Perturbed rigidly isochronous centers and their critical periods

J. Math. Anal. Appl. 453 (2017) 366–382

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Perturbed rigidly isochronous centers and their critical periods ✩ Linping Peng a , Lianghaolong Lu a , Zhaosheng Feng b,∗ a

School of Mathematics and System Sciences, Beihang University, LIMB of the Ministry of Education, Beijing 100191, China b School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA

a r t i c l e

i n f o

Article history: Received 9 September 2016 Available online 4 April 2017 Submitted by G. Chen Keywords: Perturbation Bifurcation of critical period Period bifurcation function Rigidly isochronous center

a b s t r a c t This paper investigates the bifurcation of critical periods from a cubic rigidly isochronous center under any small polynomial perturbations of degree n. It proves that for n = 3, 4 and 5, there are at most 2 and 4 critical periods induced by periodic orbits of the unperturbed cubic system respectively, and in each  case this upper bound is sharp. Moreover, for any n > 5, there are at most n−1 critical 2 periods induced by periodic orbits of the unperturbed cubic system. An example is given to show that the upper bound in the case of n = 11 can be reached. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Consider a two-dimensional differential system x˙ = f (x, y), (1.1) y˙ = g(x, y), which has a center at the origin. Let T (η) be the period of the periodic orbit passing through the point (η, 0), called the period function of system (1.1). By convention, a center is called an isochronous center if the associated period function is a constant. A center is called a rigidly isochronous center if θ˙ ≡ 1 in the polar coordinates x = r cos θ and y = r sin θ [2]. When T  (η ∗ ) = 0, T (η ∗ ) becomes a critical period. In the past decades, one has seen that much effort has been dedicated to the study of period function for planar polynomial vector fields, for example, on monotonicity [1,4,8,9,13,17,24], isochronicity [2,20,22], finiteness of critical periods [5,16,18], and local bifurcation of critical periods [6,21,24], which concerns how many critical periods can arise near the center. In recent years, considerable attention is paid to the global ✩

This research is supported by National Science Foundation of China under 11371046 and 11290141.

* Corresponding author. E-mail address: [email protected] (Z. Feng). http://dx.doi.org/10.1016/j.jmaa.2017.03.081 0022-247X/© 2017 Elsevier Inc. All rights reserved.

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Fig. 1. The phase portrait of system (1.2) in the Poincaré disk.

bifurcation, especially the bifurcation of critical periods from the periodic orbits surrounding the isochronous center under some perturbations [3,7,10–14,19,23]. In our previous work, we considered the bifurcation of critical periods from a rigidly quartic isochronous center [15], and showed that at most one critical period bifurcates from the periodic orbits of the unperturbed quadratic system based on the derived formulas of the qth period bifurcation function for any perturbed isochronous system with a center [19]. It is notable that in [11] some elegant results about the bifurcation of critical periods have been presented when the following system x˙ = −y + x2 y, y˙ = x + xy 2 ,

(1.2)

is perturbed inside L3 , where L3 is denoted by x˙ = −y + P3 (x, y), y˙ = x + Q3 (x, y), the family of vector fields with a center at the origin, where P3 (x, y) and Q3 (x, y) are homogeneous polynomials of degree 3. In the present paper, we mainly study the bifurcation of critical periods from system (1.2) under any small polynomial perturbation of degree n, and prove that there are at most 2[ n−1 2 ] critical periods induced from periodic orbits of system (1.2). The phase portrait of system (1.2) is depicted in Fig. 1. The origin is a rigidly isochronous center. It is not 2 +y 2 2 difficult to find that this system has a first integral H(x, y) = x1−x 2 with an integral factor (1−x2 )2 . For each η ∈ (0, 1) the orbit through (η, 0) lies in the annulus of periodic orbits formed by {(x, y)|H(x, y) = c, c ∈ (0, +∞)}, which starts at (0, 0) and terminates at the separatrix passing the infinite degenerate singularity on the equator. Let us state our main results as follows. Theorem 1.1. Assume that for any sufficiently small |ε|, the origin of the following system x˙ = −y + x2 y + εp(x, y), y˙ = x + xy 2 + εq(x, y),

(1.3)

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is a center, where p(x, y) and q(x, y) are polynomials of degree n as follows

p(x, y) =

n 

l m

clm x y

and q(x, y) =

l+m=2

n 

dlm xl y m .

l+m=2

Then, the following three statements are true. (1) For n = 3 and n = 4, up to the first order in ε, at most two critical periods arise from the periodic orbits of the unperturbed system (1.3)|ε=0 , respectively. Moreover, in each case this upper bound is sharp. (2) For n = 5, up to the first order in ε, at most four critical periods arise from the periodic orbits of the unperturbed system (1.3)|ε=0 . Moreover, this upper bound is sharp.   (3) For any n > 5, up to the first order in ε, at most 2 n−1 critical periods arise from the periodic 2 orbits of the unperturbed system (1.3)|ε=0 , where [x] denotes the greatest integer function. The rest of this paper is organized as follows. In Section 2, we present some important preliminaries, such as explicit formulas of the period bifurcation function for any perturbed rigidly isochronous center, and some results about the integrals. In Section 3, we demonstrate the proof of Theorem 1.1. 2. Preliminary results Consider the perturbed rigidly isochronous system

x˙ = f0 (x, y) +

∞ 

εk fmk (x, y),

k=1

y˙ = g0 (x, y) +

∞ 

(2.1) k

ε gnk (x, y),

k=1

where f0 (x, y) and g0 (x, y) are two polynomials such that (2.1)|ε=0 has a rigidly isochronous center at the origin, and fmk (x, y) and gnk (x, y) are polynomials of degrees mk and nk (mk , nk ∈ N), respectively. Here the origin is the center of system (2.1) for any sufficiently small |ε|. Making use of the polar coordinates x = r cos θ and y = r sin θ to system (2.1), we have

r˙ = F0 (r, θ) +

∞ 

εk Fk (r, θ),

k=1

θ˙ = 1 +

∞ 

(2.2)

k

ε Gk (r, θ),

k=1

where F0 (r, θ) = cos θf0 (r cos θ, r sin θ) + sin θg0 (r cos θ, r sin θ), Fk (r, θ) = cos θfmk (r cos θ, r sin θ) + sin θgnk (r cos θ, r sin θ), Gk (r, θ) =

cos θgnk (r cos θ, r sin θ) − sin θfmk (r cos θ, r sin θ) , r

for k ≥ 1. Then system (2.2) is equivalent to ∞ F0 (r, θ) + k=1 εk Fk (r, θ) dr ∞ = . dθ 1 + k=1 εk Gk (r, θ)

(2.3)

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Assume that equation (2.3) has the solution of the form

r(θ, η, ε) =

∞ 

rq (θ)εq

(2.4)

q=0

with the initial condition r(0, η, ε) = η. Define T (η, ε) = 2π +

∞ 

Tq (η)εq

(2.5)

q=1

as the period of the closed orbit of system (2.1) passing through the point (η, 0), where Tq (η) is called the q-th period bifurcation function [3]. Definition 2.1. [13] Given a system of form (2.1), we define its period function as formula (2.5), and assume that T1 (η) ≡ T2 (η) ≡ · · · ≡ Tq−1 (η) ≡ 0 and Tq (η) ≡ 0. When the equation Tq (η) = 0 has exactly k simple zeros, we say that for system (2.1), up to the order q in ε, there are k critical periods arising from periodic orbits around the isochronous center of system (2.1)|ε=0 . Let us recall the explicit formula of the period bifurcation function for the perturbed rigidly isochronous center, which will be used in the proof of our main results. Lemma 2.2. Assume that for any sufficiently small |ε|, system (2.1) has a center at the origin. Let T (η, ε) be the period, given by formula (2.5), of the closed orbit r(θ, η, ε) of system (2.1) passing through the point (η, 0), where η = r(0, η, ε). Then for any q ∈ Z+ , we have

Tq (η) = −

Gk + sgn(q − k)

0

+

k=1



p=1

j1 +j2 +···+jp ≤q−k 1≤j1 ,j2 ,··· ,jp ≤q−k−p+1

p

(−1)

1 ∂ s Gk ri (θ)ri2 (θ) · · · ris (θ) − Gk s! ∂rs 1

Gk Gj1 Gj2 · · · Gjp + sgn(q − k − (j1 + j2 + · · · + jp ))

 1≤s≤q−k−(j1 +j2 +···+jp ) i1 +i2 +···+is =q−k−(j1 +j2 +···+jp ) 1≤i1 ,i2 ,··· ,is ≤q−k−(j1 +j2 +···+jp )−s+1



− G k G j1 G j2 · · · G jp

 1≤s≤q−k i1 +i2 +···+is =q−k 1≤i1 ,i2 ,··· ,is ≤q−k−s+1



∞ 

·



 2π  q

1 ∂ s (Gk Gj1 Gj2 · · · Gjp ) ri1 (θ)ri2 (θ) · · · ris (θ) s! ∂rs

dθ,

r=r0 (θ)

where rj (θ) (0 ≤ j ≤ q) satisfy the following differential equations r0 (θ) = F0 (r0 (θ), θ), with r0 (0) = r0 (2π) = η;

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370

 rj (θ)



=

1≤s≤j i1 +i2 +···+is =j 1≤i1 ,i2 ,··· ,is ≤j−s+1



 j  1 ∂ s F0 Fk − F0 Gk ri (θ)ri2 (θ) · · · ris (θ) + s! ∂rs 1 k=1



+ sgn(j − k)

1≤s≤j−k i1 +i2 +···+is =j−k 1≤i1 ,i2 ,··· ,is ≤j−k−s+1

+

1 ∂ s (Fk − F0 Gk ) ri1 (θ)ri2 (θ) · · · ris (θ) − (Fk − F0 Gk ) s! ∂rs



∞ 



p=1

j1 +j2 +···+jp ≤j−k 1≤j1 ,j2 ,··· ,jp ≤j−k−p+1

(Fk − F0 Gk ) Gj1 Gj2 · · · Gjp

p

(−1)

+ sgn(j − k − (j1 + j2 + · · · + jp ))  · 1≤s≤j−k−(j1 +j2 +···+jp ) i1 +i2 +···+is =j−k−(j1 +j2 +···+jp ) 1≤i1 ,i2 ,··· ,is ≤j−k−(j1 +j2 +···+jp )−s+1

1 ∂ s (Fk − F0 Gk )Gj1 Gj2 · · · Gjp ri1 (θ)ri2 (θ) · · · ris (θ) s! ∂rs



− (Fk − F0 Gk ) Gj1 Gj2 · · · Gjp

r=r0 (θ)

with rj (0) = rj (2π) = 0. Lemma 2.2 can be proved in an analogous manner as shown in [19], so we omit it. Let 2π 

Im (η) = 0

1 1 η2

− sin2 θ

m dθ,

m ∈ Z,

where 0 < |η| < 1 and Z is the set of all integers. In order to present our discussions in a straightforward manner, now we introduce Lemmas 2.3 to 2.7 which are derived indirectly from Lemmas 4.1–4.3 in [14]. Lemma 2.3. For 0 < |η| < 1, k, m ∈ Z and k ≥ 0, we have 2π  0

sin2k θ 1 η2

− sin2 θ

m dθ =

k−j k    k 1 (−1)j Im−j (η). 2 j η j=0

Lemma 2.4. For any 0 < |η| < 1, m ∈ Z, the following integral equality holds:     2m η 2 − 1 Im+1 (η) = 2(m − 1)η 4 Im−1 (η) + (1 − 2m)η 2 2 − η 2 Im (η). By using Lemmas 2.3 and 2.4, it is not difficult to obtain the following corollary. Corollary 2.5. For the above integral Im (η) (m = −1, 0, 1, 2, 3), we get I−1 (η) =

(2 − η 2 )π 2η 2 π η 4 (2 − η 2 )π , I (η) = 2π, I (η) = , I (η) = , 0 1 2 1 3 η2 (1 − η 2 ) 2 (1 − η 2 ) 2

and I3 (η) =

η 6 (3η 4 − 8η 2 + 8)π 5

4 (1 − η 2 ) 2

.

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Lemma 2.6. For any 0 < |η| < 1, j ≥ 0 (j ∈ Z), the following integral equality holds: η 4j−2

Ij (η) =

j− 12

(1 − η 2 )

I1−j (η).

Lemma 2.7. For any 0 < |η| < 1, j ≥ 1 (j ∈ Z), we obtain Ij (η) =

η 2j (1 −

j− 1 η2 ) 2

Rj−1 (η 2 ),

where Rk (x) is a polynomial of degree k with respect to x. 3. Proof of Theorem 1.1 With the polar coordinates x = r cos θ, y = r sin θ, system (1.3) can be written in the form r˙ =r3 sin θ cos θ + εF (r, θ), (3.1)

θ˙ =1 + εG(r, θ), where F (r, θ) =

n 

  rl+m clm cosl+1 θ sinm θ + dlm cosl θ sinm+1 θ ,

l+m=2

G(r, θ) =

n 

  rl+m−1 dlm cosl+1 θ sinm θ − clm cosl θ sinm+1 θ ,

l+m=2

and system (3.1)|ε=0 has the periodic orbit r0 (θ) = 

1 1 η2

− sin2 θ

,

η ∈ (0, 1)

surrounding the rigidly isochronous center (0, 0). 3.1. Cases of n = 3, 4 in system (3.1) Lemma 3.1. For n = 3 and 4 in system (3.1), up to the first order in ε, at most two critical periods arise from the periodic orbits of the unperturbed system, respectively. Moreover, in each case this upper bound is sharp. Proof. 1. When n = 3 in system (3.1), as a result of the fact that   G1 (r, θ) =r d20 cos3 θ + (d11 − c20 ) cos2 θ sin θ + (d02 − c11 ) cos θ sin2 θ − c02 sin3 θ  + r2 d30 cos4 θ + (d21 − c30 ) cos3 θ sin θ + (d12 − c21 ) cos2 θ sin2 θ  + (d03 − c12 ) cos θ sin3 θ − c03 sin4 θ , and

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372

2π  0

sin2k θ cos θ  12 dθ = 0 1 2 − sin θ 2 η

for k ≥ 0, it follows from Lemma 2.2 that 2π T1 (η) = −

G1 (r0 (θ), θ)dθ 0

2π =− 0

(3.2) d30 cos θ + (d12 − c21 ) cos θ sin θ − c03 sin θ dθ. 1 2 η 2 − sin θ 4

2

2

4

Note that 2π 0

1 η2

dθ =  − sin2 θ

4π 2 η2

−1

2

. −1

By virtue of Lemma 2.3 and Corollary 2.5, and making the transformation η 2 = 1 − w2 for 0 < w < 1, (3.2) becomes

T1 (w) = T1 (η)

η 2 =1−w2

=−

 π(w − 1)  2c03 + (c03 + c21 − d12 − d30 ) w − 2d30 w2 w(w + 1)

with 0 < w < 1. Then, it gives 2π · g(w) + 1)2   2π c03 + 2c03 w + (c21 − d12 ) w2 − 2d30 w3 − d30 w4 . =− 2 2 w (w + 1)

T1 (w) = −

w2 (w

Since all coefficients of g(w) in the ascending order change their signs at most twice, by the Descartes √ Rule, we know that T1 (w) has at most two zeros in (0, 1). It follows from T1 (η) = − 1 − w2 T1 (w)/w that there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system (3.1)|ε=0 . Moreover, this upper bound can be reached. In fact, we consider the following system    x˙ = − y + x2 y + ε d12 − 408 x2 y + 11y 3 ,   y˙ =x + xy 2 + ε −256x3 + d12 xy 2 ,

(3.3)

where d12 is a real constant. Obviously, this system has a center at (0, 0), and a direct calculation shows 2π · g(w) w2 (w + 1)2   2π =− 2 11 + 22w − 408w2 + 512w3 + 256w4 2 w (w + 1)

T1 (w) = −

L. Peng et al. / J. Math. Anal. Appl. 453 (2017) 366–382

512π =− 2 w (w + 1)2



1 w− 2

373

   1 11 11 2 w− w + w+ , 4 4 32

which means that T1 (w) has w = 1/2 and w = 1/4 as its zeros in w ∈ (0, 1). Hence, we conclude that up to order 1 in ε, there are just two critical periods emerging from the periodic orbits around the isochronous center (0, 0) of the unperturbed system (3.3)|ε=0 . 2. When n = 4, from Lemma 2.2, the first order period bifurcation function of system (3.1) takes the form 2π T 1 (η) = − 0

2π − 0

2π − 0

d20 cos3 θ + (d02 − c11 ) cos θ sin2 θ dθ  12  1 2 − sin θ 2 η d30 cos4 θ + (d12 − c21 ) cos2 θ sin2 θ − c03 sin4 θ dθ 1 2 η 2 − sin θ d40 cos5 θ + (d22 − c31 ) cos3 θ sin2 θ + (d04 − c13 ) cos θ sin4 θ dθ.  32  1 2 η 2 − sin θ

Note that for any k ≥ 0 and p ≥ 0, it has 2π 0

sin2k θ cos θ dθ = 0.  2p+1  2 1 2 − sin θ η2

So we get 2π T 1 (η) = − 0

d30 cos4 θ + (d12 − c21 ) cos2 θ sin2 θ − c03 sin4 θ dθ, 1 2 η 2 − sin θ

which is equal to T1 (η) for n = 3 in system (3.1). Thus, the same conclusion can be drawn in this case too. 2 3.2. Case of n = 5 in system (3.1) Lemma 3.2. For n = 5 in system (3.1), up to the first order in ε, at most four critical periods arise from the periodic orbits of the unperturbed system. Moreover, this upper bound is sharp. Proof. In this case, the first order period bifurcation function of system (3.1) can be expressed as T1 (η) = −

2π 0

d30 cos4 θ + (d12 − c21 ) cos2 θ sin2 θ − c03 sin4 θ dθ 1 2 η 2 − sin θ

2π − 0

d50 cos6 θ + (d32 − c41 ) cos4 θ sin2 θ + (d14 − c23 ) cos2 θ sin4 θ − c05 sin6 θ dθ. 2  1 2 − sin θ 2 η

Using Lemma 2.3 and Corollary 2.5, and making the transformation η 2 = 1 − w2 for 0 < w < 1, we get

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374

T˘1 (w) := T1 (η) η2 =1−w2  (w − 1)π c05 + 2c05 w + (2c03 − 2c05 + c23 − d14 )w2 =− 3 w (w + 1) + (c03 − c05 + c21 − c23 + c41 − d12 + d14 − d30 − d32 − d50 )w3  + (−c41 − 2d30 + d32 − 2d50 )w4 + 2d50 w5 + d50 w6 . Differentiating the above function yields T˘1 (w) =−

  π 2 3 4 5 6 7 8 + 2M w + M w + M w + M w + M w + M w + 2M w + M w M , 0 0 2 3 4 5 6 8 8 w4 (1 + w)2

where M0 = 3c05 , M2 = 2c03 − c05 + c23 − d14 , M3 = 4c03 − 8c05 + 2c23 − 2d14 , M4 = 2c21 − 3c23 + 3c41 − 2d12 + 3d14 − 3d32 , M5 = −2c41 − 4d30 + 2d32 − 8d50 , M6 = −c41 − 2d30 + d32 − d50 , and M8 = 3d50 . From M3 = 2(M2 − M0 ) and M5 = 2(M6 − M8 ), we know that if M2 has the different sign with M0 , then M3 and M2 must have the same sign. This property also holds for M5 , M6 and M8 . So the sequence of coefficients M0 , 2M0 , M2 , 2(M2 − M0 ), M4 , 2(M6 − M8 ), M6 , 2M8 , M8 , changes its sign at most four times. By the Descartes Rule, T˘1 (w) has at most four positive zeros. Recall the fact that √ dT1 (η) 1 − w2 ˘ =− T1 (w). dη w Hence, in this case system (3.1) has at most four critical periods bifurcating from the periodic orbits of the unperturbed one. Next, we illustrate an example to verify this upper bound can be reached. Consider the following system

1901 5 1 y + − 5043643 + 2c21 + 3c41 − 2d12 + 3d14 − 3d32 x2 y 3 x˙ = −y + x y + ε 3 3

 c d d c 21 41 12 32 − + + y 3 + c21 x2 y , + c41 x4 y + 776798 − 3 2 3 2 

2

L. Peng et al. / J. Math. Anal. Appl. 453 (2017) 366–382





y˙ = x + xy 2 + ε d14 xy 4 + d32 x3 y 2 + 1819200x5 + d12 xy 2 +

375

 d32 c41 + x3 , − 782098 − 2 2

(3.4)

where c21 , c41 , d12 , d14 and d32 are real constants. Apparently, (0, 0) is a center. By making the transformation η 2 = 1 − w2 , a straightforward calculation yields   5457600π 1  ˘ w− · w− T1 (w) = − 4 w (1 + w)2 2  197 3 162601 2 · w4 + w + w + 60 45480

1 3



1 w− 4



1 w− 5  1901 3802 w+ , 5685 45480



˘ which means that

w = 1/2, w = 1/3, w = 1/4 and w = 1/5 are all zeros of T1 (w) in (0, 1). As a result of T˘1 (w) := T1 (η) η2 =1−w2 , we conclude that system (3.4) has exactly four critical periods. Consequently, this completes the proof of Lemma 3.2. 2 3.3. Case of any n > 5 in system (3.1) For any n > 5, we denote p =

 n−1  . Recall the fact that 2 2π 0

sin2m θ cos θ dθ = 0  2k+1  2 1 2 η 2 − sin θ

for any k ≥ 0 and m ≥ 0. Then the first order period bifurcation function of system (3.1) becomes 2π T1 (η) = −

G1 (r, θ) r=r

0 (θ)

dθ

0 p  

2π

=−

 r02k (θ) d2k+1,0 cos2k+2 θ − c0,2k+1 sin2k+2 θ

k=1 0

+

 (d2i−1,2j − c2i,2j−1 ) cos2i θ sin2j θ dθ

 2i+2j=2k+2 1≤i,j≤k p  

 k+1 

2π

=−

r02k (θ)

 Mk,t sin θ dθ, 2t

t=0

k=1 0

where the coefficients Mk,t are dependent on cij and dij as given in Theorem 1.1. Then, we have the following results. Lemma 3.3. 1. By making the transformation η 2 = 1 − w2 , then T1 (η) of system (3.1) becomes

T1 (w) := T1 (η)

η 2 =1−w2

=−

where P1 (w) is a polynomial of degree 4p of the form

1 · P1 (w), w2p−1 (1 − w2 )

(3.5)

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376

P1 (w) = a0 + a1 w2 + · · · + ap−1 w2p−2 + b0 w2p−1 + ap w2p + b1 w2p+1 + ap+1 w2p+2 + · · · + a2p−1 w4p−2 + a2p w4p , where ai (0 ≤ i ≤ 2p), b0 and b1 are dependent on cij and dij as given in Theorem 1.1. 2. The above polynomial P1 (w) can be decomposed as P1 (w) = (1 − w)2 P2 (w), where P2 (w) is a polynomial of degree 4p − 2 of the form

P2 (w) =

4p−2 

ci wi ,

i=0

where ci (0 ≤ i ≤ 4p − 2) depends on cij and dij as given in Theorem 1.1. Moreover, for any i = 2k + 1 and 1 ≤ k ≤ 2p − 2 (k = p − 1 or p), the coefficients ci s satisfy that c1 = 2c0 , ci = 2ci−1 − ci−2 and c4p−3 = 2c4p−2 . Proof. 1. From Lemma 2.3, formula (3.5) can be reduced to

T1 (η) = −

⎤⎫ ⎡    t−j t ⎬  t 1 ⎦ . Mk,t ⎣ (−1)j I (η) k−j ⎭ ⎩ η2 j t=0 j=0

⎧ p ⎨k+1   k=1

By Lemmas 2.6 and 2.7, we get ⎤ ⎡    t−j t  t 1 Mk,t ⎣ (−1)j Ik−j (η)⎦ j η2 t=0 j=0

k+1 



 (k + 1)Mk,k+1 + M k,k η2 t   t−i

k   k − i 1 k−t + It (η)(−1) Mk,k−i η2 k−t t=1 i=−1   (k + 1)Mk,k+1 2 − η2 k+1 k = π(−1) Mk,k+1 + 2π(−1) + Mk,k η2 η2   t+1    k t   η 2t 1 k−i 2 k−t 2(i+1) + Rt−1 (η )(−1) Mk,k−i η 2 t− 12 η2 k−t t=1 (1 − η ) i=−1 = I−1 (η)(−1)k+1 Mk,k+1 + I0 (η)(−1)k

k   π  (−1)k−t 2 2 )M + 2η M Rt−1 (η 2 )Rt+1 (η 2 ) (2k + η + k,k+1 k,k 2 )t− 12 η 2 η2 (1 − η t=1   k   1  2     2 1 2 k− 2 2 k−t = 1−η R1 η + 1−η R2t η . k− 1 η 2 (1 − η 2 ) 2 t=1

= (−1)k

L. Peng et al. / J. Math. Anal. Appl. 453 (2017) 366–382

By substituting η 2 = 1 − w2 into the above formula, it becomes

T1 (w) :=T1 (η)

η 2 =1−w2

=−

p  k=1

w2k−1

      1 R2k w2 + R1 w2 w2k−1 2 (1 − w )

      1 R2k w2 + R1 w2 w2k−1 w2p−2k 2p−1 2 w (1 − w ) p

=−

k=1

  2  2p−1   1 = − 2p−1 + R2p w2 , R1 w w 2 w (1 − w ) where the polynomial Ri (x) is defined as before. Moreover, we have T1 (w) = −

w2p−1

  1 a0 + a1 w2 + · · · + a2p w4p + b0 w2p−1 + b1 w2p+1 . 2 (1 − w )

Define P1 (w) := a0 + a1 w2 + · · · + a2p w4p + b0 w2p−1 + b1 w2p+1 . Then, we arrive at Part 1. 2. Note the fact that T1 (0) = lim T1 (η) η→0



= − lim

2k p 2π η 

η→0

k+1 t=0

 Mk,t sin2t θ



k=1 0

k

1 − η 2 sin2 θ

=0, which implies T1 (1) = 0. Hence, it has 1 · P1 (w) w2p−1 (1 − w2 ) 1−w = − 2p−1 · P2 (w), w (1 + w)

T1 (w) = −

where P2 (w) is a polynomial of degree 4p − 2. Suppose that P2 (w) =

4p−2 

ci wi ,

i=0

where ci is dependent on cij and dij . Then we have P1 (w) = (1 − w)2 P2 (w) =

4p−2  i=0

ci wi − 2

4p−2  i=0

ci wi+1 +

4p−2  i=0

ci wi+2

dθ

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= c0 + (−2c0 + c1 )w + (c0 − 2c1 + c2 )w2 + · · · + (ci−2 − 2ci−1 + ci )wi + · · · + (c4p−3 − 2c4p−2 )w4p−1 + c4p−2 w4p . In view of the fact that when k = p − 1 or p, the coefficients of w2k+1 in P1 (w) are equal to 0, for any i = 2k + 1 (1 ≤ k ≤ 2p − 2 and k = p − 1 or p) we have ci−2 − 2ci−1 + ci = 0. That is, c1 = 2c0 , ci = 2ci−1 − ci−2 and c4p−3 = 2c4p−2 . Hence, we arrive at Part 2. 2 Lemma 3.4. Under the transformation η 2 = 1 −w2 , the derivative of T1 (w) with respect to w can be expressed as T1 (w) = −

1 · P3 (w), w2p (1 + w)2

where P3 (w) is a polynomial of degree 4p. Moreover, we suppose that P3 (w) =

4p 

di w i .

i=0

Then the coefficients di s satisfy di+2 = 2di+1 − di for i = 2k + 1 (0 ≤ k ≤ 2p − 2 and k = p − 1), and d1 = 2d0 and d4p−1 = 2d4p . Proof. From Lemma 3.3, we have T1 (w) = −

1−w P2 (w). + w)

w2p−1 (1

A direct calculation leads to T1 (w) = −

   1 2 2 2  + 2pw (w) + (1 − w )wP (w) . 1 − 2p − 2w − w P 2 2 w2p (1 + w)2

Let     P3 (w) := 1 − 2p − 2w − w2 + 2pw2 P2 (w) + 1 − w2 wP2 (w) = (1 − 2p)c0 + 2(1 − 2p)c0 w + 2(1 − 2p)c4p−2 w4p−1 + (1 − 2p)c4p−2 w4p +

4p−4  i=0

[(3 + i − 2p)ci+2 − 2ci+1 + (−1 − i + 2p)ci ] wi+2 .

(3.6)

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On the other hand, we suppose that P3 (w) =

4p 

di w i .

i=0

Substituting it into (3.6) and equating the corresponding coefficients with respect to wi+2 on both sides, we get di+2 = (3 + i − 2p)ci+2 − 2ci+1 + (−1 − i + 2p)ci for 0 ≤ i ≤ 4p − 4, and d1 = 2d0 and d4p−1 = 2d4p . In the following, we separate our discussions into three cases. Case 1: k = 0. Let i = 0 and i = 1 in (3.7). Then we find d2 =(3 − 2p)c2 − 2c1 + (−1 + 2p)c0 , d3 =(4 − 2p)c3 − 2c2 + (−2 + 2p)c1 . From Lemma 3.3 and d1 = 2(1 − 2p)c0 , we obtain d3 = 2d2 − d1 . Case 2: for 1 ≤ k ≤ 2p − 2 and k = p − 2, p − 1 or p, we have d2k+1 =(2 + 2k − 2p)c2k+1 − 2c2k + (−2k + 2p)c2k−1 , d2k+2 =(3 + 2k − 2p)c2k+2 − 2c2k+1 + (−1 − 2k + 2p)c2k , d2k+3 =(4 + 2k − 2p)c2k+3 − 2c2k+2 + (−2 − 2k + 2p)c2k+1 . It follows from Lemma 3.3 that d2k+1 =(2 + 4k − 4p)c2k+1 + (−2 − 4k + 4p)c2k , d2k+2 =(3 + 2k − 2p)c2k+2 − 2c2k+1 + (−1 − 2k + 2p)c2k , d2k+3 =(6 + 4k − 4p)c2k+2 + (−6 − 4k + 4p)c2k+1 . Then, it has d2k+3 = 2d2k+2 − d2k+1 for 1 ≤ k ≤ 2p − 2 and k = p − 2, p − 1 or p, which implies all equalities in Lemma 3.4 hold. Case 3: k = p − 2. From Lemma 3.3 and (3.7), we get d2p−3 = − 6c2p−3 + 6c2p−4 , d2p−2 = − c2p−2 − 2c2p−3 + 3c2p−4 , d2p−1 = − 2c2p−2 + 2c2p−3 ,

(3.7)

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which implies d2p−1 = 2d2p−2 − d2p−3 . Similarly, we obtain d2p+3 = 2d2p+2 − d2p+1 for k = p. Combining the above three cases, we have completed the proof of Lemma 3.4.

2

Lemma 3.5. For any n > 5 in system (3.1), up to the first order in ε, there are at most 2p = 2 periods which arise from periodic orbits of the unperturbed system.

 n−1  2

critical

Proof. From Lemma 3.4, we have T1 (w) = −

1 P3 (w), + w)2

w2p (1

where P3 (w) is defined as before. We rewrite P3 (w) in the form 

     d1 + d1 w + d2 w2 + d3 w3 + · · · + d2p−2 w2p−2 + d2p−1 w2p−1 P3 (w) = 2   + d2p w2p + d2p+1 w2p+1 + d2p+2 w2p+2 + d2p+3 w2p+3 + · · ·   d4p−1 4p 4p−2 4p−1 . w + d4p−2 w + d4p−1 w + 2 Recall that d2k+3 = 2d2k+2 − d2k+1 . Then d2k+3 must have the same sign as d2k+2 in the case where d2k+2 has the different sign with d2k+1 for 0 ≤ k ≤ 2p − 2 and k = p − 1. We obtain that the ordered list of coefficients of P3 (w) changes its sign at most 2p times. Since √ dT1 (η) 1 − w2  =− T1 (w), dη w we conclude that for any n > 5, system (3.1) has at most 2p = 2

 n−1  2

critical periods. 2

Proof of Theorem 1.1. Theorem 1.1 follows from Lemmas 3.1–3.2 and Lemma 3.5 immediately.

2

  Remark 3.6. Although it is not always easy to verify whether the upper bound 2 n−1 of any n > 5 in 2 Theorem 1.1 can be reached, there exist some examples of n = 11 in system (1.3) to explicitly show that   they have exactly 2 n−1 = 10 critical periods. 2 For example, we consider the system

x˙ = −y + x2 y + ε

5 

c0,2i+1 y 2i+1 + c2,9 εx2 y 9 ,

i=1 2

y˙ = x + xy + ε

5  i=1

d2i+1,0 x

(3.8) 2i+1

,

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where 1 19483077797631322707649 105756648169531802843 , c0,9 = − , c0,7 = , 2362500 8329263638642150400000 30163588485617664000 105710470495617105840973 183427381589476743962753 c0,5 = − , c0,3 = , 17451790480964505600000 13961432384771604480000 214318538621851819895963 c2,9 = , 11452737503132956800000 27433499169500 48462282866675 23104554865321499 d11,0 = , d9,0 = − , d7,0 = , 4909438230081 7141001061936 2938011865482240 1489582051634199467 290772914380407223097 d5,0 = , d3,0 = . 634610562944163840 21512222472683520000 c0,11 = −

Then after the transformation η 2 = 1 − w2 , a straightforward computation yields T (w)

       1 1 2 1 3 1 1 =− w − w − w − w − w − w − 2154542034686976w10 (1 + w)2 2 3 3 4 4 5       3 4 1 2 w− w− w− · 33664719291984 + 1018357758582516w w− 5 5 5 6 + 13378257143885872w2 + 99345559792748335w3 + 450873908839114591w4 + 1247198391863333329w5 + 1984629577408083133w6 + 1931771622156374500w7  8 9 10 . + 1156673918804365500w + 395042388040800000w + 59256358206120000w Recalling that √ dT1 (η) 1 − w2  =− T1 (w), dη w we conclude that system (3.8) has exactly 10 critical periods bifurcating from the period orbits of the unperturbed system. References [1] L. Bonorino, E. Brietzke, J.P. Lukaszczyk, C.A. Taschetto, Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation, J. Differential Equations 214 (2005) 156–175. [2] J. Chavarriga, M. Sabatini, A survey of isochronous centers, Qual. Theory Dyn. Syst. 1 (1999) 1–70. [3] X.W. Chen, V.G. Romanovski, W.N. Zhang, Critical periods of perturbations of reversible rigidly isochronous centers, J. Differential Equations 251 (2011) 1505–1525. [4] C. Chicone, The monotonicity of the period function for planar Hamiltonian vector fields, J. Differential Equations 69 (1987) 310–321. [5] C. Chicone, F. Dumortier, Finitness for critical periods for plane vector fields, Nonlinear Anal. 20 (1993) 315–335. [6] C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486. [7] S.N. Chow, J.A. Sander, On the number of critical points of period, J. Differential Equations 64 (1986) 51–66. [8] S.N. Chow, D. Wang, On the monotonicity of the period function of some second order equation, Čas. Pěst. Mat. 111 (1986) 14–25. [9] A. Cima, A. Gasull, F. Mañosas, Period function for a class of Hamiltonian systems, J. Differential Equations 168 (2000) 180–199. [10] A. Cima, A. Gasull, P.R. Silva, On the number of critical periods for planar polynomial systems, Nonlinear Anal. 69 (2008) 1889–1903. [11] M. Grau, J. Villadelprat, Bifurcation of critical periods from Pleshkan’s isochrones, J. Lond. Math. Soc. 81 (2010) 142–160. [12] A. Gasull, J. Yu, On the critical periods of perturbed isochronous centers, J. Differential Equations 244 (2008) 696–715.

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