This comment focuses on some properties of the Liénard type equation in the original article and, in particular, on the relation between the condition for linearizability by a point transformation and that for the existence of periodic, isochronous solutions. Differently from what is indicated in the paper we comment upon, these conditions are generally distinct. In most cases, isochronicity does not imply linearizability by a point transformation, and linearizable cases may have non-periodic solutions. After illustrating this point with some examples, we write down explicitly the solution to the linearizable case, which may be of use for the applications.
The second-order, nonlinear ordinary differential equation
is also referred to as the Liénard type equation (LTE hereafter), because of its formal resemblance with the Liénard equation (LE), which is obtained by replacing ẋ2 by ẋ. The LE and the LTE have been widely used in the last few decades to explore basic issues in the theory of nonlinear oscillations, such as the determination of the conditions for periodic solutions, and the characterization of the corresponding period functions.
The paper by Tiwari et al.1 we comment upon (Tiw13 hereafter) provides a classification of the Lie point symmetries of the LTE (see also Ref. 2). Among other cases with more than one symmetry, a special case of (1) is found that admits an eight parameter symmetry group and can consequently be linearized by a point transformation. It is then noted that this linearizable LTE includes isochronous cases, i.e., cases with a periodic solution whose frequency does not depend on the oscillation amplitude. At this point, however, some incorrect conclusions are drawn about the exact relation between linearizability and isochronicity of the LTE. The purpose of this note is to clarify this relation and to offer some additional comments on the related issues.
(1) The abstract of Tiw13 states that “In the case of eight parameter symmetry group, the identified general equation becomes linearizable as well as isochronic,” and similar statements are made in the Introduction and in Sec. IV. This is repeated in Ref. 3, where it is said [see after Eq. (8)] that all the linearizable cases of the LTE are isochronous. Moreover, in the Conclusion of Tiw13, it is said that “We have also deduced the interesting result that the condition for isochronicity of Eq. (2) [i.e., our LTE (1)] is the same as that of the linearizability condition.”
These statements imply that the linearizability and isochronicity conditions for the LTE must coincide, but this is not the case. The linearizability condition states that (1) can be linearized by a point transformation if and only if
where K is an arbitrary constant. Condition (2) happens to have the same form of the constraint on the coefficient functions f and g that appears in a sufficient condition for isochronicity of the LTE derived by Sabatini,4 but there are two important differences: (a) in the latter condition, K needs to be positive definite and (b) other conditions on the coefficient functions need to be satisfied (see Theorem 2 of Ref. 4) to ensure that the origin is a center. Moreover, Sabatini’s condition is just a special case of a more general isochronicity condition derived in Ref. 5, which typically involves a complicated integro-differential constraint on f and g.
Thus, it is clear that the class of the linearizable LTEs does not coincide with that of the isochronous LTEs. The two classes do overlap, and their intersection is formed by the LTEs that satisfy Sabatini’s conditions for isochronicity. Out of this intersection, however, there are isochronous LTEs that are not linearizable by a point transformation, and linearizable LTEs that are not isochronous, and not even periodic. An example of the latter is obtained taking g = x and K = −1 in (2). This yields the LTE
whose general solution is
with A and B arbitrary constants.
(2) That the class of the isochronous LTEs is wider than Sabatini’s class may be illustrated using the results of a classical investigation by Loud.6 Loud showed that quadratic first order systems with a center in the origin may be reduced to the LTE
with D and F constants that satisfy a certain algebraic relation. Note that the form of the coefficient functions in (5) implies that
Then, Loud proved that there are only four couples (D, F) such that (5) has isochronous orbits around the origin,
and derived the corresponding solutions. For the first two couples, the rhs of (6) reduces to unity so that (2) is satisfied with K = 1. Thus, these two isochronous cases belong to Sabatini’s class. The two other isochronous cases do not since, for these cases, the rhs of (6) is not a positive constant, but a polynomial of x. Their coefficient functions satisfy a more complicated, second order differential constraint,
It can be shown that this constraint, together with the conditions for the origin to be a center, defines another subclass of the isochronous class identified by Chouikha et al.5
We note that the coefficient functions of the cases with two and three parameter symmetries found in Tiw13 do not satisfy condition (8). Thus, if these two LTEs include isochronous cases, they must belong to yet another isochronous class.
(3) It is worth mentioning that one of the two examples (B) solved in Sec. V of Tiw13,
is reduced to Loud’s first isochronous case (D = 0, F = 1) by a scaling of the dependent and independent variables [−λx → x, (t/ω0) → t].
(4) Although the paper contains all the information needed to write it, the general solution of the linearizable LTE was not explicitly given in Tiw13. This solution can be written as
where A, B, and ϕ are arbitrary constants. For simple forms of g(x), such that the integral on the lhs can be evaluated, and the resulting function of x can be inverted; this directly yields an explicit solution, without the need of looking for linearizing transformations. As an example, consider the LTE
which is a special case (with a = 0, b = 1) of case 8 in Theorem 3 of Ref. 5. The coefficients of (13) satisfy (2) with K = 1. The integral on the lhs of (11) is readily computed, yielding
and consequently, the solution
which oscillates with unit frequency for any value of the amplitude (the amplitude is ).
(5) The existence of a wide class of isochronous LTEs that cannot be linearized by a point transformation is not surprising because the conditions for isochronicity are usually of a local nature, they just imply that there is a finite region around a center (or a focus) that is filled with isochronous orbits. An exceptional case is that corresponding to Sabatini’s condition, since in this case the center at the origin becomes “global.” The situation is similar to that found for the LE; as noted by Iacono and Russo,7 the only LE with periodic, isochronous solutions that can be linearized by a point transformation is
with λ > 0, but this is just a member of a wide class of LEs admitting isochronous solutions (see Ref. 7 and references therein). Whether there are other types of symmetries underlying these isochronous classes is an interesting open question that deserves to be investigated.
