Critical Periods of Perturbations of Reversible Rigidly Isochronous Centers

Abstract

We study the problem of bifurcation of critical periods of a time-reversible polynomial system of degree 𝑛. We first present a new method to find the number of zeros of the period function. Then applying our results, we study the number of critical periods for some polynomial systems and obtain new results.

1. Introduction

Consider a two-dimensional analytic real differential system of the form 𝑥̇=𝐹𝑥,𝑦,𝑦̇=𝐺𝑥,𝑦, (1) where we suppose there is a nondegenerate center at the origin 𝑂. Let 𝐿(ℎ) denote the orbit passing through the point (ℎ,0) with ℎ>0 of (1) and 𝑇(ℎ) denote its period. As we all know, the isolated zeros of derivative of 𝑇(ℎ) are named critical periods. An interesting problem is to investigate the number of the critical periods. This is an important problem in the research of period functions and a lot of results have been obtained for polynomial differential systems, for example, monotonicity [1–8], finiteness of critical periods [9, 10] and isochronicity [11–13], and local bifurcation of critical periods [14]. The number of critical periods was also discussed in [15–17] for perturbations of isochronous vector fields.

Recently, the authors in [18] studied the following system: 𝑥̇=−𝑦+𝐹_𝑛𝑥,𝑦+𝜀^𝑚_𝑖=2𝑃_𝑖,̇𝑥,𝑦𝑦=𝑥+𝐾_𝑛𝑥,𝑦+𝜀^𝑚_𝑖=2𝑄_𝑖𝑥,𝑦, (2) where 𝑛,𝑚≥2, 0<𝜀≪1, 𝐹_𝑛𝑥,𝑦=^[𝑛/2]_𝑗=1𝑎_{𝑛+1−2𝑗,2𝑗−1}𝑥^{𝑛+1−2𝑗}𝑦^2𝑗−1,𝑃_𝑖𝑥,𝑦=^[(𝑖+1)/2]_𝑗=1𝑏_{𝑖+1−2𝑗,2𝑗−1}𝑥^{𝑖+1−2𝑗}𝑦^2𝑗−1,𝐾_𝑛𝑥,𝑦=^[𝑛/2]_𝑗=1𝑎_{𝑛+1−2𝑗,2𝑗−1}𝑥^𝑛−2𝑗𝑦^2𝑗,𝑄_𝑖𝑥,𝑦=^[(𝑖+2)/2]_𝑗=1𝑐_{𝑖+2−2𝑗,2𝑗−2}𝑥^{𝑖+2−2𝑗}𝑦^2𝑗−2. (3) Under the conditions above, system (2) is time-reversible and hence has a center at the origin.

As shown in [15–20], for 0<𝜀≪1, the period function 𝑇(ℎ,𝜀) of system (2) can be written as 𝑇(ℎ,𝜀)=𝑇₀+^+∞_𝑖=1𝑇_𝑖(ℎ)𝜀^𝑖, (4) where 𝑇₀ is a positive constant. The authors [18] gave expressions of 𝑇₁(ℎ) and 𝑇₂(ℎ). It is also proved in [18] that one critical period can appear for the case 𝑛=𝑚=2 and two critical periods can be found for the case 𝑛=𝑚=3.

In this paper, we consider the following system: 𝑥̇=−𝑦+𝐹_𝑛𝑥,𝑦+𝜆𝐹_𝑙𝑃𝑥,𝑦+𝜀_𝑚𝑥,𝑦+𝜆𝑃_𝑘,̇𝐾𝑥,𝑦𝑦=𝑥+_𝑛𝑥,𝑦+𝜆𝐾_𝑙𝑄𝑥,𝑦+𝜀_𝑚𝑥,𝑦+𝜆𝑄_𝑘𝑥,𝑦, (5) where 𝐹_𝑛𝑥,𝑦=^[𝑛/2]_𝑗=1𝑎_{𝑛+1−2𝑗,2𝑗−1}𝑥^{𝑛−2𝑗+1}𝑦^2𝑗−1,𝐹_𝑙𝑥,𝑦=^[𝑙/2]_𝑗=1𝑎_{𝑙+1−2𝑗,2𝑗−1}𝑥^{𝑙−2𝑗+1}𝑦^2𝑗−1,𝑃_𝑚𝑥,𝑦=^𝑚_𝑖=2^[(𝑖+1)/2]_𝑗=1𝑏_{𝑖+1−2𝑗,2𝑗−1}𝑥^{𝑖+1−2𝑗}𝑦^2𝑗−1,𝑃_𝑘𝑥,𝑦=^𝑘_𝑖=2^[(𝑖+1)/2]_𝑗=1𝑏_{𝑖+1−2𝑗,2𝑗−1}𝑥^{𝑖+1−2𝑗}𝑦^2𝑗−1,𝐾_𝑛𝑥,𝑦=^[𝑛/2]_𝑗=1𝑎_{𝑛+1−2𝑗,2𝑗−1}𝑥^𝑛−2𝑗𝑦^2𝑗,𝐾_𝑙𝑥,𝑦=^[𝑙/2]_𝑗=1𝑎_{𝑙+1−2𝑗,2𝑗−1}𝑥^𝑙−2𝑗𝑦^2𝑗,𝑄_𝑚𝑥,𝑦=^𝑚_𝑖=2^[(𝑖+2)/2]_𝑗=1𝑐_{𝑖+2−2𝑗,2𝑗−2}𝑥^{𝑖+2−2𝑗}𝑦^2𝑗−2,𝑄_𝑘𝑥,𝑦=^𝑘_𝑖=2^[(𝑖+2)/2]_𝑗=1𝑐_{𝑖+2−2𝑗,2𝑗−2}𝑥^{𝑖+2−2𝑗}𝑦^2𝑗−2, (6)(7)𝑚, 𝑛, 𝑙, 𝑘≥2, and both 𝜀 and 𝜆 are small parameters. From the above expressions, we know that the system (5) satisfies the following properties: (i) it is a time-reversible system, and the perturbations with 𝜀 are of general form; (ii) for all 𝜀, the origin is a center, and for 𝜀=0, the origin is isochronous since there exist polynomials 𝜑_𝑛(𝑥,𝑦) and 𝜑_𝑙(𝑥,𝑦) such that 𝐹_𝑛(𝑥,𝑦)=𝑥𝜑_𝑛(𝑥,𝑦), 𝐾_𝑛(𝑥,𝑦)=𝑦𝜑_𝑛(𝑥,𝑦), 𝐹_𝑙(𝑥,𝑦)=𝑥𝜑_𝑙(𝑥,𝑦), 𝐾_𝑙(𝑥,𝑦)=𝑦𝜑_𝑙(𝑥,𝑦).

Let 𝑇(ℎ,𝜀,𝜆) denote the period of the periodic orbit of system (5) passing through the point (ℎ,0). Then it has the following expansion: 𝑇(ℎ,𝜀,𝜆)=𝑇₀+^+∞_𝑖=1𝑇_𝑖(ℎ,𝜆)𝜀^𝑖, (8) where 𝑇_𝑖(ℎ,𝜆)=𝑇_𝑖0(ℎ)+^+∞_𝑗=1𝑇_𝑖𝑗(ℎ)𝜆^𝑗, (9) and since (5) becomes (2) as 𝜆=0, we have 𝑇(ℎ,𝜀,0)=𝑇(ℎ,𝜀).

Next section, we give some preliminary lemmas.

2. Preliminary Lemmas

In this section, we cite some results obtained in [18].

Lemma 1 (see [18]). For system (5), 𝑇₁(ℎ,0)=𝑇₁₀(ℎ)=−^𝑚−1_𝑖=10^2𝜋ℎ^𝑖𝑆_𝑖(𝜃)1+(1−𝑛)ℎ^𝑛−1∫^𝜃₀𝐺₀(𝛼)𝑑𝛼^{𝑖/(𝑛−1)}𝑇𝑑𝜃,₂(ℎ,0)=𝑇₂₀(ℎ)=₀^{2𝜋𝑚−1}_𝑖=1𝑟^𝑖₀(𝜃,ℎ)𝑆_𝑖(𝜃)²−^𝑚−1_𝑖=1𝑖𝑟₀^𝑖−1(𝜃,ℎ)𝑟₁(𝜃,ℎ)𝑆_𝑖(𝜃)𝑑𝜃, (10)(11) where 𝑟₀(𝜃,ℎ)=ℎ^1−𝑛+(1−𝑛)^𝜃₀𝐺₀(𝛼)𝑑𝛼^1/(1−𝑛),𝐺₀(𝜃)=^[𝑛/2]_𝑗=1𝑎_{𝑛+1−2𝑗,2𝑗−1}cos^𝑛−2𝑗(𝜃)sin^2𝑗−1(𝜃),𝑆_𝑖(𝜃)=^[(𝑖+1)/2]_𝑗∑₌₀𝑐_{𝑖+1−2𝑗,2𝑗}−𝑏_{𝑖+2−2𝑗,2𝑗−1}×cos^{𝑖+2−2𝑗}(𝜃)sin^2𝑗(𝜃)−𝑏_0,𝑖+1sin^𝑖+2(𝜃),𝑖𝑒𝑣𝑒𝑛,_{[(𝑖+1)/2]𝑗}∑₌₀𝑐_{𝑖+1−2𝑗,2𝑗}−𝑏_{𝑖+2−2𝑗,2𝑗−1}×cos^{𝑖+2−2𝑗}(𝜃)sin^2𝑗(𝜃)𝑟,𝑖𝑜𝑑𝑑,₁(𝜃,ℎ)=^𝜃₀^𝑚−1_𝑖=1𝑟₀^𝑖+1(𝛼,ℎ)𝐺_𝑖(𝛼)−𝐺₀(𝛼)×^𝑚−1_𝑖=1𝑟₀^𝑛+𝑖(𝛼,ℎ)𝑆_𝑖(𝛼)⋅exp^𝜃_𝛼𝑛𝑟₀^𝑛−1𝛽,ℎ𝐺₀𝛽𝑑𝛽𝑑𝛼, (12)(13)(14)(15) moreover, if 𝑛 is odd, 𝑆₀(𝜃)≡0, and 𝑇₁(ℎ,0)=−^{[(𝑚−1)/2]}_𝑘=10^2𝜋ℎ^2𝑘𝑆_2𝑘(𝜃)1+(1−𝑛)ℎ^𝑛−1∫^𝜃₀𝐺₀(𝛼)𝑑𝛼^{2𝑘/(𝑛−1)}𝑑𝜃. (16)

Lemma 2 (see [18]). Let 𝑛=𝑚=2 and 𝜆=0 in (5). Then 𝑇₁(ℎ,0)=𝐷₀₂−3𝐷₂₀𝜋+2𝐷₀₂−𝐷₂₀𝜋ℎ²+4𝐷₂₀−𝐷₀₂𝜋ℎ+2(1−ℎ)𝐷₂₀ℎ²+2𝐷₀₂−𝐷₂₀ℎ+𝐷₂₀−𝐷₀₂𝜋ℎ²√,1−2ℎ (17) where ℎ∈(0,1/2) and 𝐷_𝑖𝑗 are constants depending on the coefficients appearing in 𝐹_𝑛, 𝐾_𝑛, 𝑃_𝑚, and 𝑄_𝑚. Furthermore, for sufficiently small 𝜀,(i)the center 𝑂 preserves the isochronicity when 𝐷₂₀=𝐷₀₂=0; period function 𝑇(ℎ,𝜀,0) is increasing (resp., decreasing) for ℎ∈(0,1/2) when 𝐷₂₀=0 and 𝐷₀₂>0 (resp., < 0);(ii)there is at most one critical period in (0,1/2) when 𝐷₂₀≠0. Moreover, there is exactly one critical period in (0,1/2) if and only if 𝑟_∗𝐷∶=⁰²𝐷₂₀<−3. (18)

Lemma 3 (see [18]). Let 𝑛=𝑚=3 and 𝜆=0 in (5). Then 𝑇₁(ℎ,0)=1−1−ℎ²1+1−ℎ²1−ℎ²𝜋×2𝐵₀₃−2𝐷₃₀+𝐵₀₃+𝐵₂₁−𝐷₁₂−𝐷₃₀×1−ℎ²+2𝐷₃₀ℎ², (19) where ℎ∈(0,1) and 𝐵_𝑖𝑗, 𝐷_𝑖𝑗 are constants depending on the coefficients appearing in 𝐹_𝑛, 𝐾_𝑛, 𝑃_𝑚, and 𝑄_𝑚. Furthermore, for sufficiently small 𝜀, one has the following results.(i)The center 𝑂 preserves the isochronicity when 𝐷₂₀=𝐵₁₁−𝐷₀₂𝐷=0,₃₀=𝐵₀₃=𝐵₂₁−𝐷₁₂=0. (20)(ii)If (20) do not hold, there are at most two critical periods in (0,1) and the maximum is achievable.

3. Main Results

As in [18], one can make the polar coordinates 𝑥=𝑟cos(𝜃), 𝑦=𝑟sin(𝜃), so that system (5) becomes 𝑟̇=𝑟^𝑛𝐺₀(𝜃)+𝜆𝑟^𝑙𝐺₀(𝜃)+𝜀^𝑚−1_𝑖=1𝑟^𝑖+1𝐺_𝑖(𝜃)+𝜆^𝑘−1_𝑖=1𝑟^𝑖+1𝐺_𝑖(𝜃,̇)𝜃=1+𝜀^𝑚−1_𝑖=1𝑟^𝑖𝑆_𝑖(𝜃)+𝜆^𝑘−1_𝑖=1𝑟^𝑖𝑆_𝑖(𝜃), (21) where 𝐺₀(𝜃) and 𝑆_𝑖(𝜃) are given by (13) and (14), 𝐺₀(𝜃)=^[𝑙/2]_𝑗=1𝑎_{𝑙+1−2𝑗,2𝑗−1}cos^𝑙−2𝑗(𝜃)sin^2𝑗−1(𝜃),𝐺_𝑖(𝜃)=^[(𝑖+2)/2]_𝑗∑₌₁𝑏_{𝑖+2−2𝑗,2𝑗−1}+𝑐_{𝑖+3−2𝑗,2𝑗−2}×cos^{𝑖+3−2𝑗}(𝜃)sin^2𝑗−1(𝜃),𝑖even,_{[(𝑖+2)/2]𝑗}∑₌₁𝑏_{𝑖+2−2𝑗,2𝑗−1}+𝑐_{𝑖+3−2𝑗,2𝑗−2}×cos^{𝑖+3−2𝑗}(𝜃)sin^2𝑗−1(𝜃)+𝑐_0,𝑖+1sin^𝑖+2(𝜃),𝑖odd,𝐺_𝑖(𝜃)=^[(𝑖+2)/2]_𝑗∑₌₁𝑏_{𝑖+2−2𝑗,2𝑗−1}+𝑐_{𝑖+3−2𝑗,2𝑗−2}×cos^{𝑖+3−2𝑗}(𝜃)sin^2𝑗−1(𝜃),𝑖even,_{[(𝑖+2)/2]𝑗}∑₌₁𝑏_{𝑖+2−2𝑗,2𝑗−1}+𝑐_{𝑖+3−2𝑗,2𝑗−2}×cos^{𝑖+3−2𝑗}(𝜃)sin^2𝑗−1(𝜃)+𝑐_0,𝑖+1sin^𝑖+2(𝜃),𝑖odd,𝑆_𝑖(𝜃)=^[(𝑖+1)/2]_𝑗∑₌₀𝑐_{𝑖+1−2𝑗,2𝑗}−𝑏_{𝑖+2−2𝑗,2𝑗−1}×cos^{𝑖+2−2𝑗}(𝜃)sin^2𝑗(𝜃)−𝑏_0,𝑖+1sin^𝑖+2(𝜃),𝑖even,_{[(𝑖+1)/2]𝑗}∑₌₀𝑐_{𝑖+1−2𝑗,2𝑗}−𝑏_{𝑖+2−2𝑗,2𝑗−1}×cos^{𝑖+2−2𝑗}(𝜃)sin^2𝑗(𝜃),𝑖odd. (22)(23)(24)(25) Obviously, 𝜃̇≡1 when 𝜀=0, which implies that the unperturbed system (5) |_𝜀=0 has an isochronous center at 𝑂 which is called a rigidly or uniformly isochronous center. We have the following fundamental result.

Theorem 4. Let (8) and (9) hold. Then, for system (5), one has 𝑇₁₁(ℎ)=−^𝑚−1_𝑖=10^2𝜋𝑖𝑟^𝑖−1₀₀(𝜃,ℎ)𝑟₀₁(𝜃,ℎ)𝑆_𝑖(𝜃)−𝑑𝜃_{𝑘−1𝑖=102𝜋}𝑟^𝑖₀₀(𝜃,ℎ)𝑆_𝑖(𝜃)𝑑𝜃, (26) where 𝑟₀₀()=ℎ𝜃,ℎ1+(1−𝑛)ℎ^𝑛−1∫^𝜃₀𝐺₀(𝛼)𝑑𝛼^1/(𝑛−1),𝑟₀₁(𝜃,ℎ)=^𝜃₀𝑟^𝑙₀₀(𝛼,ℎ)𝐺₀(𝛼)⋅exp^𝜃_𝛼𝑛𝑟^𝑛−1₀₀𝛽,ℎ𝐺₀𝛽𝑑𝛽𝑑𝛼. (27)(28)

Proof. We follow the idea of proving Lemma 1 which is Theorem 2.1 given in [18].
From (21), we have 𝑑𝑟=𝑟𝑑𝜃^𝑛𝐺₀(𝜃)+𝜆𝑟^𝑙𝐺₀(𝜃)+𝜀^𝑚−1_𝑖=1𝑟^𝑖+1𝐺_𝑖(𝜃)+𝜆^𝑘−1_𝑖=1𝑟^𝑖+1𝐺_𝑖(𝜃×)1+𝜀^𝑚−1_𝑖=1𝑟^𝑖𝑆_𝑖(𝜃)+𝜆^𝑘−1_𝑖=1𝑟^𝑖𝑆_𝑖(𝜃)⁻¹. (29)
Let 𝑟(𝜃,𝜀,𝜆,ℎ) be the solution of the above equation satisfying the initial condition 𝑟(0,𝜀,𝜆,ℎ)=ℎ. It can be written as a series of 𝜀𝑟(𝜃,𝜀,𝜆,ℎ)=^+∞_𝑖=0𝑟_𝑖(𝜃,𝜆,ℎ)𝜀^𝑖, (30) where 𝑟₀(0,𝜆,ℎ)=ℎ, 𝑟_𝑖(0,𝜆,ℎ)=0, 𝑖≥1.
Substituting (30) into differential equation (29) and comparing the coefficients of 𝜀⁰, we have 𝑑𝑟𝑑𝜃⁰(𝜃,𝜆,ℎ)=𝑟^𝑛₀(𝜃,𝜆,ℎ)𝐺₀(𝜃)+𝜆𝑟^𝑙₀(𝜃,𝜆,ℎ)𝐺₀(𝜃). (31) We can write 𝑟₀(𝜃,𝜆,ℎ)=𝑟₀₀(𝜃,ℎ)+𝑟₀₁(𝜃,ℎ)𝜆+⋯, where 𝑟₀₀(0,ℎ)=ℎ, 𝑟_0𝑖(0,ℎ)=0, 𝑖≥1. Then, substituting 𝑟₀(𝜃,𝜆,ℎ) into (31), then comparing the coefficients of 𝜆⁰ and 𝜆, we can obtain 𝑑𝑟₀₀(𝜃,ℎ)𝑑𝜃=𝑟^𝑛₀₀(𝜃,ℎ)𝐺₀(𝜃),𝑑𝑟𝑑𝜃⁰¹(𝜃,ℎ)=𝑛𝑟^𝑛−1₀₀(𝜃,ℎ)𝑟₀₁(𝜃,ℎ)𝐺₀(𝜃)+𝑟^𝑙₀₀(𝜃,ℎ)𝐺₀(𝜃). (32) Solving the above ODEs associated with the initial values 𝑟₀₀(0,ℎ)=ℎ and 𝑟₀₁(0,ℎ)=0, we can get (27) and (28). By (21) we have 𝑇(ℎ,𝜀,𝜆)=₀^2𝜋1𝜃̇𝑑𝜃=₀^2𝜋𝑑𝜃1+𝜀∑^𝑚−1_𝑖=1𝑟^𝑖𝑆_𝑖(𝜃)+𝜆∑^𝑘−1_𝑖=1𝑟^𝑖𝑆_𝑖(𝜃)=2𝜋−₀^{2𝜋𝑚−1}_𝑖=1𝑟^𝑖₀(𝜃,𝜆,ℎ)𝑆_𝑖(𝜃)+𝜆^𝑘−1_𝑖=1𝑟^𝑖₀(𝜃,𝜆,ℎ)𝑆_𝑖(𝜃𝜀)𝑑𝜃⋅𝜀+𝑂². (33) Therefore, 𝑇₁(ℎ,𝜆)=−₀^{2𝜋𝑚−1}_𝑖=1𝑟^𝑖₀(𝜃,𝜆,ℎ)𝑆_𝑖(𝜃)+𝜆^𝑘−1_𝑖=1𝑟^𝑖₀(𝜃,𝜆,ℎ)𝑆_𝑖(𝜃)𝑑𝜃=−^𝑚−1_𝑖=10^2𝜋𝑟^𝑖₀₀(𝜃,ℎ)𝑆_𝑖(𝜃)−𝑑𝜃_{𝑚−1𝑖=102𝜋}𝑖𝑟^𝑖−1₀₀(𝜃,ℎ)𝑟₀₁(𝜃,ℎ)𝑆_𝑖(𝜃)+𝑑𝜃_{𝑘−1𝑖=102𝜋}𝑟^𝑖₀₀(𝜃,ℎ)𝑆_𝑖(𝜃)𝜆𝜆𝑑𝜃+𝑂². (34) It implies that 𝑇₁₀(ℎ)=−^𝑚−1_𝑖=10^2𝜋𝑟^𝑖₀₀(𝜃,ℎ)𝑆_𝑖(𝜃)𝑇𝑑𝜃,₁₁(ℎ)=−^𝑚−1_𝑖=10^2𝜋𝑖𝑟^𝑖−1₀₀(𝜃,ℎ)𝑟₀₁(𝜃,ℎ)𝑆_𝑖(𝜃)+𝑑𝜃_{𝑘−1𝑖=102𝜋}𝑟^𝑖₀₀(𝜃,ℎ)𝑆_𝑖(𝜃)𝑑𝜃. (35) Thus, (26) is proved. This ends the proof of Theorem 4.

With the same method as for 𝑟₀₁(𝜃,ℎ), we can compute 𝑟_0𝑖(𝜃,ℎ) for 𝑖≥2 and give an expression of 𝑇_1𝑖(ℎ) for 𝑖≥2 from the proof of Theorem 4, which are omitted here.

Define 𝑀₀(ℎ)=𝑑𝑇₁₀(ℎ)𝑑ℎ,𝑀₁(ℎ)=𝑑𝑇₁₁(ℎ).𝑑ℎ (36)

Another fundamental result is as follows.

Theorem 5. Let 𝑇(ℎ,𝜀,𝜆) be defined as before with ℎ∈(0,𝐾). Then for 0<|𝜀|≪|𝜆|≪1, one has the following. (i) The period function 𝑇(ℎ,𝜀,𝜆) of system (5) is increasing (resp., decreasing) in ℎ∈(0,𝐾) if 𝑀₀(ℎ)>0 (resp., < 0) for all ℎ∈(0,𝐾). (ii) If 𝑀₀(ℎ) is not identically zero, the number of critical periods of  𝑇(ℎ,𝜀,𝜆) in (0,𝐾) is not more than the number of zeros (take the multiplicity into consideration) of 𝑀₀(ℎ) in (0,𝐾). And there are exactly 𝑘 critical periods if 𝑀₀(ℎ) has exactly 𝑘 simple positive zeros.  (iii) If 𝑀₀(ℎ)≡0, the number of critical periods of 𝑇(ℎ,𝜀,𝜆) in (0,𝐾) is not larger than the number of zeros of 𝑀₁(ℎ) in (0,𝐾). And 𝑘 critical periods can appear if 𝑀₁(ℎ) has 𝑘 simple zeros. Similarly, the period function 𝑇(ℎ,𝜀,𝜆) of (5) is increasing (resp., decreasing) in ℎ∈(0,𝐾) if 𝑀₁(ℎ)>0 (resp., < 0).

Proof. From (8) and (9) 𝜕𝑇(ℎ,𝜀,𝜆)=𝜕ℎ^+∞_𝑖=1𝜕𝑇_𝑖(ℎ,𝜆)𝜀𝜕ℎ^𝑖(),Γ()==𝜀Γℎ,𝜀,𝜆ℎ,𝜀,𝜆𝜕𝑇₁(ℎ,𝜆)+𝜕ℎ^+∞_𝑖=1𝜕𝑇_𝑖+1(ℎ,𝜆)𝜀𝜕ℎ^𝑖,𝜕𝑇₁(ℎ,𝜆)=𝜕ℎ^+∞_𝑖=0𝑑𝑇_1𝑖(ℎ)𝜆𝑑ℎ^𝑖=𝑀₀(ℎ)+^+∞_𝑖=1𝑑𝑇_1𝑖(ℎ)𝜆𝑑ℎ^𝑖=Γ₁(ℎ,𝜆). (37)
If 𝑀₀(ℎ)>0, ℎ∈(0,𝐾), then there exists a 𝜆₀ such that for 0<|𝜆|<𝜆₀, 𝜕𝑇₁(ℎ,𝜆)/𝜕ℎ has the same sign with 𝑑𝑇₁₀(ℎ)/𝑑ℎ. For 0<𝜀≪𝜆, the sign of the function 𝜕𝑇(ℎ,𝜀,𝜆)/𝜕ℎ is the same as 𝜕𝑇₁(ℎ,𝜆)/𝜕ℎ. Then the conclusion (i) is proved.
Further, suppose that ℎ₁,ℎ₂,…,ℎ_𝑘 are the 𝑘 zeros of 𝑀₀(ℎ) with the multiplicity 𝑚₁,𝑚₂,…,𝑚_𝑘, respectively. We only need to prove that the number of critical periods of 𝑇(ℎ,𝜀,𝜆) is less than or equal to ∑^𝑘_𝑗=1𝑚_𝑗. For the purpose, it suffices to prove that 𝜕𝑇(ℎ,𝜀,𝜆)/𝜕ℎ has at most ∑^𝑘_𝑗=1𝑚_𝑗 roots. Thus, we only need to prove that 𝜕𝑇(ℎ,𝜀,𝜆)/𝜕ℎ has at most 𝑚_𝑗 zeros near ℎ_𝑗, 𝑗=1,2,…,𝑘. At first, we prove that 𝜕𝑇₁(ℎ,𝜆)/𝜕ℎ has at most 𝑚_𝑗 zeros near ℎ_𝑗 for 𝜆 small. If it is not the case, then 𝜕𝑇₁(ℎ,𝜆)/𝜕ℎ has at least 𝑚_𝑗+1 zeros near ℎ_𝑗; that is, there exists 𝜆_𝑛→0 such that 𝜕𝑇₁(ℎ,𝜆_𝑛)/𝜕ℎ has at least 𝑚_𝑗+1 zeros ℎ_𝑖,𝑛  (𝑖=1,2,…,𝑚_𝑗+1) near ℎ_𝑗, where ℎ_𝑖,𝑛→ℎ_𝑗  (𝑛→∞). By Rolls theorem, 𝜕²𝑇₁(ℎ,𝜆_𝑛)/𝜕ℎ² has at least 𝑚_𝑗 zeros near ℎ_𝑗,…, and  𝜕^𝑚𝑗+1𝑇₁(ℎ,𝜆_𝑛)/𝜕ℎ^𝑚𝑗+1 has at least one zero ℎ^∗_𝑖,𝑛 near ℎ_𝑗. Thus, 𝜕^𝑚𝑗+1𝑇₁(ℎ^∗_𝑖,𝑛,𝜆_𝑛)/𝜕ℎ^𝑚𝑗+1=0. Letting 𝑛→∞, we have 𝜕^𝑚𝑗+1𝑇₁(ℎ_𝑗,0)/𝜕ℎ^𝑚𝑗+1=𝜕^𝑚𝑗𝑀₀(ℎ_𝑗)/𝜕ℎ^𝑚𝑗=0. This is a contradiction. Thus, 𝜕𝑇₁(ℎ,𝜆)/𝜕ℎ has at most 𝑚_𝑗 zeros near ℎ_𝑗. Using the same method, for sufficiently small 𝜀, Γ(ℎ,𝜀,𝜆) or 𝜕𝑇/𝜕ℎ has at most 𝑚_𝑗 zeros near ℎ_𝑗. Thus, 𝑇(ℎ,𝜀,𝜆) has at most 𝑚_𝑗 critical periods for each ℎ_𝑗. Thus, we have proved the first part of (ii).
For the second part of (ii), assume that ℎ₁,ℎ₂,…,ℎ_𝑘 are the 𝑘 simple zeros of 𝑀₀(ℎ); that is, 𝑀₀(ℎ_𝑗)=0 and 𝑀^′₀(ℎ_𝑗)≠0, 𝑗=1,2,…,𝑘. Then Γ₁ℎ_𝑗,0=𝜕𝑇₁(ℎ,𝜆)𝜕ℎ_{(ℎ,𝜆)=(ℎ𝑗,0)}=0,𝜕Γ₁(ℎ,𝜆)𝜕ℎ_{(ℎ,𝜆)=(ℎ𝑗,0)}=𝜕²𝑇₁(ℎ,𝜆)𝜕ℎ²_{(ℎ,𝜆)=(ℎ𝑗,0)}≠0. (38) By the Implicit Function Theorem, there is a unique function ℎ_𝑗(𝜆) such that Γ₁(ℎ_𝑗(𝜆),𝜆)=0 and ℎ_𝑗(0)=ℎ_𝑗, 𝑗=1,2,…,𝑘. Therefore ℎ₁(𝜆),ℎ₂(𝜆),…,ℎ_𝑘(𝜆) are 𝑘 simple zeros of 𝜕𝑇₁(ℎ,𝜆)/𝜕ℎ. By the same method, there is a unique function ℎ^∗_𝑗(𝜀,𝜆) such that Γ(ℎ^∗_𝑗(𝜀,𝜆),𝜀)=0, and ℎ^∗_𝑗(0,𝜆)=ℎ_𝑗(𝜆). The second part of (ii) is proved.
When 𝑀₀≡0, then 𝜕𝑇₁(ℎ,𝜆)𝜕ℎ=Γ₁(ℎ,𝜆)=^+∞_𝑖=1𝑑𝑇_1𝑖(ℎ)𝜆𝑑ℎ^𝑖=𝑑𝑇₁₁(ℎ)𝑑ℎ𝜆+^+∞_𝑖=2𝑑𝑇_1𝑖(ℎ)𝜆𝑑ℎ^𝑖=𝜆Γ₁(),Γℎ,𝜆₁(ℎ,𝜆)=𝑀₁(ℎ)+^+∞_𝑖=1𝑑𝑇_1,𝑖+1(ℎ)𝜆𝑑ℎ^𝑖. (39) Then, we can prove conclusion (iii) in the same way as proving conclusion (ii). The proof is completed.

4. Application

In this section, we apply Theorems 4 and 5 to the cases 𝑛=𝑚=2, 𝑙=𝑘=3; 𝑛=𝑚=3, 𝑙=𝑘=4, and 𝑛=𝑙=2, 𝑚=𝑘=3, respectively.

Case 1 (𝑛=𝑚=2,  𝑙=𝑘=3). In this case, system (5) becomes 𝑥̇=−𝑦+𝑎₁₁𝑥𝑦+𝜆𝑎₂₁𝑥²𝑦+𝜀𝑏₁₁𝑥𝑦+𝜆𝑏₁₁𝑥𝑦+𝑏₂₁𝑥²𝑦+𝑏₀₃𝑦³,𝑦̇=𝑥+𝑎₁₁𝑦²+𝜆𝑎₂₁𝑥𝑦²+𝜀𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝜆𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝑐₃₀𝑥³+𝑐₁₂𝑥𝑦², (40) where 𝑎₁₁≠0. For system (40), 𝑂 is a center because of the symmetry.
We only need to consider the case of 𝑎₁₁>0 in (40); otherwise, we can use the transformation 𝑥→−𝑥,𝑦→−𝑦 to change (40) into the same form with opposite signs to the coefficients. Further, suppose 𝑎₁₁=1, otherwise, system (40) can be simplified as the form of system (41) by the transformation 𝑢=𝑎₁₁𝑥, 𝑣=𝑎₁₁𝑦. In this case, system (40) becomes 𝑥̇=−𝑦+𝑥𝑦+𝜆𝑎₂₁𝑥²𝑦+𝜀𝑏₁₁𝑥𝑦+𝜆𝑏₁₁𝑥𝑦+𝑏₂₁𝑥²𝑦+𝑏₀₃𝑦³,𝑦̇=𝑥+𝑦²+𝜆𝑎₂₁𝑥𝑦²+𝜀𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝜆𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝑐₃₀𝑥³+𝑐₁₂𝑥𝑦². (41) It is easy to conclude that system (41) ∣_𝜀=0 has a first integral of the form 𝐻=2𝑥,𝑦1+𝜆𝑎₂₁𝑥+1+𝜆𝑎₂₁𝑦²−11+𝜆𝑎₂₁²1−1+𝜆𝑎₂₁𝑥². (42)

Theorem 6. For system (41), one has (i)𝑇₁₀(ℎ)=𝑐₀₂−𝑏₁₁−3𝑐₂₀𝜋+2𝑐₀₂−𝑏₁₁−𝑐₂₀𝜋ℎ²+4𝑐₂₀−𝑐₀₂+𝑏₁₁𝜋ℎ+2(1−ℎ)𝑐₂₀ℎ²+2𝑐₀₂−𝑏₁₁−𝑐₂₀ℎ+𝑐₂₀−𝑐₀₂+𝑏₁₁𝜋×ℎ²√1−2ℎ⁻¹, (43) where ℎ∈(0,1/2). For sufficiently small 𝜀 and 𝜆, there is at most one critical period if 𝑀₀(ℎ) is not identically zero and 𝑐₂₀≠0; there is exactly one critical period if and only if 𝜌∶=(𝑐₀₂−𝑏₁₁)/𝑐₂₀<−3. Otherwise, the period function 𝑇(ℎ,𝜀,𝜆) is increasing for ℎ∈(0,1/2) (resp., decreasing) if 𝑐₂₀>0 (resp., < 0); (ii) if 𝑀₀(ℎ)=𝑇^′₁₀(ℎ)≡0, then for 0<ℎ≪1, 𝑇₁₁(ℎ)=−𝐴₂ℎ²+𝐴₃ℎ³+𝐴₄ℎ⁴+𝐴₅ℎ⁵+𝐴₆ℎ⁶+𝑂ℎ⁶𝜋, (44) where 𝐴₂1=−4𝑐₀₂−𝑏₁₁−34𝑐₂₀+34𝑐₃₀+14𝑐₁₂−𝑏₁₁−34𝑏₀₃,𝐴₃=2𝐴₂,𝐴₄=−238𝑐₂₀+338𝑐₃₀−78𝑐₀₂−𝑏₁₁−218𝑏₀₃+98𝑐₁₂−𝑏₂₁,𝐴₅9=−2𝑏₀₃−112𝑐₂₀+52𝑐₁₂−𝑏₂₁+212𝑐₃₀−32𝑐₀₂−𝑏₁₁,𝐴₆=−67564𝑐₂₀−9916𝑏₀₃−16564𝑐₀₂−𝑏₁₁+32316𝑐₃₀+6916𝑐₁₂−𝑏₂₁. (45)There can appear three critical periods in ℎ∈(0,1/2) if 𝑇₁₁(ℎ) is not identically zero.

Proof. For system (41), from (13), (14), (22)–(25), we have 𝐺₀(𝜃)=sin(𝜃),𝐺₀(𝜃)=𝑎₂₁(𝜃)(𝜃),𝐺cossin₁(𝜃)=𝑐₂₀+𝑏₁₁cos²(𝜃)sin(𝜃)+𝑐₀₂sin³(𝜃),𝐺₁(𝜃)=𝑏₁₁+𝑐₂₀cos²(𝜃)sin(𝜃)+𝑐₀₂sin³(𝜃),𝐺₂(𝜃)=𝑏₂₁+𝑐₃₀cos³(𝜃)(𝜃)+sin𝑏₀₃+𝑐₁₂cos(𝜃)sin³(𝜃),𝑆₁(𝜃)=𝑐₂₀cos³(𝜃)+𝑐₀₂−𝑏₁₁cos(𝜃)sin²(𝜃),𝑆₁(𝜃)=𝑐₂₀cos³(𝜃)+𝑐₀₂−𝑏₁₁cos(𝜃)sin²(𝜃),𝑆₂(𝜃)=𝑐₃₀cos⁴(𝜃)+𝑐₁₂−𝑏₂₁cos²(𝜃)sin²(𝜃)−𝑏₀₃sin⁴(𝜃). (46) From (27), we have 𝑟₀₀()=ℎ𝜃,ℎ(𝜃).1−ℎ+ℎcos (47) From (10), we have 𝑇₁₀(ℎ)=−^𝑚−1_𝑖=10^2𝜋ℎ^𝑖𝑆_𝑖(𝜃)1+(1−𝑛)ℎ^𝑛−1∫^𝜃₀𝐺₀(𝛼)𝑑𝛼^{𝑖/(𝑛−1)}𝑑𝜃=−₀^2𝜋ℎ𝑆₁(𝜃)1−ℎ∫^𝜃₀sin(𝛼)𝑑𝛼𝑑𝜃=−₀^2𝜋ℎ𝑐₂₀cos³(𝜃)+𝑐₀₂−𝑏₁₁cos(𝜃)sin²(𝜃)1−ℎ+ℎcos(𝜃)𝑑𝜃=−₀^2𝜋ℎcos(𝜃)(𝜃)×𝑐1−ℎ+ℎcos₂₀cos²(𝜃)+𝑐₀₂−𝑏₁₁sin²(𝜃)𝑑𝜃=−₀^2𝜋1−1−ℎ(𝜃)×𝑐1−ℎ+ℎcos₂₀cos²(𝜃)+𝑐₀₂−𝑏₁₁sin²(𝜃=𝑐)𝑑𝜃₀₂−𝑏₁₁−3𝑐₂₀2𝑐𝜋+⁰²−𝑏₁₁−𝑐₂₀𝜋ℎ²+4𝑐₂₀−𝑐₀₂+𝑏₁₁𝜋ℎ+2(1−ℎ)𝑐₂₀ℎ²+2𝑐₀₂−𝑏₁₁−𝑐₂₀ℎ+𝑐₂₀+𝑏₁₁−𝑐₀₂𝜋ℎ²√.1−2ℎ (48) Then, from [18], we have 𝑑𝑇₁₀(ℎ)=()𝐹(𝛿)𝜋𝑑ℎ1−𝛿𝛿³(1−𝛿)³, (49) where 𝛿=√1−2ℎ and 𝐹(𝛿)=𝑐₂₀𝛿⁴+4𝑐₂₀𝛿³+2𝑐₂₀+4𝑐₀₂−4𝑏₁₁𝛿²+4𝑐₂₀𝛿+𝑐₂₀. (50) Support that 𝑐₂₀≠0. By (50) we have four zeros of 𝐹(𝛿) as follows: 𝛿_±=−1−1−𝜌±1−𝜌−21−𝜌,𝛿_±=−1+1−𝜌±1−𝜌−21−𝜌, (51) where 𝜌 is given by 𝜌=(𝑐₀₂−𝑏₁₁)/𝑐₂₀. It is not difficult to examine that 𝛿_± is not in (0,1) and 𝛿₊ is not in (0,1). In addition, 𝛿₋∈(0,1) if and only if 𝜌<−3. Thus, there is at most one critical period in (0,1/2). If 𝜌≥−3, 𝐹(𝛿) has no zero in ℎ∈(0,1/2). From the expression of 𝑑𝑇₁₀(ℎ)/𝑑ℎ, we can easily check that 𝑑𝑇₁₀(ℎ)/𝑑ℎ>0 (resp., < 0) when 𝑐₂₀>0 (resp., < 0). Therefore, for sufficiently small 𝜀 and 𝜆, 𝜕𝑇(ℎ,𝜀,𝜆)/𝜕ℎ>0 (resp., < 0) if 𝑐₂₀>0 (resp., < 0). The proof of conclusion (i) is completed.
For conclusion (ii), let 𝑀₀(ℎ)≡0, which gives 𝑐₂₀=𝑐₀₂−𝑏₁₁=0 or 𝑆₁(𝜃)=0. From (26), we have that 𝑇₁₁(ℎ)=−₀^2𝜋𝑟₀₀(𝜃,ℎ)𝑆₁(𝜃)𝑑𝜃−₀^2𝜋𝑟²₀₀(𝜃,ℎ)𝑆₂(𝜃)𝑑𝜃=−₀^2𝜋ℎ𝑐₂₀cos³(𝜃)+𝑐₀₂−𝑏₁₁cos(𝜃)sin²(𝜃)(𝜃)−1−ℎ+ℎcos𝑑𝜃_02𝜋ℎ²𝑐₃₀cos⁴(𝜃)+𝑐₁₂−𝑏₂₁cos²(𝜃)sin²(𝜃)−𝑏₀₃sin⁴(𝜃)(1−ℎ+ℎcos(𝜃))⁻²𝑑𝜃. (52)
Then we expand 𝑇₁₁(ℎ) at ℎ=0 by letting 𝑇₁₁(ℎ)=−𝐴₂ℎ²+𝐴₃ℎ³+𝐴₄ℎ⁴+𝐴₅ℎ⁵+𝐴₆ℎ⁶+𝑂ℎ⁶𝜋. (53) By (52), we have 𝐴₂1=−4𝑐₀₂−𝑏₁₁−34𝑐₂₀+34𝑐₃₀+14𝑐₁₂−𝑏₂₁−34𝑏₀₃,𝐴₃=2𝐴₂,𝐴₄=−238𝑐₂₀+338𝑐₃₀−78𝑐₀₂−𝑏₁₁−218𝑏₀₃+98𝑐₁₂−𝑏₂₁,𝐴₅9=−2𝑏₀₃−112𝑐₂₀+52𝑐₁₂−𝑏₂₁+212𝑐₃₀−32𝑐₀₂−𝑏₁₁,𝐴₆=−67564𝑐₂₀−9916𝑏₀₃−16564𝑐₀₂−𝑏₁₁+32316𝑐₃₀+6916𝑐₁₂−𝑏₂₁. (54) Note that 𝑇^′₁₁(ℎ)=−2𝐴₂ℎ+3𝐴₃ℎ²+4𝐴₄ℎ³+5𝐴₅ℎ⁴+6𝐴₆ℎ⁵+𝑂ℎ⁵𝜋, (55) and that 𝜕𝐴det²,𝐴₄,𝐴₅,𝐴₆𝜕𝑐₂₀,𝑐₀₂−𝑏₁₁,𝑏₀₃,𝑐₁₂−𝑏₂₁=−34−238−34−675−1644−78−218−1653644338−92323116498−991669=164117516384≠0. (56) Therefore, 𝐴₂,𝐴₄,𝐴₅,𝐴₆ can be taken as free parameters. Hence, we can change the sign of 𝐴₂,𝐴₄,𝐴₅,𝐴₆, satisfying 0<−𝐴₂≪𝐴₄≪−𝐴₅≪𝐴₆, (57) which ensures that 𝑇^′₁₁(ℎ) has three positive zeros in ℎ near ℎ=0. Thus, conclusion (ii) is proved.

We remark that for cubic system (40), we obtain three critical periods using 𝑇₁₁(ℎ), which is one more than the results in Lemma 2.3 obtained in [18].

Case 2 (𝑛=𝑚=3, 𝑙=𝑘=4). In this case, system (40) becomes 𝑥̇=−𝑦+𝑎₂₁𝑥²𝑦+𝜆𝑎₃₁𝑥³𝑦+𝜆𝑎₁₃𝑥𝑦³+𝜀𝑏₁₁𝑥𝑦+𝑏₂₁𝑥²𝑦+𝑏₀₃𝑦³+𝜆𝑏₁₁𝑥𝑦+𝑏₂₁𝑥²𝑦+𝑏₀₃𝑦³+𝑏₃₁𝑥³𝑦+𝑏₁₃𝑥𝑦³,𝑦̇=𝑥+𝑎₂₁𝑥𝑦²+𝜆𝑎₃₁𝑥²𝑦²+𝜆𝑎₁₃𝑦⁴+𝜀𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝑐₃₀𝑥³+𝑐₁₂𝑥𝑦²+𝜆𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝑐₃₀𝑥³+𝑐₁₂𝑥𝑦²+𝑐₄₀𝑥⁴+𝑐₂₂𝑥²𝑦²+𝑐₀₄𝑦⁴, (58) where we suppose 𝑎₂₁≠0. This is a new system which is not studied in [18]. For the case 𝑎₂₁>0, by the rescaling 𝑢=√𝑎₂₁𝑥,𝑣=√𝑎₂₁𝑦, (59) system (58) can be written as 𝑢̇=−𝑣+𝑢²𝑣+𝜆𝐴₃₁𝑢³𝑣+𝜆𝐴₁₃𝑢𝑣³+𝜀𝐵₁₁𝑢𝑣+𝐵₂₁𝑢²𝑣+𝐵₀₃𝑣³+𝜆𝐵₁₁𝑢𝑣+𝐵₂₁𝑢²𝑣+𝐵₀₃𝑣³+𝐵₃₁𝑢³𝑣+𝐵₁₃𝑢𝑣³,𝑣̇=𝑢+𝑢𝑣²+𝜆𝐴₃₁𝑢²𝑣²+𝜆𝐴₁₃𝑣⁴+𝜀𝐷₂₀𝑢²+𝐷₀₂𝑣²+𝐷₃₀𝑢³+𝐷₁₂𝑢𝑣²+𝜆𝐷₂₀𝑢²+𝐷₀₂𝑣²+𝐷₃₀𝑢³+𝐷₁₂𝑢𝑣²+𝐷₄₀𝑢⁴+𝐷₂₂𝑢²𝑣²+𝐷₀₄𝑣⁴, (60) where 𝐴₃₁=𝑎₃₁𝑎₂₁√𝑎₂₁,𝐴₁₃=𝑎₁₃𝑎₂₁√𝑎₂₁,𝐵₁₁=𝑏₁₁√𝑎₂₁,𝐵₂₁=𝑏₂₁𝑎₂₁,𝐵₀₃=𝑏₀₃𝑎₂₁,𝐵₁₁=𝑏₁₁√𝑎₂₁,𝐵₂₁=𝑏₂₁𝑎₂₁,𝐵₀₃=𝑏₀₃𝑎₂₁,𝐵₃₁=𝑏₃₁𝑎₂₁√𝑎₂₁,𝐵₁₃=𝑏₁₃𝑎₂₁√𝑎₂₁,𝐷₂₀=𝑐₂₀√𝑎₂₁,𝐷₀₂=𝑐₀₂√𝑎₂₁,𝐷₃₀=𝑐₃₀𝑎₂₁,𝐷₁₂=𝑐₁₂𝑎₂₁,𝐷₂₀=𝑐₂₀√𝑎₂₁,𝐷₀₂=𝑐₀₂√𝑎₂₁,𝐷₃₀=𝑐₃₀𝑎₂₁,𝐷₁₂=𝑐₁₂𝑎₂₁,𝐷₄₀=𝑐₄₀𝑎₂₁√𝑎₂₁,𝐷₂₂=𝑐₂₂𝑎₂₁√𝑎₂₁,𝐷₀₄=𝑐₀₄𝑎₂₁√𝑎₂₁. (61) If 𝑎₂₁<0, system (58) can be simplified as (60) similarly by the change 𝑢=√−𝑎₂₁𝑥, 𝑣=√−𝑎₂₁𝑦.

Theorem 7. For system (60), one has (i)𝑇₁₀(ℎ)=1−1−ℎ²1+1−ℎ²1−ℎ²𝜋×2𝐵₀₃−2𝐷₃₀+𝐵₀₃+𝐵₂₁−𝐷₁₂−𝐷₃₀×1−ℎ²+2𝐷₃₀ℎ², (62) where ℎ∈(0,1). If 𝑀₀(ℎ) is not identically zero, there are at most two critical periods in ℎ∈(0,1) for sufficiently small 𝜀 and 𝜆, and the maximum can be achievable; (ii) if 𝑀₀(ℎ)≡0, one has 𝑇₁₁(ℎ)=1−1−ℎ²1+1−ℎ²1−ℎ²𝜋×2𝐵₀₃−2𝐷₃₀+𝐵₀₃+𝐵₂₁−𝐷₁₂−𝐷₃₀×1−ℎ²+2𝐷₃₀ℎ², (63) and there are at most two critical periods in ℎ∈(0,1) if 𝑇₁₁(ℎ) is not identically zero.

Proof. From (13), (14), (22)–(25), we have that for system (60) 𝐺₀(𝜃)=cos(𝜃)sin(𝜃),𝐺₀(𝜃)=𝐴₃₁cos²(𝜃)sin(𝜃)+𝐴₁₃sin³(𝜃),𝐺₁(𝜃)=𝐵₁₁+𝐷₂₀cos²(𝜃)sin(𝜃)+𝐷₀₂sin³(𝜃),𝐺₂(𝜃)=𝐵₂₁+𝐷₃₀cos³(𝜃)(𝜃)+𝐵sin₀₃+𝐷₁₂cos(𝜃)sin³(𝜃),𝐺₁(𝜃)=𝐵₁₁+𝐷₂₀cos²(𝜃)sin(𝜃)+𝐷₀₂sin³(𝜃),𝐺₂(𝜃)=𝐵₂₁+𝐷₃₀cos³(𝜃)(𝜃)+sin𝐵₀₃+𝐷₁₂cos(𝜃)sin³(𝜃),𝑆₁(𝜃)=𝐷₂₀cos³(𝜃)+𝐷₀₂−𝐵₁₁cos(𝜃)sin²(𝜃),𝑆₂(𝜃)=𝐷₃₀cos⁴(𝜃)+𝐷₁₂−𝐵₂₁cos²(𝜃)sin²(𝜃)−𝐵₀₃sin⁴(𝜃),𝑆₁(𝜃)=𝐷₂₀cos³(𝜃)+𝐷₀₂−𝐵₁₁cos(𝜃)sin²(𝜃),𝑆₂(𝜃)=𝐷₃₀cos⁴(𝜃)+𝐷₁₂−𝐵₂₁cos²(𝜃)sin²(𝜃)−𝐵₀₃sin⁴(𝜃),𝑆₃(𝜃)=𝐷₄₀cos⁵(𝜃)+𝐷₂₂−𝐵₃₁cos³(𝜃)sin²(𝜃)+𝐷₀₄−𝐵₁₃sin⁴(𝜃)cos(𝜃). (64) From (27) and (28), we have 𝑟₀₀()=ℎ𝜃,ℎ1−ℎ²sin²(𝜃),𝑟₀₁(𝜃,ℎ)=𝑟³₀₀(𝜃,ℎ)1−1−ℎ²sin²(𝜃)ℎ. (65) Hence, from (10), 𝑇₁₀(ℎ)=−^𝑚−1_𝑖=10^2𝜋ℎ^𝑖𝑆_𝑖(𝜃)1+(1−𝑛)ℎ^𝑛−1∫^𝜃₀𝐺₀(𝛼)𝑑𝛼^{𝑖/(𝑛−1)}𝑑𝜃=−₀^2𝜋ℎ𝑆₁(𝜃)1−ℎ²sin²(𝜃)𝑑𝜃−₀^2𝜋ℎ²𝑆₂(𝜃)1−ℎ²sin²(𝜃)=𝑑𝜃1−1−ℎ²1+1−ℎ²1−ℎ²𝜋×2𝐵₀₃−2𝐷₃₀+𝐵₀₃+𝐵₂₁−𝐷₁₂−𝐷₃₀1−ℎ²+2𝐷₃₀ℎ². (66)
Thus, 𝑑𝑇₁₀(ℎ)/𝑑ℎ=−(2𝜋𝐹(𝛿)/𝛿³(1+𝛿)²)1−𝛿², where 𝛿=1−ℎ², 𝐹(𝛿)=𝐷₃₀𝛿⁴+2𝐷₃₀𝛿³+(𝐷₁₂−𝐵₂₁)𝛿²−2𝐵₀₃𝛿−𝐵₀₃. If 𝐷₃₀=0, 𝐹(𝛿)=(𝐷₁₂−𝐵₂₁)𝛿²−2𝐵₀₃𝛿−𝐵₀₃, we can control the coefficients 𝐷₁₂−𝐵₂₁, −2𝐵₀₃, −𝐵₀₃, which gives two zeros of 𝐹(𝛿) in (0,1). If 𝐷₃₀≠0, according to the expression of 𝐹(𝛿), we suppose that 𝛿₁,𝛿₂,𝛿₃,𝛿₄ are four zeros of 𝐹(𝛿). From the relationship of root and coefficients, we have 𝛿₁+𝛿₂+𝛿₃+𝛿₄=−2; thus not all of the zeros are in (0,1). Then from [18], and it is impossible for 𝐹(𝛿) to have 3 zeros in (0,1). Thus, the first part of conclusion (i) is proved.
Next, we give an example to show that the zeros in (0,1) can be achievable. We can choose 𝐷₃₀=1300, 𝐷₁₂−𝐵₂₁=−2553, 𝐵₀₃=116, then, 𝐹(𝛿)=1300𝛿⁴+2600𝛿³−2553𝛿²−232𝛿−116=(2𝛿−1)(5𝛿−2)(130𝛿²+377𝛿+58), so, 𝐹(𝛿) has two simple zeros 2/5 and 1/2 in (0,1). Thus, 𝑇(ℎ,𝜀,𝜆) has exactly two critical periods in (0,1). The proof of conclusion (i) is completed.
For the conclusion (ii), if 𝑀₀(ℎ)≡0, we get 𝐷₃₀=𝐵₀₃=𝐷₁₂−𝐵₂₁=0 or 𝑆₂(𝜃)=0.
Hence, from (26), we have 𝑇₁₁(ℎ)=−₀^2𝜋𝑟₀₁(𝜃,ℎ)𝑆₁(𝜃)𝑑𝜃−₀^2𝜋𝑟₀₀(𝜃,ℎ)𝑆₁(𝜃)−𝑑𝜃_02𝜋𝑟²₀₀(𝜃,ℎ)𝑆₂(𝜃)𝑑𝜃−₀^2𝜋𝑟³₀₀(𝜃,ℎ)𝑆₃(𝜃)=𝑑𝜃1−1−ℎ²1+1−ℎ²1−ℎ²𝜋×2𝐵₀₃−2𝐷₃₀+𝐵₀₃+𝐵₂₁−𝐷₁₂−𝐷₃₀×1−ℎ²+2𝐷₃₀ℎ². (67) So, 𝑑𝑇₁₁(ℎ)/𝑑ℎ=−(2𝜋𝐹(𝛿)/𝛿³(1+𝛿)²)1−𝛿², where 𝛿=1−ℎ², 𝐹(𝛿)=𝐷₃₀𝛿⁴+2𝐷₃₀𝛿³+(𝐷₁₂−𝐵₂₁)𝛿²−2𝐵₀₃𝛿−𝐵₀₃. From the proof above, we can get result (ii). This ends the proof of Theorem 7.

Corollary 8. For system (58) with 𝑎₂₁>0, one obtains 𝑇₁₀(ℎ)=1−1−𝑎₂₁ℎ²1+1−𝑎₂₁ℎ²1−𝑎₂₁ℎ²𝜋×2𝑏₀₃−2𝑐₃₀+𝑏₀₃+𝑏₂₁−𝑐₁₂−𝑐₃₀×1−𝑎₂₁ℎ²+2𝑐₃₀𝑎₂₁ℎ²𝑎₂₁⁻¹, (68) where ℎ∈(0,1/√𝑎₂₁). For sufficiently small 𝜀 and 𝜆, there are at most two critical periods for 𝑇(ℎ,𝜀,𝜆) in ℎ∈(0,1) and the maximum number can be achievable.

Proof. Let 𝑇(ℎ,𝜀,𝜆) denote the period function of (60) and 𝑇(ℎ,𝜀,𝜆) denote the period function of (58). Then we have ℎ=√𝑎₂₁ℎ, and 𝑇(ℎ,𝜀,𝜆)∣_𝜀=𝜆=0=𝑇(ℎ,𝜀,𝜆)∣_𝜀=𝜆=0. Hence by Theorem 7 and (61), we have for system (58), 𝑇₁₀(ℎ)=1−1−𝑎₂₁ℎ²1+1−𝑎₂₁ℎ²1−𝑎₂₁ℎ²𝜋×2𝑏₀₃−2𝑐₃₀+𝑏₀₃+𝑏₂₁−𝑐₁₂−𝑐₃₀×1−𝑎₂₁ℎ²+2𝑐₃₀𝑎₂₁ℎ²𝑎₂₁⁻¹, (69) which gives the formula of 𝑇₁₀(ℎ). Other conclusions in the corollary can also be gotten from results in Theorem 7 similarly. This ends the proof of Corollary 8.

Case 3 (𝑛=𝑙=2, 𝑚=𝑘=3). In this case, system (5) becomes 𝑥̇=−𝑦+𝑎₁₁𝑥𝑦+𝜆𝑎₁₁𝑏𝑥𝑦+𝜀₁₁𝑥𝑦+𝑏₂₁𝑥²𝑦+𝑏₀₃𝑦³+𝜆𝑏₁₁𝑥𝑦+𝑏₂₁𝑥²𝑦+𝑏₀₃𝑦³,𝑦̇=𝑥+𝑎₁₁𝑦²+𝜆𝑎₁₁𝑦²+𝜀𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝑐₃₀𝑥³+𝑐₁₂𝑥𝑦²+𝜆𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝑐₃₀𝑥³+𝑐₁₂𝑥𝑦². (70) Without loss of generality, we suppose 𝑎₁₁=1, otherwise, system (70) can be simplified as the form of system (71) by the transformation 𝑢=𝑎₁₁𝑥, 𝑣=𝑎₁₁𝑦. In this case, system (70) becomes 𝑥̇=−𝑦+𝑥𝑦+𝜆𝑎₁₁𝑏𝑥𝑦+𝜀₁₁𝑥𝑦+𝑏₂₁𝑥²𝑦+𝑏₀₃𝑦³+𝜆𝑏₁₁𝑥𝑦+𝑏₂₁𝑥²𝑦+𝑏₀₃𝑦³,𝑦̇=𝑥+𝑦²+𝜆𝑎₁₁𝑦²+𝜀𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝑐₃₀𝑥³+𝑐₁₂𝑥𝑦²+𝜆𝑐₂₀𝑥²+𝑐₀₂𝑦²+𝑐₃₀𝑥³+𝑐₁₂𝑥𝑦². (71)

Theorem 9. For system (71) and sufficiently small 𝜀 and 𝜆, one has(i)𝑇₁₀(ℎ)=−𝜋𝐼₁+𝐼₂, (72) where 𝐼₁=𝑐₀₂+𝑏₁₁−3𝑐₂₀+2𝑐₀₂+𝑏₁₁−𝑐₂₀ℎ²+4𝑐₂₀−𝑐₀₂+𝑏₁₁ℎ+2(1−ℎ)𝑐₂₀ℎ²+2𝑐₀₂+𝑏₁₁−𝑐₂₀ℎ+𝑐₂₀−𝑐₀₂+𝑏₁₁ℎ²√,𝐼1−2ℎ₂=𝑐₃₀−𝑏₀₃−𝑐₁₂+𝑏₂₁𝐼₂₁+𝑐₁₂−𝑏₂₁+2𝑏₀₃𝐼₂₂−𝑏₀₃𝐼₂₃,𝐼₂₁=−4ℎ⁶−10ℎ⁵+78ℎ⁴−140ℎ³−112ℎ²−7(1−2ℎ)^5/2ℎ²+12(1−2ℎ)^5/2ℎ−6(1−2ℎ)^5/2+6−42ℎ(1−2ℎ)^5/2ℎ²⁻¹,𝐼₂₂=−22ℎ⁴+ℎ³−7ℎ²+5ℎ−1−(1−2ℎ)^5/2(1−2ℎ)^5/2,𝐼₂₃=2ℎ²4ℎ³−8ℎ²+5ℎ−1(1−2ℎ)^7/2. (73) There can exist three critical periods for system (71) if 𝑇₁₀(ℎ) is not identically zero;(ii)if 𝑀₁₀(ℎ)=𝑇^′₁₀(ℎ)≡0, one has 𝑇₁₁(ℎ)=−𝜋𝐼₃+𝐼₄, (74) where 𝐼₃=𝑐₀₂−𝑏₁₁+3𝑐₂₀+2𝑐₀₂−𝑏₁₁−𝑐₂₀ℎ²+4𝑐₂₀−𝑐₀₂+𝑏₁₁ℎ+2(1−ℎ)𝑐₂₀ℎ²+2𝑐₀₂−𝑏₁₁−𝑐₂₀ℎ+𝑐₂₀−𝑐₀₂+𝑏₁₁ℎ²√1−2ℎ⁻¹,𝐼₄=𝑐₃₀−𝑏₀₃−𝑐₁₂+𝑏₂₁𝐼₂₁+𝑐₁₂−𝑏₂₁+2𝑏₀₃𝐼₂₂−𝑏₀₃𝐼₂₃. (75)There can appear three critical periods for system (71) if 𝑇₁₁(ℎ) is not identically zero.

Proof. From (13), (14), (22)–(25), we have 𝐺₀(𝜃)=sin(𝜃),𝐺₀(𝜃)=𝑎₁₁(𝜃),𝐺sin₁(𝜃)=𝑐₂₀+𝑏₁₁cos²(𝜃)sin(𝜃)+𝑐₀₂sin³(𝜃),𝐺₂(𝜃)=𝑏₂₁+𝑐₃₀cos³(𝜃)(𝜃)+𝑏sin₀₃+𝑐₁₂cos(𝜃)sin³(𝜃),𝑆₁(𝜃)=𝑐₂₀cos³(𝜃)+𝑐₀₂−𝑏₁₁cos(𝜃)sin²(𝜃),𝑆₂(𝜃)=𝑐₃₀cos⁴(𝜃)+𝑐₁₂−𝑏₂₁cos²(𝜃)sin²(𝜃)−𝑏₀₃sin⁴(𝜃),𝐺₁(𝜃)=𝑏₁₁+𝑐₂₀cos²(𝜃)sin(𝜃)+𝑐₀₂sin³(𝜃),𝐺₂(𝜃)=𝑏₂₁+𝑐₃₀cos³(𝜃)(𝜃)+sin𝑏₀₃+𝑐₁₂cos(𝜃)sin³(𝜃),𝑆₁(𝜃)=𝑐₂₀cos³(𝜃)+𝑐₀₂−𝑏₁₁cos(𝜃)sin²(𝜃),𝑆₂(𝜃)=𝑐₃₀cos⁴(𝜃)+𝑐₁₂−𝑏₂₁cos²(𝜃)sin²(𝜃)−𝑏₀₃sin⁴(𝜃). (76) From (10), we can get 𝑇₁₀(ℎ)=−₀^2𝜋ℎ𝑆₁(𝜃)1−ℎ+ℎcos(𝜃)𝑑𝜃−₀^2𝜋ℎ²𝑆₂(𝜃)(1−ℎ+ℎcos(𝜃))²𝑑𝜃=−₀^2𝜋ℎ𝑐₂₀cos³(𝜃)+𝑐₀₂−𝑏₁₁×cos(𝜃)sin²(𝜃)×(1−ℎ+ℎcos(𝜃))⁻¹−𝑑𝜃_02𝜋ℎ²𝑐₃₀cos⁴(𝜃)+𝑐₁₂−𝑏₂₁×cos²(𝜃)sin²(𝜃)−𝑏₀₃sin⁴(𝜃×((𝜃)1−ℎ+ℎcos))⁻²𝐼𝑑𝜃=−𝜋₁+𝐼₂, (77) where 𝐼₁=𝑐₀₂−𝑏₁₁−3𝑐₂₀+2𝑐₀₂−𝑏₁₁−𝑐₂₀ℎ²+4𝑐₂₀−𝑐₀₂+𝑏₁₁ℎ+2(1−ℎ)𝑐₂₀ℎ²+2𝑐₀₂−𝑏₁₁−𝑐₂₀ℎ+𝑐₂₀−𝑐₀₂+𝑏₁₁ℎ²√1−2ℎ⁻¹,𝐼₂=𝑐₃₀−𝑏₀₃−𝑐₁₂+𝑏₂₁𝐼₂₁+𝑐₁₂−𝑏₂₁+2𝑏₀₃𝐼₂₂−𝑏₀₃𝐼₂₃,𝐼₂₁=−4ℎ⁶−10ℎ⁵+78ℎ⁴−140ℎ³−112ℎ²−7(1−2ℎ)^5/2ℎ²+12(1−2ℎ)^5/2ℎ−6(1−2ℎ)^5/2×()+6−42ℎ1−2ℎ^5/2ℎ²⁻¹,𝐼₂₂=−22ℎ⁴+ℎ³−7ℎ²+5ℎ−1−(1−2ℎ)^5/2(1−2ℎ)^5/2,𝐼₂₃=2ℎ²4ℎ³−8ℎ²+5ℎ−1(1−2ℎ)^7/2. (78) By the same method as the proof of Theorem 6, we have conclusion (i).
For conclusion (ii), if 𝑀₁₀(ℎ)=𝑇^′₁₀(ℎ)≡0, giving 𝑆₁(𝜃)=𝑆₂(𝜃)=0, thus from (26), 𝑇₁₁(ℎ)=−₀^2𝜋𝑟₀₀(𝜃,ℎ)𝑆₁(𝜃)−𝑑𝜃_02𝜋𝑟²₀₀(𝜃,ℎ)𝑆₂(𝜃)𝑑𝜃=−₀^2𝜋ℎ𝑐₂₀cos³(𝜃)+𝑐₀₂−𝑏₁₁cos(𝜃)sin²(𝜃×((𝜃)1−ℎ+ℎcos))⁻¹−𝑑𝜃_02𝜋ℎ²𝑐₃₀cos⁴(𝜃)+𝑐₁₂−𝑏₂₁×cos²(𝜃)sin²(𝜃)−𝑏₀₃sin⁴(𝜃×((𝜃)1−ℎ+ℎcos))⁻²𝐼𝑑𝜃=−𝜋₃+𝐼₄, (79) where 𝐼₃=𝑐₀₂−𝑏₁₁+3𝑐₂₀+2𝑐₀₂−𝑏₁₁−𝑐₂₀ℎ²+4𝑐₂₀−𝑐₀₂+𝑏₁₁ℎ+2(1−ℎ)𝑐₂₀ℎ²+2𝑐₀₂−𝑏₁₁−𝑐₂₀ℎ+𝑐₂₀−𝑐₀₂+𝑏₁₁ℎ²√1−2ℎ⁻¹,𝐼₄=𝑐₃₀−𝑏₀₃−𝑐₁₂+𝑏₂₁𝐼₂₁+𝑐₁₂−𝑏₂₁+2𝑏₀₃𝐼₂₂−𝑏₀₃𝐼₂₃. (80) We can get conclusion (ii) with the same method as the proof of Theorem 6. This ends the proof of Theorem 9.

Acknowledgment

The project was supported by National Natural Science Foundation of China (11271261).