Journal of Nonlinear Mathematical Physics

Volume 27, Issue 2, January 2020, Pages 185 - 198

The Lie symmetry group of the general Liénard-type equation

Ágota Figula, Gábor Horváth, Tamás Milkovszki, Zoltán Muzsnay
University of Debrecen, Institute of Mathematics, Pf. 400, Debrecen, 4002, Hungary,,,,
Received 8 June 2019, Accepted 13 August 2019, Available Online 27 January 2020.
10.1080/14029251.2020.1700623How to use a DOI?
second order ordinary differential equation; Liénard-type equation; Levinson–Smith-type equation; Lie group; symmetry analysis; Lie algebra of infinitesimal symmetries

We consider the general Liénard-type equation u¨=k=0nfku˙k for n ≥ 4. This equation naturally admits the Lie symmetry t. We completely characterize when this equation admits another Lie symmetry, and give an easily verifiable condition for this on the functions f0,..., fn. Moreover, we give an equivalent characterization of this condition. Similar results have already been obtained previously in the cases n = 1 or n = 2. That is, this paper handles all remaining cases except for n = 3.

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1. Introduction

In this paper we consider the Liénard-type second order ordinary differential equation

where the dot denotes differentiating by the independent variable t representing the time. Eq. (1.1) is a special case of the Levinson–Smith-type equation u¨=g1(u,u˙)u˙+g0(u) [1, G.3, p. 198–199], for which existence and uniqueness of a limit cycle have been established under certain conditions [2,3].

Eq. (1.1) is a common generalization of the Rayleigh-type equation u¨+F(u˙)+u=0 when F is a polynomial, the classical Liénard-type equation u¨=f1(u)u˙+f0(u), and the quadratic Liénard-type equation u¨=f2(u)u˙2+f1(u)u˙+f0(u). These equations come up quite often in Physics or Biology. Rayleigh-type systems play an important role in the theory of sound [4] or in the theory of non-linear oscillations [5, Chapter 2.2.4]. Classical Liénard-type equations arise in the model of the van der Pol oscillator applied in physical and biological sciences [6], but electric activity of the heart rate [7] or nerve impulses are modelled by a Liénard-type model, as well [8,9] or [10, Chapter 7]. In [11] the population of Easter Island is modelled, and the system of differential equations is then reduced to a second order quadratic Liénard-type equation. One can even find applications in economy [1215].

Symmetry analysis is a very useful tool developed to understand and solve differential equations. Several examples come from Physics (see e.g. [16, 17] for comprehensive studies on the topic), and an increasing number of examples from Biology (see e.g. [11, 1820]). Finding some symmetries for a differential equation can be used to derive an appropriate change of coordinates which then helps to eliminate the independent variables or to decrease the order of the system.

In many cases mentioned above (e.g. the Fitzhugh–Nagumo model [8, 9] or the model for the population of Easter Island [11]) the model is based on a first order system of two equations equivalent to a second order Liénard equation. In such a case, one might benefit to consider the equivalent second order system, which would only admit a finite dimensional Lie symmetry group instead of an infinite one. If this Lie group is at least two-dimensional, then pulling the symmetries back to the original system could yield two independent symmetries of the original system, and solutions can be determined by quadratures. This method has been applied successfully in several situations in the past (see e.g. [11, 18, 20] for some recent examples in Biology). This motivates to study the Lie symmetries of (1.1).

Pandey, Bindu, Senthilvelan and Lakshmanan [21,22] considered the classical Liénard equation

where f1 and f0 are arbitrary, infinitely many times differentiable functions. They classified when (1.2) has a 1, 2, 3, or 8 dimensional Lie symmetry group depending on f0 and f1. Then Tiwari, Pandey, Senthilvelan and Lakshmanan [23, 24] classified the dimension of the Lie symmetry group of quadratic Liénard-type equations without u˙ term, and then more generally [25] the mixed quadratic Liénard-type equation
where f0, f1 and f2 are arbitrary, infinitely many times differentiable functions. Further, Paliathanasis and Leach [26] showed how one can simplify (1.3) by removing f2 from (1.3) in the case f1 = 0. The question naturally arises: what are the Lie symmetries if the right-hand side of (1.3) is a higher order polynomial in u˙?

In this paper we consider (1.1) for n ≥ 4 and for differentiable functions fk depending only on u, and not on t. Note, that (1.1) is autonomous, therefore the tangential Lie algebra of the Lie group of all its symmetries always contains the 1-dimensional subalgebra generated by the vector field t. Determining another generator of would then lead to a solution by quadratures of (1.1), and of any first order system equivalent to it. In Theorem 3.1 (see Section 3 for details) we completely characterize the case when (1.1) admits a more than 1 (in fact, 2) dimensional symmetry group. In particular, we give conditions (3.13.4) such that the symmetry group is 2-dimensional if and only if these conditions hold.

Here, conditions (3.13.3) are natural, but the meaning of the system (3.4) seems less intuitive, even though the system (3.4) is easily verifiable for a particular choice of F. In Theorem 4.1 (see Section 4 for details) we provide a necessary and sufficient condition for f0,... , fn to satisfy (3.4). It turns out that f0,... , fn satisfy (3.4) if and only if they are expressible by F and some constants.

2. The symmetry condition

We formulate the symmetry condition for (1.1) in this section. Consider (1.1) on the plane (t, u), where t is the independent variable, and u is the dependent variable. Further, the computations will be slightly easier if we consider the right-hand side as an infinite sum kfku˙k=k=fku˙k, where fk = 0 if k < 0 or k > n.

The general form of an infinitesimal generator of a symmetry of (1.1) has the form


Let D denote the total derivation by t, that is Dξ=ξt+u˙ξu, Dη=ηt+u˙ηu. We use the convention of writing partial derivatives into the lower right index. Then the first prolongation of X is


Further, let

be the spray corresponding to the differential equation (1.1). The vector field (2.1) is an infinitesimal symmetry of (1.1) if and only if its first prolongation X1 satisfies the Lie bracket condition
on the space (t, u, u˙ ) (cf. [16, Chapter 4, §3]). Substituting X1 and S1 into (2.2) we obtain
therefore the symmetry condition is

3. Lie symmetry algebra

We consider (1.1) for n ≥ 4 and for differentiable functions fk depending only on u, and not on t. In Theorem 3.1 we completely characterize the case when (1.1) admits more than 1 (in fact, 2) dimensional symmetry group. We prove following

Theorem 3.1.

Consider (1.1) for some n ≥ 4 and for f0,... , fn : I → ℝ being differentiable functions such that fn is not constant zero on the open interval I ⊆ ℝ. Then the Lie symmetry algebra ℒ of (1.1) is exactly 1 dimensional, unless there exists a constant a ∈ ℝ, an open interval UI, and a three-times differentiable function F : U → ℝ such that for all uU we have

and with the notation g(u)=(n2)F(u)(n1)F(u) the following hold:

Further, if both F1 and F2 satisfy conditions (3.13.4), then F1 = F2.

Remark 3.1.

(Generator of the symmetry algebra of (1.1).) In case conditions (3.13.4)

  1. 1.

    do not hold, then the symmetry algebra is generated by t,

  2. 2.

    hold, then the 2 dimensional Lie symmetry algebra is generated

    1. (a)

      by t and tt+g(u)u if a = 0, or

    2. (b)

      by t and eatt+aeatg(u)u if a ≠ 0.


The left-hand side of (2.3) has to be zero for all (t, u, u˙). As ξ, η, fk (0 ≤ kn) do not depend on u˙, (2.3) is a polynomial in u˙. Thus, (2.3) holds if and only if each of its coefficients is zero. Since fn is not constant 0 on the interval I, there exists an open interval U′ ⊆ I such that fn(u) ≠ 0 for uU′. Thus, from the coefficient of u˙n+1 by n ≥ 4 we obtain


In particular, ξ only depends on t and not on u. Substituting ξu = 0 into (2.3) and considering the coefficients, we obtain that


In the following we analyze the system (3.5). Note, that ξ = c, η = 0 (for any c ∈ ℝ) satisfies (3.5). Further, if η = 0 then from (3.5e) we have ξt = 0 and ξ = c for some c ∈ ℝ. Thus, in the following we assume that η is not constant 0.

Consider (3.5e) first. Now, fn is nonzero on the open interval U′, therefore either fn(u) > 0 for all uU′ or fn(u) < 0 for all uU′. Choose ε ∈ {1, −1} such that εfn(u) > 0 for all uU′, that is | fn| = εfn. Then we have


Multiplying by 1n1|fn|2nn1 yields


Here, the left-hand side is the u-derivative of |fn|1n1η, hence


Let F : U′ → ℝ be a function defined as in (3.3), that is

and let g be defined as in Theorem 3.1. that is

Thus we obtained that


Note, that g(u) cannot be constant 0 on U′ (otherwise both F and F=|fn|1n1 were constant 0), thus there exists an open interval UU′ such that F(u) ≠ 0 for all uU, and hence g(u) ≠ 0 (uU). By substituting (3.6) into (3.5d) we have


In particular, for k = n − 1 we have fn(u)g(u) ≠ 0 for all uU, thus


Now, fn, fn−1, and g only depend on u, and ξtt and ξt only depend on t. Therefore there exists a ∈ ℝ such that

or else ξt = 0 implying η = 0 by (3.6), a contradiction. Thus,
for some c ∈ ℝ. Substituting (3.9) and (3.10) into (3.5), one obtains

Thus, either f0, f1, f2,... fn,g,F satisfy the conditions (3.13.4), or else c = 0, implying ξt = 0 and η = 0, a contradiction.

Finally, assume that F1 and F2 both satisfy conditions (3.23.3) on U. Then let g1 and g2 be defined from F1 and F2, and let a1 and a2 be defined using (3.7). Let ξ1, ξ2 be such that (ξi)t = eait, let ηi = eaitgi(u), and let Xi=ξit+ηiu for i = 1,2. Further, let ξ = ξ1ξ2, η = η1η2, and X=ξt+ηu. Now, if F1 and F2 both satisfy conditions (3.23.3) and (3.4), then both X1 and X2 are elements of the Lie symmetry algebra , and thus X=X1X2=(ξ1ξ2)t+(η1η2)u, as well. However, we have proved that if either ξt ≠ 0 or η ≠ 0, then they are of the form (3.9) and (3.10). Thus, ea1tea2t is of the form ceat for some a,c ∈ ℝ, that is a1 = a2 and c = 0. Then ξt = 0, implying η = 0 by (3.6), which yields g1 = g2. Finally, by F′1 = F′2, the definition of g immediately implies

and hence F1 = F2. This finishes the proof of Theorem 3.1.

4. Equivalent description of the conditions

In this paragraph we provide a necessary and sufficient condition for f0,... , fn to satisfy (3.4). We have the following

Theorem 4.1.

Let U ⊆ ℝ be an open interval, f0,... , fn, F : U → ℝ be functions satisfying (3.13.3), g as introduced in Theorem 3.1 and let ε, ν ∈ {1,−1}, a ∈ ℝ be real constants such that |fn| = εfn, |F| = νF. Then f0,... , fn, F, g satisfy (3.4) if and only if there exist real constants bk and functions Ak, Bk : U → ℝ (0 ≤ kn) where bn=ε(n1n2)1n, An is constant 0,

such that f0,... , fn are of the form

In particular, if a = 0, then Bk(u) = 0 for all uU (0 ≤ kn), and thus f0,... , fn, F, g satisfy (3.4) if and only if there exist real constants b′k (0 ≤ kn) such that


Further, if F is positive on U, then


First, in Section 4.1 we show that Ak (0 ≤ kn) defined by (4.1) satisfy a recursive system of differential equations. The details are contained in Lemma 4.1. Then in Section 4.2 we consider the case a = 0, when (3.4) results in homogeneous equations for fk. Finally, in Section 4.3 we prove Theorem 4.1 by considering the general case a ≠ 0 and applying the method of variation of parameters.

4.1. Auxiliary functions

Let us use the notations of Theorem 4.1.

Lemma 4.1.

Let An(u) = 0. For all 0 ≤ kn − 1 a particular solution of the ordinary differential equation

where bk+1 is an arbitrary real constant, is the function Ak defined by (4.1) in Theorem 4.1.


We prove (4.1) by induction on m = nk. For m = 1 we have

and a particular solution is

This proves (4.1) for k = n − 1. Assume now, that (4.1) holds for an integer m = nk, 1 ≤ mn, that is


Putting this into (4.6) for k = n − (m + 1) one obtains


By integrating, one can obtain a particular solution as


Hence, (4.1) holds for k = n − (m + 1) and by induction it holds for all integers 0 ≤ kn − 1.

4.2. The homogeneous case

Assume a = 0. Now, (3.4) takes the form


Note, that (4.7d) for k = 0,1 gives (4.7a) and (4.7b). Now, g(u) ≠ 0 for uU, hence the solution of (4.7d) is

for some bk ∈ ℝ (0 ≤ kn − 1, k ≠ 2). For k = 2 eq. (4.7c) has the form (−g′ + f2g) = 0, thus
for some b2 ∈ ℝ. This proves Theorem 4.1 in the case a = 0 by selecting bk=bk(n1n2)k1(0kn,k2) and b2=1+b2n1n2.

4.3. The general (inhomogeneous) case


[Proof of Theorem 4.1] If a ≠ 0, then (3.4d) is an inhomogeneous linear differential equation for fk, and by (4.8) its general solution (by variation of parameters) is

for some function Bk = Bk(u) and constant bk ∈ ℝ. Write fk in the form fk = (bk + Bk)hk, where

Putting (4.10) into (3.4d) we obtain


Now, hk is a particular solution of the homogeneous differential equation (4.7d), thus


Since hk(u) ≠ 0 for all uU, Bk is a particular solution of the differential equation


We prove by induction on m = nk that Bk = Ak (3 ≤ kn − 1) by showing that Bk satisfies the recursive system of differential equations (4.6) of Lemma 4.1. Let m = 1, that is k = n − 1. From (3.3) we have fn = ε(F′)n−1. Applying (4.11) we obtain


Comparing (4.12) and (4.6) for m = 1, we find B′n−1 = A′n−1 by choosing bn=ε(n1n2)1n. Hence, a particular solution Bn−1 of the differential equation B′n−1 = A′n−1 is An−1. Therefore, for k = n − 1 we have Bk = Ak.

Assume that for an integer 4 ≤ kn − 1 Bk = Ak holds, thus from (4.10) for k = nm + 1 we have


Putting this into (4.11) for k = nm we obtain


Comparing (4.6) for k = nm and (4.13) we see that B′nm = A′nm. Hence a particular solution Bnm of the differential equation B′nm = A′nm is Anm. Therefore for k = nm one has Bk = Ak. By induction, for all 3 ≤ kn − 1 one has Bk = Ak. Thus (4.3a) holds for 3 ≤ kn − 1, and (4.2a) is proved.

We continue by proving (4.3b) and (4.2b), that is we show the condition on fk for k = 2. Now, f2 is the solution of the inhomogeneous linear differential equation (3.4c). The general solution of (3.4c) by (4.9) and by variation of parameters has the form

for some function B2 = B2(u) and constant b2 ∈ ℝ. Putting (4.14) into (3.4c), then using (1g)g=gg, and the fact that g+b2g is the solution to the homogeneous differential equation (4.7c), one has
that is B′2 = −3af3g. Using the form of f3 given by (4.10) and the definition of g we obtain

Now, A3 = B3 by (4.2a), thus comparing (4.15) and (4.6) for k = 2 yields B′2 = νA′2. Hence, a particular solution B2 of the differential equation B′2 = νA′2 has the form B2 = νA2. Thus, (4.3b) and (4.2b) hold.

Now, we obtain the condition on fk for k = 1. The function f1 is the solution of the inhomogeneous linear differential equation (3.4b). The general solution of (3.4b) by (4.8) and by variation of parameters is

for some function B1 = B1(u) and constant b1 ∈ ℝ. Putting (4.16) into (3.4b), and using the fact that |F|1nn2 is the solution to the homogeneous differential equation (4.7b), one obtains

As f2 has the form (4.14) and B2 = νA2 by (4.2b), we obtain that


Comparing (4.17) and (4.6) for k = 1 we obtain


Hence a particular solution B1 of the differential equation (4.18) has the form


Therefore, (4.3a) holds for k = 1, and (4.2c) is proved.

Finally, we prove that (4.3a) holds for k = 0. For k = 0 the function f0 is the solution of the inhomogeneous linear differential equation (3.4a). The general solution of (3.4a) by (4.8) and by variation of parameters is

for some function B0 = B0(u) and constant b0 ∈ ℝ. Putting (4.20) into (3.4a), and using the fact that (n2n1)|F|nn2(F)1 is the solution to the homogeneous differential equation (4.7a), we obtain

As f1 has the form (4.16), and B1 has the form (4.19), we obtain


Comparing (4.21) and (4.6) for k = 0 we have


Therefore, a particular solution B0 of (4.22) is


Hence, (4.3a) holds for k = 0, and (4.2d) is proved. This finishes the proof of Theorem 4.1

5. Open problems

Several questions arise after determining the symmetries of (1.1). Indeed, if the Lie group of symmetries is at least two dimensional, then one can apply the two-dimensional solvable Lie group to obtain the solutions of (1.1).

Problem 5.1.

Determine the solutions of (1.1) provided fk (0 ≤ kn, n ≥ 4) satisfy the conditions of Theorem 3.1.

The only remaining case for (1.1) not covered by Theorem 3.1 or by [2126] is when n = 3. Then one cannot immediately conclude ξu = 0 from (2.3), because the u˙4 term of (2.3) is identically 0. In fact, for n = 3 the symmetry condition translates to


A potential simplification of the system (5.1) might be to eliminate f2 from (5.1) by using a coordinate change v = G(u), for some bijective, two-times differentiable G, for which G″(u) = G′(u) f2(u) is satisfied (see e.g. [26]). This, however, still does not give an immediate answer as to what the solutions of (5.1) are.

Problem 5.2.

Determine all symmetries (and solutions) of the autonomous differential equation

where f0, f1, f2, f3 are arbitrary continuous functions in u.


The research was supported by the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 318202. The first, third and fourth authors were partially supported by the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 317721. The second author was partially supported by the Hungarian Scientific Research Fund (OTKA) grant no. K109185 and grant no. FK124814. The first and fourth authors were partially supported by the EFOP-3.6.1-16-2016-00022 project. These projects have been supported by the European Union, co-financed by the European Social Fund.


[1]R.E. Mickens, An Introduction to Nonlinear Oscillations, Cambridge University Press, Cambridge etc., 1981, pp. XIV 224.
[2]N. Levinson, On the existence of periodic solutions for second order differential equations with a forcing term, J. Math. Phys., Mass. Inst. Techn, Vol. 22, 1943, pp. 41-48.
[4]J.W. Strutt, The Theory of Sound, 2nd ed., Dover Publications, New York, 1945, pp. 504. Two volumes in one. XIII, 480 p.; XII,
[5]A.D. Polyanin and V.F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed., CRC Press, Boca Raton, FL, 2003. 2nd ed. edn
[12]J.R. Faria, J.C. Cuestas, and L.A. Gil-Alana. Unemployment and entrepreneurship: a cyclical relation? (Discussion papers. Nottingham Trent University, Bottingham Business School, Economics Division, 2008/2)
[13]R.M. Goodwin, A growth cycle, C.H. Feinstein (editor), Socialism, Capitalism and Economic Growth, Cambridge University Press, Cambridge, 1975, pp. 54-58.
[16]A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khorꞌkova, I.S. Krasilꞌshchik, A.V. Samokhin, Y.N. Torkhov, A.M. Verbovetsky, and A.M. Vinogradov. Symmetries and conservation laws for differential equations of mathematical physics, Translations of Mathematical Monographs, Vol. 182 (American Mathematical Society, Providence, RI, 1999). Edited and with a preface by Krasilꞌshchik and Vinogradov, Translated from the 1997 Russian original by Verbovetsky [A. M. Verbovetskiĭ] and Krasilꞌshchik
[17]P.J. Olver, Applications of Lie Groups to Differential Equations, second edn, Springer-Verlag, New York, Vol. 107, 1993. Graduate Texts in Mathematics
Journal of Nonlinear Mathematical Physics
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© 2020 The Authors. Published by Atlantis and Taylor & Francis
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Cite this article

AU  - Ágota Figula
AU  - Gábor Horváth
AU  - Tamás Milkovszki
AU  - Zoltán Muzsnay
PY  - 2020
DA  - 2020/01/27
TI  - The Lie symmetry group of the general Liénard-type equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 185
EP  - 198
VL  - 27
IS  - 2
SN  - 1776-0852
UR  -
DO  - 10.1080/14029251.2020.1700623
ID  - Figula2020
ER  -