This paper presents the analytical and approximate solutions of the coupled chaotic Van der Pol-Duffing systems, by using the He\u27s parameter-expansion method (PEM). One iteration is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. From the obtained results, we can conclude that the suggest method, is of utter simplicity, and can be easily extended to all kinds of non-linear equations

Khader, M. M.

Sweilam, N. H.

English

Digital Commons @ PVAMU (Prairie View A&M Univ)

Applications and Applied Mathematics: An International Journal (AAM) Volume 5 Issue 3 Special Issue 1 – August 2010 Article 8 8-2010 A Note on He’s Parameter-Expansion Method of Coupled Van der Pol–Duffing Oscillators N. H. Sweilam Cairo University M. M. Khader Benha University Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Numerical Analysis and Computation Commons, and the Partial Differential Equations Commons Recommended Citation Sweilam, N. H. and Khader, M. M. (2010). A Note on He’s Parameter-Expansion Method of Coupled Van der Pol–Duffing Oscillators, Applications and Applied Mathematics: An International Journal (AAM), Vol. 5, Iss. 3, Article 8. Available at: https://digitalcommons.pvamu.edu/aam/vol5/iss3/8 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. 93  Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466  Special Issue No. 1 (August 2010)  pp. 94 – 100  Applications and Applied Mathematics:  An International Journal (AAM)     A Note on He’s Parameter-Expansion Method of  Coupled Van der Pol–Duffing Oscillators   N. H. Sweilam   Department of Mathematics  Faculty of Science Cairo University  Giza, Egypt n_sweilam@yahoo.com  M. M. Khader Department of Mathematics  Faculty of Science  Benha University  Egypt mohamedmbd@yahoo.com   Received: February 6, 2010; Accepted: March 16, 2010    Abstract  This paper presents the analytical and approximate solutions of the coupled chaotic Van der Pol-Duffing systems, by using the He's parameter-expansion method (PEM). One iteration is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. From the obtained results, we can conclude that the suggest method, is of utter simplicity, and can be easily extended to all kinds of non-linear equations.  Keywords:  Coupled chaotic Van der Pol-Duffing systems; Parameter expansion method.  MS Classification: 35A25, 65Z05.     1. Introduction  In recent years, extensive investigations have been carried out to analyze chaotic synchronization dynamics of coupled non-linear oscillators Boccaletti et al. (2002), Bowong et al. (2006). The interest devoted to such topics is due to the potential applications of synchronization in 1Sweilam and Khader: A Note on He’s Parameter-Expansion MethodPublished by Digital Commons @PVAMU, 201094  Sweilam et al. communication engineering (using chaos to mask the information signals Pecora and Caroll (1990), Perez and Cedeira (1995), electrical and automation engineering, biology, and chemistry (Kuramoto (1984), Winfree (1980).  Recently, considerable attention has been directed towards analytical solutions for nonlinear equations without small parameters. There are many analytical methods and numerical schemes to handle non-linear problems, such as variational iteration method He (2006), Sweilam and Khader (2007), Sweilam et al. (2008), homotopy perturbation method He (2006) Turgut and Yildirim (2007), the exp-function method He and Wu (2006), standard Lindstedt-Poincaré He (2002) and parameter-expansion method (Darvishi et al. (2008), Xu (2007), Sweilam and Khader (2009). The parameter-expansion method (PEM) which proposed by He is proved to be a very effective and convenient way for handling the non-linear problems.  The organization of the paper is as follows. The next Section deals with the model and the problem statement. In Section 3, the application of the He's parameter-expansion method for solving the coupled chaotic Van der Pol-Duffing systems is presented. A conclusion is given in the last section.  The organization of the paper is as follows. The next section deals with the model and the problem statement. In Section 3, the application of the He’s parameter-expansion method for solving artificial model problem and the coupled chaotic Van der Pol–Duffing systems is presented. A conclusion is given in the last section.   2. Problem Statement   In the chaotic state, the main feature is the high-sensitivity to initial conditions. This is the result of the combined effects of the cubic and intrinsic nonlinearities, and of the extrinsic periodic drive. Consequently, a very small difference in the initial conditions will lead to different time histories or orbits. If two systems are launched with two initial conditions 1IC  and 2IC , they will circulate on different degenerated chaotic orbits. The goal of the synchronization in this case is to call one of the systems (slave) from its degenerated chaotic orbit to that of the other system (master). For this aim, the master system is described by the component x  while the slave system has the corresponding component y. The enslavement is carried out by coupling the slave to the master through the following scheme Yamapi and Filatrella (2008):   ),0Tx)H(tK(yt)cos(0E3αyyy)2yμ(1yt),cos(0E3αxxx)2xμ(1x (1)  (dots denote differentiation with respect to time). The quantities μ  and α  are two positive coefficients. 0E and ε are respectively the amplitude and frequency of the external excitation. Where K is the feedback coupling coefficient, 0T  the onset time of the synchronization process and H(z)  is the Heaviside function defined as:   0.zfor             1,0;zfor              0,H(z)2Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 8https://digitalcommons.pvamu.edu/aam/vol5/iss3/8AAM: Intern. J., Special Issue No. 1 (August 2010)                                                                                                        95 Thus, for 0Tt , the two oscillators are completely uncoupled; at 0Tt  the coupling turns on and stays on for all subsequent t. We let the system evolve unperturbed before the coupling is turned on to avoid a transient that could influence the beginning of the synchronization process. This will be relevant in the next session where we have estimated the synchronization time, not just the asymptotic properties of the system. Practically, this type of unidirectional coupling between the master system and the slave system can be done through a linear resistor and a buffer. The buffer acts a signal-driving element that isolates the master system variable from the slave system variable, thereby providing a one-way coupling. In the absence of the buffer the system represents two identical Van der Pol–Duffing oscillators coupled by a common resistor cR , when both the master and slave systems will mutually affect each other. It is worth to note that the sign of K has an intuitive meaning: K > 0 corresponds to the attractive case between the master and the slave, while K < 0 corresponds to repulsion. A natural expectation would then be: for some value of positive K the system achieves synchronization, while for negative values of K one would not expect synchronization.  3. Implementation of He’s PEM  In this section, the basic idea of the parameter-expansion method will be illustrates by solving the coupled system of nonlinear ordinary differential equations (1). To implement PEM, we re-write the system (1) in the following form:   t)..cos(0EK.x3α.yy)2yμ.(1K).y1(yt),.cos(0E3α.xx)2xμ.(11.xx (2)  According to PEM, the solutions and coefficients K α, μ,  K),(1 1, and 0E of  (2) are expanded into a series of an artificial parameter, p  respectively, in the forms:     ...,2y2p1py0yy         ...,2x2p1px0xx (3)   ....2d2p1pd 0E    ...,2c2p1pcK            ...,2b2p1pbα      ...,2a2p1paμ        ...,2s2p1ps2sK1 ...,2r2p1r p2r  1      (4)  3Sweilam and Khader: A Note on He’s Parameter-Expansion MethodPublished by Digital Commons @PVAMU, 201096  Sweilam et al. Now, by substituting from (3)-(4) into (2) respectively, and equating the terms with the identical powers of p, we can obtain a series of linear equations. These linear equations are easy to solving by using Mathematica software. Here we only write the first few linear equations:  1,(0)0y      0,(0)0y            0,0y2s 0y1,(0)0      x0,(0)0x            0,0x2r 0x (5)   .0(0)1y(0)1y 0,)t(c1d0x1c30y1b0y)20y(11a0y1s1y2s 1y,0(0)1x(0)1x            0,)t(c1d30x1b0x)20x(11a0x1r1x2r 1xosos (6)  By considering the initial conditions, the solutions of (5) are:   cos(st).(t)0y      cos(rt),(t)0x (7)  By substituting the result into the first equation of the system (6), we have:   0,t)cos(1d(rt)3cos1b(rt)3sinr 1acos(rt)1r1x2r 1x   or  0,t)cos(1dsin(3rt)r 1a41sin(rt)r 1a43cos(3rt)1b41)cos(rt)1b431r(1x2r 1x (8)  No secular term in 1x  requires that:   .01b431r (9)  If the first-order approximation is enough, then, setting p = 1 in Eqs. (3) - (4), we have:     .1d0E   , 1b   ,1a    , 1r2r1     ,1x0xx (10)  Solving the equations (9) – (10) simultaneously, we have:   .3421r (11)  The approximate period can be expressed in the form:   .344r2πT (12) 4Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 8https://digitalcommons.pvamu.edu/aam/vol5/iss3/8AAM: Intern. J., Special Issue No. 1 (August 2010)                                                                                                        97  To obtain (t)1x , from the relations (8) - (11) we have:   t),cos(0Esin(3rt)4μrsin(rt)4r 3cos(3rt)4α1x2r 1x (13)  Solving (13) yields:  t),cos(220sin(3rt)32r                cos(rt)t 83cos(3rt)232rsin(rt)32r9cos(rt) )220232((t)1xrErEr (14)  from the relations (7), (10) and (14), the first order solution for x  is:  ,t)cos(220sin(3rt)32r cos(rt)t 83                       cos(3rt)232rsin(rt)32r9cos(rt) )220232(cos(rt)x(t)rErEr (15)  where r is defined in (11).   Also, by substituting the result into the second equation of the system (6), we have:   0,t)cos(1dt)cos(1c(st)3cos1b(st)3sin1acos(st)11y2s 1y sss   or  0.t)cos(1d)st3cos(41)st3sin(41)stsin(413)cos(st)1c1b431(1y2s 1ybsasas (16)  No secular term in 1y  requires that:   .01c1b431s (17)  If the first-order approximation is enough, then, setting p = 1 in equations (3) - (4) , we have:     .1d0E    ,1cK  , 1b   ,1a    , 1s2sK1     ,1y0yy (18)  By solving equations (17)–(18) simultaneously, we have: 5Sweilam and Khader: A Note on He’s Parameter-Expansion MethodPublished by Digital Commons @PVAMU, 201098  Sweilam et al.  .α3421s (19)  The approximate period can be expressed in the form (12).  To obtain (t)1y , from relations (16)-(19) we have:  t),cos(0Esin(3st)4μssin(st)4s 3cos(3st)4α1y2s 1y (20)  solving (20) yields: t),cos(220sin(3st)32s-                 cos(st)t 83cos(3st)232ssin(st)32s9-cos(st) )220232((t)1ysEsEs (21)  from the relations (7), (18) and (21), the first order solution for y is:  .t)cos(220sin(3st)32s cos(st)t 83                      cos(3st)232ssin(st)32s9cos(st) )220232(cos(st)y(t)sEsEs (22)  where s is defined in (19).  Figs. 1-2 Illustrate the behavior of the solution (x(t), y(t)) with different values of the coefficients , ,  ,  and 0E  in the system (1).  .20 40 60 80 100t-150-100-5050100150x20 40 60 80 100t-40-30-20-10102030x Figure 1:    Left: =5;    =0.01;     =2.463;     0E =5;                      Right:  =1;    =0.1;      =2.463;      0E =1; 6Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 8https://digitalcommons.pvamu.edu/aam/vol5/iss3/8AAM: Intern. J., Special Issue No. 1 (August 2010)                                                                                                        99 20 40 60 80 100t-150-100-5050100150y20 40 60 80 100t-40-30-20-10102030y Figure 2:    Left: =5;    =0.01;    =2.463;     0E =5;                     Right:   =1;    =0.1;      =2.463;      0E =1;   4. Conclusion  From the presented results of the example in this paper, we conclude that He's parameter-expansion method is an extremely simple and a powerful tool to solve this particular problem of mutually coupled biological systems described by coupled chaotic Van der Pol-Duffing systems. The method might become a promising and powerful new method for many other applications in searching for periodic solutions of non-linear oscillations without any difficulty. The suggest method by help of Mathematica, is of utter simplicity, and can be easily extended to all kinds of non-linear equations. The PEM can be easily comprehended with only a basic knowledge of advanced calculus.     REFERENCES  Boccaletti, S., Kurths, J., Osipov, G., Valladeres, D.L. and Zhou, C.S. (2002). The synchronization of chaotic systems, Phys. Rev., Vol. 366, pp. 1-101. Bowong, S., Kakmeni, M., Dimi, J.L. and Koina, R. (2006). Synchronizing chaotic dynamics with uncertainties using a predictable synchronization delay design, Communications in non- linear Science and Numerical Simulation, Vol. 11(8), pp. 973-987. Darvishi, M.T., Karami, A. and Shin, B.C. (2008). Application of He's parameter-expansion method for oscillators with smooth odd nonlinearities, Physics Letters A, Vol. 372, pp. 5381-5384. He, J.H. (2006). Some asymptotic methods for strongly non-linear equations, International Journal of Modern Physics B, Vol. 20(10), pp. 1141-1199. He, J.H. and Wu, Xu-Hong (2006). Exp-function method for non-linear wave equations, Topological Methods in Non-linear Analysis, Chaos, Solitons and Fractals, Vol. 30, pp. 700-708.  He, J.H. (2002). Modified Lindstedt-Poincaré methods for some strongly non-linear oscillations. Part I: expansion of a constant, Int. J. Non-linear Mech., Vol. 37, pp. 309-314. Kuramoto, Y. 1984. Chemical Oscillations, Waves and Turbulence, Berlin: Springer-Verlag.  Xu, Lan (2007). Application of He's parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire, Physics Letters A, Vol. 368, pp. 259-262. Pecora, L. M., Caroll, T.L. (1990). Synchronization in chaotic systems, Phys. Rev. Lett., Vol. 64, pp. 821-824. Perez, G. and Cedeira, H.A. (1995). Extracting messages masked by chaos, Phys. Rev. Lett., Vol. 74, pp. 1970-1973. 7Sweilam and Khader: A Note on He’s Parameter-Expansion MethodPublished by Digital Commons @PVAMU, 2010100  Sweilam et al. Sweilam, N.H. and Khader, M.M. (2007). Variational iteration method for one dimensional non-linear thermo-elasticity, Chaos, Solitons and Fractals, Vol. 32, pp. 145-149. Sweilam, N.H.  and Khader, M.M. (2009). Application of He's parameter-expansion method for  the nonlinear differential equations, Int. J. of Nonlinear Science & Numerical Simulation, Vol.  10(2), pp. 265-272. Sweilam, N.H., Khader, M.M.  and Al-Bar, R. (2008).  Non-linear focusing Manakov systems by variational iteration method and Adomian decomposition method, Journal of Physics: Conference Series, Vol. 96, pp. 1-7. Turgut, O . and Yildirim, A. (2007). A note on He's homotopy perturbation method for Van der Pol oscillator with very strong nonlinearity, Chaos, Solitons and Fractals,Vol. 34, pp. 989-991. Winfree, A.T. 1980.  The Geometry of Biological Time, New York: Springer-Verlag. Yamapi, R. and Filatrella, G. (2008). Strange attractors and synchronization dynamics of coupled Van der Pol-Duffing oscillators, Communications in Nonlinear Science and Numerical Simulation, Vol. 13, pp. 1121-1130.   8Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 8https://digitalcommons.pvamu.edu/aam/vol5/iss3/8

A Note on He’s Parameter-Expansion Method of Coupled Van der Pol–Duffing Oscillators

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A Note on He’s Parameter-Expansion Method of Coupled Van der Pol–Duffing Oscillators

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