An algorithmic approach to small limit cycles of nonlinear differential systems: The averaging method revisited

Abstract

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.

Introduction

Bounding the number of limit cycles for systems of polynomial differential equations is a long standing problem in the field of dynamical systems. As is well known [Hilbert (1902); Ilyashenko (2002)], the second part of the 16th Hilbert's problem asks for “the maximal number $H(n)$ and relative configurations of limit cycles” for planar polynomial differential systems of degree n:

$$\dot{x}=f_n(x,y),\dot{y}=g_n(x,y).$$

Solving this problem seems to be hopeless at the present state of knowledge, even for the quadratic systems ($n=2$). While it has not been possible to obtain uniform upper bounds for $H(n)$ in the near future, there has been success in finding lower bounds. Some known results are as follows: Chen and Wang (1979) showed $H(2)\ge 4$ and Li et al. (2009) showed $H(3)\ge 13$. Christopher and Lloyd (1995) proved that $H(n)$ grows at least as rapidly as $n^2\log n$. For the latest development about $H(n)$, we refer the reader to Li (2003); Christopher and Li (2007); Han and Li (2012).

Recall that a limit cycle of system (1) is an isolated periodic orbit. It is the ω-(forward) or α-(backward) limit set of nearby orbits. One classical way of producing limit cycles is by perturbing a differential system which has a center. In this case the perturbed system displays limit cycles that bifurcate, either from the center, or from some of the periodic orbits of the period annulus surrounding the center; see, for instance, Pontrjagin (1934), the book of Christopher and Li (2007), and the hundreds of references quoted there.

In this paper we study the maximal number of small-amplitude limit cycles (or small limit cycles) that bifurcate from the centers of the unperturbed systems. The main technique is based on the averaging method. We point out that the method of averaging is a classic and mature tool for studying isolated periodic solutions of nonlinear differential systems in the presence of a small parameter. The method has a long history that started with the classical works of Lagrange and Laplace, who provided an intuitive justification of the method. The first formalization of this theory was done in 1928 by Fatou. Important practical and theoretical contributions to the averaging method were made in the 1930s by Bogoliubov-Krylov, and in 1945 by Bogoliubov. The ideas of averaging method have extended in several directions for finite and infinite dimensional differentiable systems. We refer to the books of Sanders et al. (2007) and Llibre et al. (2015b) for a modern exposition of this subject.

We remark that most of these previous results developed the averaging method up to first order in a small parameter ε, and at most up to third order. Giné et al. (2013); Llibre et al. (2014) developed the averaging method at any order to study isolated periodic solutions of nonsmooth but continuous differential systems. Recently, the averaging method has also been extended to study isolated periodic solutions of discontinuous differential systems; see Llibre et al. (2015a); Itikawa et al. (2017). In practice, the evaluation of the averaged functions is a computational problem that require powerful computational resources since the computational complexity is exponential in the averaging order. In view of this, our objective in this paper is to present an algorithmic approach to develop the averaging method at any order and to further study periodic solutions of nonlinear continuous differential systems.

There are several other methods beside the averaging method for studying Hopf bifurcations, i.e., small limit cycles that bifurcate from a singular point [Yu and Chen (2008); Han and Yu (2012)]. One of these is the focus values (or Liapunov constants) method. Various algorithms for computing focus values have been developed [Wang (1991); Romanovski (1993); Gasull and Torregrosa (2001); Wang (2004); Yu and Chen (2008)]. Such methods give no qualitative information about the bifurcated limit cycles. In contrast, using averaged functions, we can estimate the size (i.e., radius) of the bifurcated limit cycles as a function of ε for $|\epsilon |>0$ sufficiently small [Benterki and Llibre, 2017, Theorem 2]. It is worth noting that the focus values method (based on the perturbation technique) involves scaling of time and space [Yu (1998); Yu and Chen (2008)]. Chow and Mallet-Paret (1976) noted that the averaging method only involves space scaling; see also our discussion below in the paragraph after Theorem 2 in Section 2.

Overview of Paper. The paper is an essential improvement of our ISSAC'19 paper [Huang and Yap (2019)]. The main improvement is in Section 3, where we present the explicit conditions on the parameters of a perturbed differential system for the vanishing of the unperturbed term of the normal form of averaging (see Corollary 4) and the expressions of the first two averaged functions for the general class of perturbations (see Corollary 6). Lemma 3 together with some remarks in Section 3 are also new improvements. More precisely, the structure of our paper is as follows. In Section 2, we introduce the basic results on the averaging method for planar differential systems before presenting our main results in Section 3. We give our algorithms and briefly describe their implementation in Maple in Section 4. Its application is illustrated in Section 5 using several examples including a cubic polynomial differential system known as Collins First Form and a class of generalized Kukles polynomial differential systems of degree 6. Some results on quadratic differential systems and the relations between the averaging method and the Melnikov function method for studying the limit cycles are also given in Section 5. The Maple code of our three algorithms can be downloaded from https://github.com/Bo-Math/limitcycles. We end with some discussions in Section 6.