Some general solutions to the Van der Pol equation. I. The (G0/G) Method. Solomon M. Antoniou SKEMSYS Scientific Knowledge Engineering and Management Systems Corinthos 20100, Greece solomon
[email protected] Abstract We apply the G0 /G method to find some general solutions to the Van der Pol equation x00 + µ(x2 − λ)x0 + αx = 0, where x≡x(t) is the unknown function while α, λ and µ are free parameters. We first convert the equation into a second order nonlinear differential equation √ using the substitution x = w, where w is a new function. We then apply the G0 /G method to the resulting equation. We find two families of solutions. The Van der Pol equation, which is a Li´enard (type of) equation, is being transformed into a second kind Abel equation, which can be solved in principle using the Panayotounakos algorithm.
Keywords: Van der Pol equation, nonlinear differential equations, exact solutions, the G0 /G method, Li´enard equation, Panayotounakos algorithm.
1
Introduction
Van der Pol equation is one of the classical paradigms in chaos theory (see for example Arrowsmith and Place [2], Jordan and Smith [4], Lichtenberg and Lieberman [6] and Wiggins [11]). 1
In this paper we apply the G0 /G method (see for example Alzaidy [1]) with constant expansion coefficients to find general solutions to the Van der Pol equation x00 + µ(x2 − λ)x0 + αx = 0 where the prime(s) denote differentiation with respect to time,√while λ, µ and α are parameters of the theory. Under the substitution x = w, where w is a new function, we first transform Van der Pol equation into another, second order nonlinear differential equation. We then apply the G0 /G method to the resulting equation. We find two families of solutions. The paper is organized as follows: In Section 2 we solve Van der Pol’s equation using the G0 /G method. In Section 3 we transform Van der Pol’s equation into an Abel equation of the second kind, which can in principle be solved using the so-called Panayotounakos algorithm (Panayotounakos [8], Panayotounakos, Zarmpoutis and Sotiropoulos [9]).
2
The Van der Pol equation and its solutions
We consider Van der Pol’s differential equation x00 + µ(x2 − λ)x0 + αx = 0
(2.1)
where x≡x(t) is the unknown function and t is the independent variable. The subscripts denote derivatives with respect to the independent variable t. Under the substitution √ (2.2) x = w, w = w(t) since
w0 1 00 1 (w0 )2 00 x = √ , x = √ w − 2 w 2 w 2 w Van der Pol’s equation (2.1) takes on the form √ w0 1 (w0 )2 1 √ w00 − + µ(w − λ) √ + α w = 0 2 w 2 w 2 w √ which, after multiplying by 2w w, becomes 0
(2.3)
1 w·w00 − (w0 )2 + µw2 w0 − λµww0 + 2αw2 = 0 (2.4) 2 The above equation will be solved using the G0 /G expansion method. We consider the expansion G0 w = a0 + a1 (2.5) G 2
where G≡G(t) is a function (to be determined) and a0 , a1 are constant coefficients (to be determined). The function G(t) is supposed to satisfy the second order linear differential equation G00 + mG0 + nG = 0
(2.6)
where m and n are constant coefficients (to be determined). Differentiation of (2.5) with respect to t gives w 0 = a1
G00 G
−
G0 2
(2.7)
G
and then we substitute G00 /G by −m(G0 /G) − n, using (2.6). We thus obtain G0 G0 2 w = −a1 n − a1 m − a1 G G 0
(2.8)
Differentiation of the above equation with respect to time and making the substitution G00 /G→ − m(G0 /G) − n, we obtain w00 = a1 n(m + 2) + a1 m2
G0 G
G0 2 G0 3 + 3a1 m + 2a1 G G
(2.9)
Substitution of (2.5), (2.8) and (2.9) into (2.4) and rearranging, we obtain the equation a21
3 2
− µa1
G0 4 G
+ a1 [a1 (λµ + 2m − 2µa0 ) − µma21 + 2a0 ]
G0 3 G
G0 2 + a1 [−µna21 + a1 (2α + mλµ − n + m2 /2 − 2µma0 ) + a0 (λµ − µa0 + 3m)] G G0 + a1 [a1 (nλµ − 2µna0 + 2n) + a0 (4α + λµm + m2 − µma0 )] G + a1 [n(m + 2)a0 − µna20 − n2 a1 /2 + nλµa0 ] + 2αa20 = 0 (2.10) Equating the coefficients of the different powers of (G0 /G)k , k = 0, 1, 2, 3, 4 to zero, we obtain the following system of algebraic equations (a1 6=0) 3 − µa1 = 0 2 a1 (λµ + 2m − 2µa0 ) − µma21 + 2a0 = 0 3
−µna21 + a1 (2α + mλµ − n + m2 /2 − 2µma0 ) + a0 (λµ − µa0 + 3m) = 0 a1 (nλµ − 2µna0 + 2n) + a0 (4α + λµm + m2 − µma0 ) = 0 a1 [n(m + 2)a0 − µna20 − n2 a1 /2 + nλµa0 ] + 2αa20 = 0
(2.11)
Solving the above system of algebraic equations, we obtain the following two families of solutions Solution 1. a0 = −
3 3 , a1 = , m = −2(λµ + 1), n = 2λµ + 1, α = 2λµ + 1 2µ 2µ
(2.12)
Solution 2. a0 = −
3λ α−1 3λ , a1 = , m = −(α + 1), n = α, µ = α−1 α−1 2λ
(2.13)
Since we have determined the coefficients m and n, we can evaluate the function G from (2.6) and then the ratio G0 /G. I. For the Solution 1, equation (2.6) becomes G00 − 2(λµ + 1)G0 + (2λµ + 1)G = 0
(2.14)
equation (2.14) admits the general solution G(t) = [Acosh(λµt) + Bsinh(λµt)]×exp[(1 + λµ)t] We thus obtain
(2.15)
G0 H0 = (1 + λµ) + G H
(2.16)
H = Acosh(λµt) + Bsinh(λµt)
(2.17)
where H≡H(t) is given by
On the other hand, equation (2.5) gives w=−
3 3 G0 + 2µ 2µ G
(2.18)
Therefore, using (2.16) and (2.18), we have w=
H0 3 λµ + 2µ H 4
(2.19)
We thus find that the first family of solutions of the Van der Pol equation is given by s H0 3 λµ + (2.20) x(t) = 2µ H In this case we should take into account the relation α = 1 + 2λµ. II. For the Solution 2, equation (2.6) becomes G00 − (α + 1)G0 + αG = 0
(2.21)
which admits general solution G = C1 eαt + C2 et and thus
G0 αeαt + Cet = αt , G e + Cet On the other hand, equation (2.5) gives w=−
(2.22) C2 C1
(2.23)
3λ G0 3λ + α−1 α−1 G
(2.24)
C=
Therefore using (2.23) and (2.24), we have eαt w = 3λ αt e + Cet
(2.25)
We thus find that the second family of solutions of the Van der Pol equation is given by r eαt x(t) = 3λ αt (2.26) e + Cet In this case we should also take into account the relation µ = (α − 1)/2λ, i.e. α = 1 + 2λµ. We thus arrive at the following Theorem: Theorem. The Van der Pol equation x00 + µ(x2 − λ)x0 + αx = 0 √ under the substitution x = w transforms into the equation 1 w·w00 − (w0 )2 + µw2 w0 − λµww0 + 2αw2 = 0 2 5
(2.27)
(2.28)
0 The above equation is solved using the GG − method, considering the ex 0 pansion w = a0 + a1 GG where G≡G(t) is a function satisfying the linear second order differential equation G00 + mG0 + nG = 0 and a0 , a1 , m, n are constant coefficients. There are two families of solutions The first family is given by s H0 3 λµ + , H = Acosh(λµt)+Bsinh(λµt), α = 1+2λµ (2.29) x(t) = 2µ H The second family is given by r x(t) =
3
3λ
eαt , α = 1 + 2λµ eαt + Cet
(2.30)
Van der Pol equation and Panayotounakos algorithm - Another solution
The van der Pol equation is a special type of Li´enard equation (see for example Polyanin and Zaitsev [10], or Harko, Lobo and Mak [3] for a recent review) and can therefore be transformed into an Abel equation of the second kind (for an account of Abel’s equations see Kamke [5], Murphy [7] or Polyanin and Zaitsev [10]). We introduce a function u(x) by (this is a standard transformation for the Li´enard equations) u(x) =
dx dt
(3.1)
Differentiation of the above function with respect to t, we have d2 x d u(x) = 2 dt dt and then using the chain rule
du dx dx dt
=
d2 x dt2
and (3.1) again, we get
u·u0x = x00tt
(3.2)
Therefore Van der Pol’s equation becomes uu0x + µ(x2 − λ)u + αx = 0 6
(3.3)
which is an Abel differential equation of the second kind. The above equation can be converted to the standard form of an Abel equation of the second kind. We change independent variable from x to v by Z 1 v = − µ(x2 − λ)dx = − µx3 + λµx (3.4) 3 Since
du du dv du = = (−µ(x2 − λ)) = −µ(x2 − λ)u0v dx dv dx dv equation (3.3) takes on the form u0x ≡
uu0v − u =
αx µ(x2 − λ)
(3.5)
(3.6)
where it is abvious that we have to express the right hand side of the above equation in terms of the new variable v. In fact, solving equation (3.4), i.e. 1 v = − µx3 + λµx 3
(3.7)
with respect to x, we find (this is one of the three solutions) x=
p 1 1/3 2λµ f + 1/3 , f = (−12v + 4 9v 2 − 4λ3 µ3 )µ2 2µ f
(3.8)
and thus equation (3.6) takes on the form uu0v − u =
2α(f + 4λµf 1/3 ) f 4/3 + 4λµ2 f 2/3 + 16λ2 µ4
(3.9)
There is by now an algorithm of finding closed-form solutions of (3.9), called Panayotounakos algorithm (references [8] and [9] and Appendix). Therefore u(v) is supposed to be a known function, thanks to Panayotounakos algorithm. Once we have determined u(v) and since the relation between v and x is given by (3.7), the general solution can be found from (3.1).
A
Appendix
The Panayotounakos Algorithm In this Appendix we provide some details about the Panayotounakos algorithm, which solves Abel’s equation of the second kind. 7
The solution of the differential equation (Abel’s equation of the second kind) yy 0 − y = f (x),
y≡y(x),
y0 =
dy dx
(A.1)
is given by 1 1 y(x) = (x + 2C) r(x) + 3 3 where C is a constant and r(x) are the roots of the cubic equation r3 (t) + pr(t) + q = 0,
t = ln|x + 2C|
(A.2)
(A.3)
The quantities p and q are given by 1 p = b − a3 , 3 a = −4,
a 3 1 q = c − ab + 2 3 3
b = 3 + 4[g(t) + f (t)]e−t ,
c = −4[g(t) + 2f (t)]e−t
(A.4) (A.5)
and g(t) is a function defined by g(t) =
[(tsin(t) + cos(t))ci(t) + cos2 (t)][4tci(t) + cos(t)]e−t − 2f (t) (A.6) 2[2tci(t)]3
The function ci(t) is the known cosine integral function defined by Zt ci(t) = γ + ln(t) +
cosu − 1 du u
(A.7)
0
where γ is the Euler-Mascheroni constant.
References [1] J. F. Alzaidy: ”The G0 /G - Expansion Method for Finding Traveling Wave Solutions of Some Nonlinear Pdes in Mathematical Physics”. Int. J. Mod. Eng. Res. Vol 3 Issue 1 (2013) 369-376 [2] D. K. Arrowsmith and C. M. Place: ”Dynamical Systems. Differential equations, maps and chaotic behaviour”. Chapman and Hall, 1995
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[3] T. Harko, F. S. N. Lobo and M. K. Mak: ”A class of exact solutions of the Linard type ordinary non-linear differential equation”. arXiv:1302.0836v3 [math-phys] 20 Jan 2014 [4] D. W. Jordan and P. Smith: ”Nonlinear Ordinary Differential Equations” Second Edition, Clarendon Press, Oxford 1995 [5] E. Kamke: ”Differentialgleichungen: L¨osungsmethoden und L¨osungen. Band 1. Gew¨ohnliche Differentialgleichungen . Chelsea, N.Y. 1948 [6] A. J. Lichtenberg and M. A. Lieberman: ”Regular and Chaotic Dynamics”. Second Edition, Springer-Verlag, NY 1992 [7] G. M. Murphy: ”Ordinary Differential Equations and their Solutions” D. Van Nostrand, N.Y. 1960 [8] D. E. Panayotounakos : ”Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (Part I. Abels equations)” Appl. Math. Letters 18 (2005) 155-162 [9] D. E. Panayotounakos, Th. I. Zarmpoutis and P. Sotiropoulos: ”The General Solutions of the Normal Abel’s Type Nonlinear ODE of the Second Kind”. Int. J. Appl. Math. 43:3 (2013) IJAM-43-3-01. [10] A. D. Polyanin and V. F. Zaitsev: ”Handbook of Exact Solutions for Ordinary Differential Equations”. Chapman and Hall/CRC, Second Edition 2003 [11] S. Wiggins: ”Introduction to Applied Nonlinear Dynamical Systems and Chaos”. Springer-Verlag, NY 1990
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