (or variational method) [4,5], homotopy perturbation method [6–12], The variational iteration method [13–15], various Lindstedt–Poincaré methods [16–19], variational methods [20,21], parameter-expanding method [22,23], linearized harmonic balance method [24,25], various tan h methods [26], Adomian Padé approximation [27] and more [28]. 2. The iteration perturbation method The mathematical models of many physical and engineering problems can be expressed in the nonlinear system in the form: x˙ = g1 (x, y, t, x˙ , y˙ )
(2)
y˙ = g2 (x, y, t, x˙ , y˙ )
(3)
∗ Corresponding author. Tel.: +90 0 232 388 18 93. E-mail address: [email protected] (T. Öziş). 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.03.005
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1985
For example, if we introduce the variable y = dx/dt, then Eq. (1) can be replaced by equivalent system x˙ = y
(4)
y˙ = −x − εf (x, y, t).
(5)
These nonlinear systems may be solved, with the combination of certain perturbation techniques, by “the method of elimination” similar to the method of elimination for linear differential systems. As it is well known, the objective of the method of elimination for linear differential systems is to eliminate dependent variable in succession until there remains only a single equation containing only one dependent variable. After the solution of remaining equation has been found, the other dependent variables can be found in turn, using the original differential equation or those that have appeared in the elimination process. Particularly, if we choose appropriate trial function x0 , one can easily determine the y0 by using Eq. (4) and if we substitute x0 , y0 , for example, into the right-hand side of the Eq. (5), we can determine next iterate y1 and after x1 by the aid of Eq. (4), and so on. The convergence of the iteration is set up by applying appropriate perturbation techniques at each iteration steps. The iteration perturbation method was first proposed by He [29] as a new technique coupling iteration method and perturbation methods. Later, He [2,3,30] expands this approach to solve various nonlinear equations in the form: x¨ + x + εf (x, x˙ , x¨) to search for the amplitude and the periodic solutions. In this paper, we extend and applied this approach for van der Pol’s equations with a forcing term, forcing oscillator with quadratic type damping and van der Pol oscillator with excitation term. 2.1 Example 1. Consider a forcing oscillator with quadratic type damping. This equation can be written as x¨ + x + ε 1 − x˙ 2 x˙ − ε sin t = 0.
(6)
Eq. (6) is equivalent to the two-dimensional nonautonomous system (7)
x˙ = y
y˙ = −x − ε 1 − y2 y + ε sin t.
(8)
If we choose x = A cos wt, then from (7) we have x˙ = −Aw sin wt = y.
(9)
Substituting x = A cos wt and y = −Aw sin wt into the right-hand side of Eq. (8), we have 2 y˙ = −A cos wt − ε 1 − A2 w2 sin wt (−Aw sin wt) + ε sin t
(10)
or εA 3 w 3 3 sin 3wt + ε sin t. y˙ = −A cos wt + εAw sin wt 1 − A2 w2 +
4
4
(11)
Integrating (11) yields y=−
A sin wt w
− εA cos wt 1 −
3 4
A 2 w2
−
εA 3 w 2 12
cos 3wt − ε cos t.
(12)
Comparing (9) and (12), we set
−
A w
= −Aw, and so the frequency results in, w = 1.
We, now, re-write y: 3 εA3 cos 3t y = −A sin t − εA cos t 1 − A2 − − ε cos t,
4
12
(13)
or y = −A sin t + ε cos t −A +
3A 3 4
!
−1 −
εA3 cos 3t 12
.
(14)
No secular term requires that,
−A+
3A3 4
− 1 = 0.
(15)
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1986
Then we obtain the amplitude, A = 1.492258.
(16)
Thus, Eq. (14) reduces to y = −A sin t −
εA3 cos 3t 12
.
(17)
Integrating (17), we obtain; x = A cos t −
εA3 sin 3t 36
,
(18)
which is the first-order approximate periodic solution of Eq. (5). Thus, in the first approximation, we obtain that the amplitude equal to A = 1.492258, and the frequency is 1. In order to improve its accuracy we can choose the trail functions in the form x = a cos wt + b sin wt + c cos 3wt + d sin 3wt
(19)
y = −aw sin wt + bw cos wt − 3cw sin 3wt + 3dw cos 3wt
(20)
as in Ref. [30]. Proceeding in the usual way, we can determine a, b, c, d and w, but it is laborious to do by hand. But, they can be implemented on computer with the help of symbolic computations. The first-order periodic solution (18), for ε = 0.1, takes the form; x = 1.492258 cos t − 0.009230585 sin 3t.
(21)
Abd El-Latif [31], by applying the method of time transformation, calculated the second-order explicit expansion of the periodic solution of Eq. (5) for ε = 0.1, as: x = 1.482714 cos t + 0.031153 sin t − 0.10384 sin 3t + 0.001157 cos 3t − 0.000246 cos 5t.
The second-order perturbation method solution gives; x = 1.482981 cos t + 0.032995 sin t − 0.010384 sin 3t + 0.0011466 cos 3t − 0.000217 cos 5t.
Thus, our first order approximation solution, Eq. (21), is compatible with that obtained by Abd El-Latif [31] and with that obtained by the perturbation method. Example 2. In this example, we consider the forcing van der Pol oscillator with equation, x¨ + x − ε 1 − x2 x˙ − ε sin t = 0.
(22)
Eq. (22) is equivalent to the two-dimensional nonautonomous system; (23)
x˙ = y
y˙ = −x + ε 1 − x2 y + ε sin t.
(24)
If we choose x = A cos wt, then from (23) we have x˙ = −Aw sin wt = y.
(25)
Substituting x = A cos wt and y = −Aw sin wt into the right-hand side of Eq. (24), we have y˙ = −A cos wt + ε 1 − A2 cos2 wt (−Aw sin wt) + ε sin t
(26)
or y˙ = −A cos wt + εAw sin wt −1 +
A2
!
4
+
εA 3 w 4
sin 3wt + ε sin t.
(27)
Integrating (27) yields y=−
A sin wt w
+ εA cos wt 1 −
A2
4
!
−
εA 3 12
cos 3wt − ε cos t.
Comparing (28) with (25) results in
−
A w
= −Aw, so we obtain the frequency, w = 1.
(28)
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1987
We, now, re-write y: y = −A sin t + εA cos t 1 −
A2
εA3 cos 3t
!
4
−
12
− ε cos t,
(29)
or y = −A sin t + ε cos t A −
!
A3
−1 −
4
εA3 cos 3t 12
.
(30)
No secular term requires, A−
A3
4
− 1 = 0.
Then we obtain the amplitude A = −2.382976.
(31)
Thus, Eq. (30) reduces to y = −A sin t −
εA3 cos 3t 12
.
(32)
Integrating (32), we obtain; x = A cos t −
εA3 sin 3t
(33)
36
which is the first-order approximate periodic solution of Eq. (22). Thus in the first approximation, we obtain that the amplitude equal to A = −2.382976 and the frequency is 1. The first-order periodic solution (33), for ε = 0.1, takes the form; x = −2.382976 cos t + 0.03758863 sin 3t.
(34)
Abd El-Latif [31], by applying the method of time transformation, calculated the second-order explicit expansion of the periodic solution of Eq. (22) for ε = 0.1, as: x = −2.378146 cos t − 0.126863 sin t + 0.042287 sin 3t − 0.002836 cos 3t.
Thus, our first-order approximation solution is compatible with that obtained by Abd El-Latif [31]. Example 3. Our third example is van der Pol oscillator with excitation term given by the equation: x¨ + εx˙ x2 − 1 + x cos t + x = 0.
(35)
We, again, write nonlinear two-dimensional nonautonomous system as: (36)
x˙ = y
y˙ = −εy x − 1 + x cos t − x. 2
(37)
If we, again, begin with x = A cos wt, then from (36) we have x˙ = −Aw sin wt = y.
(38)
Substituting x = A cos wt and y = −Aw sin wt into the right-hand side of Eq. (37) we have y˙ = εAw sin wt A2 cos2 wt − 1 + A cos wt cos t − A cos wt
(39)
or y˙ = −A cos wt + εAw sin wt
A2
4
!
−1 +
εA 3 w 4
sin 3wt +
εA 2 w 4
sin (2w + 1) t +
εA 2 w 4
sin (2w − 1) t.
(40)
Integrating (40) yields: y=−
A sin wt w
− εA cos wt
A2
4
!
−1 −
εA3 cos 3wt 12
−
εA2 w cos (2w + 1) t εA2 w cos (2w − 1) t − . 4 (2w + 1) 4 (2w − 1)
(41)
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1988
Comparing (41) with (38) results in
−
A w
= −Aw, so we obtain the frequency, w = 1.
We, now, re-write y: y = −A sin t − εA cos t
A2
4
!
−1 −
εA3 cos 3t 12
−
εA2 cos 3t 12
−
εA2 cos t 4
,
(42)
or y = −A sin t − εA cos t
A2
4
+
A
4
!
−1 −
εA2 cos 3t 12
(1 + A ) .
(43)
No secular term requires that, A2
4
+
A
4
− 1 = 0.
Then we obtain the amplitude A = 1.561553.
Thus, Eq. (43) reduces to y = −A sin t −
εA2 cos 3t 12
(1 + A).
(44)
Integrating (44) we obtain x = A cos t −
εA2 (1 + A) sin 3t 36
(45)
which is the first-order approximate periodic solution of Eq. (35). Thus in the first approximation we obtain that the amplitude equal to A = 1.561553 and the frequency is 1. The first-order periodic solution (45), for ε = 0.1, takes the form; x = 1.561553 cos t − 0.017350592 sin 3t.
(46)
Abd El-Latif [31], by applying the method of time transformation, calculated the second-order explicit expansion of the periodic solution of Eq. (35), for ε = 0.1, as: x = 1.571772 cos t + 0.119542 sin t − 0.002124 sin 3t + 0.001059 cos 3t − 0.000312 cos 5t.
The result obtained by second-order perturbation method is; x = 0.567314 cos t + 0.12176 sin t − 0.001449 sin 3t + 0.001125 cos 3t − 0.000375 cos 5t.
Thus, our first-order approximation solution is compatible with that obtained by Abd El-Latif [31] and the perturbation solution. 3. Conclusion In this paper, we applied the iteration perturbation method to variety of forcing van der Pol’s equations. We conclude from the results obtained that Iteration Perturbation method is extremely simple in its principle, easy to apply, and gives a very good accuracy even with the first-order approximation and simplest trial functions. Comparison made with other known results show that the method provides a powerful mathematical tool to the determination of limit cycles of more complex nonlinear systems. References [1] [2] [3] [4] [5]
A.H. Nayfeh, Problems in Perturbations, Wiley, New York, 1985. J.H. He, Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern Phys. B 10 (2006) 1141–1199. J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation. de-Verlag im Internet GmbH, 2006. J.H. He, Determination of limit cycles for strongly nonlinear oscillators, Phys. Rev. Lett 90 (17) (2003) 174301. T. Öziş, A. Yıldırım, Determination of frequency–amplitude relation for duffing- harmonic oscillator by the energy balance method, Chaos Solitons Fractals 54 (7–8) (2007) 1184–1187. [6] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Eng 178 (1999) 257–262. [7] J.H. He, Homotopy perturbation method: A new nonlinear analytical technique, Appl. Math. Comput. 135 (2003) 73–79. [8] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Non-linear Sci. Numer. Simul. 6 (2) (2005) 207–208.
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1989
[9] T. Öziş, A. Yıldırım, A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity, Chaos Solitons Fractals 34 (3) (2007) 989–991. [10] T. Öziş, A. Yıldırım, A comparative study of he’s homotopy perturbation method for determining frequency–amplitude relation of a nonlinear oscillator with discontinuities, Int. J. Non-linear Sci. Numer. Simul. 8 (2) (2007) 243–248. [11] A. Yıldırım, T. Öziş, Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method, Phys. Lett. A 369 (1–2) (2007) 70–76. [12] A. Belendez, A. Hernandez, T. Belendez, E. Fernandez, M.L. Alvarez, C. Neipp, Application of He’s homotopy perturbation method to the duffingharmonic oscillator, Int. J. Non-linear Sci. Numer. Simul. 8 (1) (2007) 79–88. [13] J.H. He, Variational iteration method: a kind of nonlinear analytical technique: Some examples, Internat. J. Non-Linear Mech. 34 (4) (1999) 699–708. [14] V. Marinca, An approximate solution for 1-D weakly nonlinear oscillations, Int. J. Non-linear Sci. Numer. Simul. 3 (2002) 107–120. [15] T. Öziş, A. Yıldırım, A study of nonlinear oscillators with u1/3 force by He’s variational iteration method, J. Sound Vibration 306 (1–2) (2007) 372–376. [16] J.H. He, Modified Lindstedt–Poincare methods for some strongly nonlinear oscillations. Part I: Expansion of a constant, Int. J. Non-Linear Mech. 37 (2) (2002) 309–314. [17] J.H. He, Modified Lindstedt–Poincaré methods for some strongly nonlinear oscillations:Part II: A new transformation, Int. J. Non-Linear Mech. 37 (2) (2002) 315–320. [18] T. Öziş, A. Yıldırım, Determination of periodic solution for a u1/3 force by He’s modified Lindstedt–Poincaré method, J. Sound Vibration 301 (1–2) (2007) 415–419. [19] T. Öziş, A. Yıldırım, Determination of limit cycles by modified straightforward expansion for nonlinear oscillators, Chaos Solitons Fractals 32 (2) (2007) 445–448. [20] H.M. Liu, Variational approach to nonlinear electrochemical system, Int. J. Non-linear Sci. Numer. Simul. 5 (1) (2004) 95–96. [21] J.H. He, Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fractals 19 (4) (2004) 847–851. [22] L. Xu, He’s parameter-expanding method for strongly nonlinear oscillators, J. Comput. Appl. Math. 207 (1) (2007) 148–154. [23] D.H. Shou, J.H. He, Application of parameter-expanding method to strongly nonlinear oscillators, Int. J. Non-linear Sci. Numer. Simul 8 (1) (2007) 121–124. [24] A. Belendez, A. Hernandez, T. Belendez, A. Marquez, C. Neipp, An improved ‘heuristic’ approximation for the period of a nonlinear pendulum: Linear analysis of a classical nonlinear problem, J. Non-linear Sci. Numer. Simul 8 (3) (2007) 329–334. [25] A. Belendez, D.I. Mendez, T. Belendez, A. Hernandez, M.L. Alvarez, Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable, J. Sound Vibration (2008), doi:10.1016/j.jsv.2008.01.021. [26] H.A. Abdusalam, On an improved complex tan h-function method, Int. J. Non-linear Sci. Numer. Simul. 6 (2) (2005) 99–106. [27] T.A. Abbasy, M.A. El-Tawil, H.K. Saleh, The solution of KdV and mKdV equations using Adomian Pade approximation, Int. J. Non-linear Sci. Numer. Simul. 5 (4) (2004) 327–340. [28] L. Geng, X.C. Cai, He’s frequency formulation for nonlinear oscillators, European J. Phys. 28 (5) (2007) 923–931. [29] J.H. He, Iteration perturbation method for strongly nonlinear oscillators, J. Vibration Control 7 (5) (2001) 631–642. [30] J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26 (2005) 827–833. [31] G.M. Abd El-Latif, Parametric excitation for forcing van der Pol oscillator, Appl. Math. Comput. 147 (2004) 255–265.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Generating the periodic solutions for forcing van der Pol oscillators by the Iteration Perturbation method Turgut Öziş ∗ , Ahmet Yıldırım Department of Mathematics, Faculty of Science, Ege University, Campus, 35100, Bornova-İzmir, Turkey
article
info
Article history: Received 30 December 2007 Accepted 5 March 2008 Keywords: Nonlinear systems Forcing nonlinear oscillators van der Pol’s equation
a b s t r a c t In this paper, the iteration perturbation method proposed by He [J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation. de-Verlag im Internet GmbH, 2006; J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos Solitons Fractals 26 (2005) 827–833] is used to generate periodic solutions of van der Pol oscillator with a forcing term, forcing oscillator with quadratic type damping and van der Pol oscillator with excitation term. The comparison of the obtained results verifies its convenience and effectiveness. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction In this paper, we will consider the second-order differential equation x¨ + x + εf (x, x˙ , t) = 0
(1)
where ε need not be small, and f (x, x˙ , t) is nonlinear analytic function of the displacement x, the velocity x˙ , and the time t. Equations of this kind arise directly in various engineering applications. One of the most studied equations within this class is the forcing van der Pol’s equation. In the present work, we study how to obtain the amplitude and period for a system described by Eq. (1) where f will be defined later. There are many approaches for approximating solutions to nonlinear oscillatory systems. The most widely studied approximation methods are the perturbation methods [1]. In the past few decades, new perturbation methods and nonperturbative methods are proposed. A review of the recently developed analytical methods is given in review article and the comprehensive book by He [2,3]. There also exist considerable number of works dealing with the problem of approximate determination of limit cycles for strongly nonlinear oscillators by using different methodologies. For example, the energy method (or variational method) [4,5], homotopy perturbation method [6–12], The variational iteration method [13–15], various Lindstedt–Poincaré methods [16–19], variational methods [20,21], parameter-expanding method [22,23], linearized harmonic balance method [24,25], various tan h methods [26], Adomian Padé approximation [27] and more [28]. 2. The iteration perturbation method The mathematical models of many physical and engineering problems can be expressed in the nonlinear system in the form: x˙ = g1 (x, y, t, x˙ , y˙ )
(2)
y˙ = g2 (x, y, t, x˙ , y˙ )
(3)
∗ Corresponding author. Tel.: +90 0 232 388 18 93. E-mail address: [email protected] (T. Öziş). 1468-1218/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2008.03.005
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1985
For example, if we introduce the variable y = dx/dt, then Eq. (1) can be replaced by equivalent system x˙ = y
(4)
y˙ = −x − εf (x, y, t).
(5)
These nonlinear systems may be solved, with the combination of certain perturbation techniques, by “the method of elimination” similar to the method of elimination for linear differential systems. As it is well known, the objective of the method of elimination for linear differential systems is to eliminate dependent variable in succession until there remains only a single equation containing only one dependent variable. After the solution of remaining equation has been found, the other dependent variables can be found in turn, using the original differential equation or those that have appeared in the elimination process. Particularly, if we choose appropriate trial function x0 , one can easily determine the y0 by using Eq. (4) and if we substitute x0 , y0 , for example, into the right-hand side of the Eq. (5), we can determine next iterate y1 and after x1 by the aid of Eq. (4), and so on. The convergence of the iteration is set up by applying appropriate perturbation techniques at each iteration steps. The iteration perturbation method was first proposed by He [29] as a new technique coupling iteration method and perturbation methods. Later, He [2,3,30] expands this approach to solve various nonlinear equations in the form: x¨ + x + εf (x, x˙ , x¨) to search for the amplitude and the periodic solutions. In this paper, we extend and applied this approach for van der Pol’s equations with a forcing term, forcing oscillator with quadratic type damping and van der Pol oscillator with excitation term. 2.1 Example 1. Consider a forcing oscillator with quadratic type damping. This equation can be written as x¨ + x + ε 1 − x˙ 2 x˙ − ε sin t = 0.
(6)
Eq. (6) is equivalent to the two-dimensional nonautonomous system (7)
x˙ = y
y˙ = −x − ε 1 − y2 y + ε sin t.
(8)
If we choose x = A cos wt, then from (7) we have x˙ = −Aw sin wt = y.
(9)
Substituting x = A cos wt and y = −Aw sin wt into the right-hand side of Eq. (8), we have 2 y˙ = −A cos wt − ε 1 − A2 w2 sin wt (−Aw sin wt) + ε sin t
(10)
or εA 3 w 3 3 sin 3wt + ε sin t. y˙ = −A cos wt + εAw sin wt 1 − A2 w2 +
4
4
(11)
Integrating (11) yields y=−
A sin wt w
− εA cos wt 1 −
3 4
A 2 w2
−
εA 3 w 2 12
cos 3wt − ε cos t.
(12)
Comparing (9) and (12), we set
−
A w
= −Aw, and so the frequency results in, w = 1.
We, now, re-write y: 3 εA3 cos 3t y = −A sin t − εA cos t 1 − A2 − − ε cos t,
4
12
(13)
or y = −A sin t + ε cos t −A +
3A 3 4
!
−1 −
εA3 cos 3t 12
.
(14)
No secular term requires that,
−A+
3A3 4
− 1 = 0.
(15)
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1986
Then we obtain the amplitude, A = 1.492258.
(16)
Thus, Eq. (14) reduces to y = −A sin t −
εA3 cos 3t 12
.
(17)
Integrating (17), we obtain; x = A cos t −
εA3 sin 3t 36
,
(18)
which is the first-order approximate periodic solution of Eq. (5). Thus, in the first approximation, we obtain that the amplitude equal to A = 1.492258, and the frequency is 1. In order to improve its accuracy we can choose the trail functions in the form x = a cos wt + b sin wt + c cos 3wt + d sin 3wt
(19)
y = −aw sin wt + bw cos wt − 3cw sin 3wt + 3dw cos 3wt
(20)
as in Ref. [30]. Proceeding in the usual way, we can determine a, b, c, d and w, but it is laborious to do by hand. But, they can be implemented on computer with the help of symbolic computations. The first-order periodic solution (18), for ε = 0.1, takes the form; x = 1.492258 cos t − 0.009230585 sin 3t.
(21)
Abd El-Latif [31], by applying the method of time transformation, calculated the second-order explicit expansion of the periodic solution of Eq. (5) for ε = 0.1, as: x = 1.482714 cos t + 0.031153 sin t − 0.10384 sin 3t + 0.001157 cos 3t − 0.000246 cos 5t.
The second-order perturbation method solution gives; x = 1.482981 cos t + 0.032995 sin t − 0.010384 sin 3t + 0.0011466 cos 3t − 0.000217 cos 5t.
Thus, our first order approximation solution, Eq. (21), is compatible with that obtained by Abd El-Latif [31] and with that obtained by the perturbation method. Example 2. In this example, we consider the forcing van der Pol oscillator with equation, x¨ + x − ε 1 − x2 x˙ − ε sin t = 0.
(22)
Eq. (22) is equivalent to the two-dimensional nonautonomous system; (23)
x˙ = y
y˙ = −x + ε 1 − x2 y + ε sin t.
(24)
If we choose x = A cos wt, then from (23) we have x˙ = −Aw sin wt = y.
(25)
Substituting x = A cos wt and y = −Aw sin wt into the right-hand side of Eq. (24), we have y˙ = −A cos wt + ε 1 − A2 cos2 wt (−Aw sin wt) + ε sin t
(26)
or y˙ = −A cos wt + εAw sin wt −1 +
A2
!
4
+
εA 3 w 4
sin 3wt + ε sin t.
(27)
Integrating (27) yields y=−
A sin wt w
+ εA cos wt 1 −
A2
4
!
−
εA 3 12
cos 3wt − ε cos t.
Comparing (28) with (25) results in
−
A w
= −Aw, so we obtain the frequency, w = 1.
(28)
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1987
We, now, re-write y: y = −A sin t + εA cos t 1 −
A2
εA3 cos 3t
!
4
−
12
− ε cos t,
(29)
or y = −A sin t + ε cos t A −
!
A3
−1 −
4
εA3 cos 3t 12
.
(30)
No secular term requires, A−
A3
4
− 1 = 0.
Then we obtain the amplitude A = −2.382976.
(31)
Thus, Eq. (30) reduces to y = −A sin t −
εA3 cos 3t 12
.
(32)
Integrating (32), we obtain; x = A cos t −
εA3 sin 3t
(33)
36
which is the first-order approximate periodic solution of Eq. (22). Thus in the first approximation, we obtain that the amplitude equal to A = −2.382976 and the frequency is 1. The first-order periodic solution (33), for ε = 0.1, takes the form; x = −2.382976 cos t + 0.03758863 sin 3t.
(34)
Abd El-Latif [31], by applying the method of time transformation, calculated the second-order explicit expansion of the periodic solution of Eq. (22) for ε = 0.1, as: x = −2.378146 cos t − 0.126863 sin t + 0.042287 sin 3t − 0.002836 cos 3t.
Thus, our first-order approximation solution is compatible with that obtained by Abd El-Latif [31]. Example 3. Our third example is van der Pol oscillator with excitation term given by the equation: x¨ + εx˙ x2 − 1 + x cos t + x = 0.
(35)
We, again, write nonlinear two-dimensional nonautonomous system as: (36)
x˙ = y
y˙ = −εy x − 1 + x cos t − x. 2
(37)
If we, again, begin with x = A cos wt, then from (36) we have x˙ = −Aw sin wt = y.
(38)
Substituting x = A cos wt and y = −Aw sin wt into the right-hand side of Eq. (37) we have y˙ = εAw sin wt A2 cos2 wt − 1 + A cos wt cos t − A cos wt
(39)
or y˙ = −A cos wt + εAw sin wt
A2
4
!
−1 +
εA 3 w 4
sin 3wt +
εA 2 w 4
sin (2w + 1) t +
εA 2 w 4
sin (2w − 1) t.
(40)
Integrating (40) yields: y=−
A sin wt w
− εA cos wt
A2
4
!
−1 −
εA3 cos 3wt 12
−
εA2 w cos (2w + 1) t εA2 w cos (2w − 1) t − . 4 (2w + 1) 4 (2w − 1)
(41)
T. Öziş, A. Yıldırım / Nonlinear Analysis: Real World Applications 10 (2009) 1984–1989
1988
Comparing (41) with (38) results in
−
A w
= −Aw, so we obtain the frequency, w = 1.
We, now, re-write y: y = −A sin t − εA cos t
A2
4
!
−1 −
εA3 cos 3t 12
−
εA2 cos 3t 12
−
εA2 cos t 4
,
(42)
or y = −A sin t − εA cos t
A2
4
+
A
4
!
−1 −
εA2 cos 3t 12
(1 + A ) .
(43)
No secular term requires that, A2
4
+
A
4
− 1 = 0.
Then we obtain the amplitude A = 1.561553.
Thus, Eq. (43) reduces to y = −A sin t −
εA2 cos 3t 12
(1 + A).
(44)
Integrating (44) we obtain x = A cos t −
εA2 (1 + A) sin 3t 36
(45)
which is the first-order approximate periodic solution of Eq. (35). Thus in the first approximation we obtain that the amplitude equal to A = 1.561553 and the frequency is 1. The first-order periodic solution (45), for ε = 0.1, takes the form; x = 1.561553 cos t − 0.017350592 sin 3t.
(46)
Abd El-Latif [31], by applying the method of time transformation, calculated the second-order explicit expansion of the periodic solution of Eq. (35), for ε = 0.1, as: x = 1.571772 cos t + 0.119542 sin t − 0.002124 sin 3t + 0.001059 cos 3t − 0.000312 cos 5t.
The result obtained by second-order perturbation method is; x = 0.567314 cos t + 0.12176 sin t − 0.001449 sin 3t + 0.001125 cos 3t − 0.000375 cos 5t.
Thus, our first-order approximation solution is compatible with that obtained by Abd El-Latif [31] and the perturbation solution. 3. Conclusion In this paper, we applied the iteration perturbation method to variety of forcing van der Pol’s equations. We conclude from the results obtained that Iteration Perturbation method is extremely simple in its principle, easy to apply, and gives a very good accuracy even with the first-order approximation and simplest trial functions. Comparison made with other known results show that the method provides a powerful mathematical tool to the determination of limit cycles of more complex nonlinear systems. References [1] [2] [3] [4] [5]
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