SOME UPPER ESTIMATES OF THE NUMBER OF LIMIT CYCLES OF ...
MOSCOW MATHEMATICAL JOURNAL Volume 1, Number 4, October–December 2001, Pages 583–599
SOME UPPER ESTIMATES OF THE NUMBER OF LIMIT CYCLES OF PLANAR VECTOR FIELDS ´ WITH APPLICATIONS TO LIENARD EQUATIONS YU. ILYASHENKO AND A. PANOV To the memory of Ivan Georgievich Petrovskii, great personality and great mathematician
Abstract. We estimate the number of limit cycles of planar vector fields through the size of the domain of the Poincar´e map, the increment of this map, and the width of the complex domain to which the Poincar´e map may be analytically extended. The estimate is based on the relationship between the growth and zeros of holomorphic functions [IYa], [I]. This estimate is then applied to getting the upper bound of the number of limit cycles of the Li´enard equation x˙ = y − F (x), y˙ = −x through the (odd) power of the monic polynomial F and magnitudes of its coefficients. 2000 Math. Subj. Class. 34Cxx, 34Mxx. Key words and phrases. Limit cycles, Poincar´ e map, Li´ enard equation.
1. Introduction 1.1. Hilbert’s 16th Problem and Li´ enard equations. Questions related to bounds of the number of limit cycles of planar vector fields go back to the second half of Hilbert’s 16th Problem. Smale suggested to study a special class, namely, Li´enard equations with polynomials of odd degree, where the Poincar´e map is globally defined and nonidentical [S]. These equations were studied by many authors; in [LMP] small perturbations of linear center are examined; small amplitude limit cycles are discussed in [LL]. All these approaches are based on different kinds of perturbations techniques. Below we study equations far from integrable ones and limit cycles distant from equilibrium points. We use the method developed in [IYa] for linear equations and based on the growth-and-zeros theorem for holomorphic functions, see 2.1 below. Using this theorem, we obtain upper bounds for the number of limit cycles of planar vector fields (Theorems 1 and 3 below). The main result is Theorem 2, which gives a bound for the number of limit cycles the Li´enard Received October 30, 2001; in revised form December 19, 2001. The authors were supported by part by the grants RFBR 95-00-0455, CRDF RM1-2086; the first author was also supported by the grants NSF DMS 997-0372, NSF 0010404. c
2001 Independent University of Moscow
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equation x˙ = y − F (x), y˙ = −x (1.1) provided that the polynomial F is monic, and the magnitude of the coefficients is no greater than C. The estimate depends on this C as well as on the degree n of the polynomial. The estimate of Theorem 2 is triple exponential, and does not pretend to be realistic (Smale conjectures that it is polynomial in n and does not depend on C [S]). Yet it is the only known estimate of the number of limit cycles of Li´enard equations with an arbitrary odd degree of the polynomial. For n ≤ 3, the Li´enard equation has no more than one limit cycle [LMP]. In [LMP] the authors suggested that the number of limit cycles of (1.1) is no greater than [n/2] + 1. This estimate is proved for small perturbations of a linear center of the form: x˙ = y − εF (x), y˙ = −x/ε 1. For almost 25 years this conjecture remains neither proved nor disproved. But there is no other result that confirms this conjecture up to now. Recently the same method was applied by the first author to bound the number of limit cycles of the Abel equation [I]. Let v be an analytic vector field in the real plane, that may be extended to C2 . For any set D in a metric space denote by U ε (D) the ε-neighborhood of D. The metrics in C and C2 are given by: ρ(z, w) = |z − w|,
z, w ∈ C;
ρ(z, w) = max(|z1 − w1 |, |z2 − w2 |),
z, w ∈ C2 .
Denote by |D| the length of the segment D. For any larger segment D0 ⊃ D, let ρ(D, ∂D0 ) be the Hausdorff distance between D and ∂D0 . 1.2. Upper bounds of the number of limit cycles. Consider the system x ∈ R2 .
x˙ = v(x),
(1.2)
Theorem 1. Let Γ be a cross-section of the vector field v, D ⊂ Γ a segment. Let P be the Poincar´e map of (1.2) defined on D, and D ⊂ D0 = P (D). Suppose that P may be analytically extended to U = U ε (D) ⊂ C, ε < 1, and P (U ) ⊂ U 1 (D0 ) ⊂ C.
(1.3)
Then the number #LC(D) of limit cycles that cross D admits an upper estimate: −1
#LC(D) ≤ e2|D|ε
log
|D0 | + 2 . ρ(D, ∂D0 )
The same is true for P replaced by P −1 . Theorem 1 is derived in §2 from the Growth-and-Zeros theorem [IYa], [I]. An application of Theorem 1 to the Li´enard equation provides the following Theorem 2. Consider the Li´enard equation x˙ = y − F (x), y˙ = −x, F (x) = xn +
n−1 X 1
aj xj , |aj | ≤ C, C ≥ 4, n ≥ 5, (1.4)
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and suppose that n is odd. Then the number L(n, C) of limit cycles of (1.4) admits the following upper estimate: L(n, C) ≤ exp(exp C 14n ). Remark. The assumption F (0) = 0 does not reduce the generality; it may be fulfilled by a shift y 7→ y + a. The assumption that F is monic may be fulfilled by rescaling in x, y and reversing time if necessary. Assumption: n ≥ 5 does not reduce the generality. Indeed, for n = 3, L(n, C) = 1, and n is odd. 1.3. Plan of the proof of Theorem 2. In order to deduce Theorem 2 from Theorem 1, we need the following data: • D, a subdomain of the inverse Poincar´e map that is crossed by all the limit cycles of (1.4); • ρ(D, ∂D0 ), where D0 = P −1 (D) ⊂ D; • ε, the radius of the neighborhood of D to which the inverse Poincar´e map P −1 may be extended with restriction (1.3). The main step is the description of D. Suppose that (1.4) has at least one limit cycle, or else there is nothing to discuss. The cycles of (1.4) form a nest around the origin. Denote by E the most interior cycle of this nest. Let Γ be a section x = 0, y ≤ 0. It is transversal to (1.4) everywhere except for 0. Let E = E ∩ Γ. In order to specify D, we analyze infinite singular points of (1.4), see [LMP]. It appears that (1.4) has two unbounded repelling domains D+ , D− shown in Figure 2 below. Consider any orbit starting from a point of D+ , say A. It hits Γ at a point B. After that the orbit winds around zero, never crossing the cycle E. All the periodic solutions of (1.4) cross Γ inside the segment [0, P (B)] that belongs to [0, P (B)]. Indeed, orbits with initial condition below P (B) on Γ are absorbed by D+ in the reversed time after one circuit around zero. We will choose D ⊂ [0, P (B)]. This will give an upper estimate for the length of D as soon as |P (B)| is estimated through C. This is done in 3.4. The gap ρ(D, ∂D0 ) is estimated from below by |B − P (B)|, which is, in turn, estimated like follows. Take a point A0 ∈ D+ , see Fig. 2, and consider an arc of the orbit from the moment when it starts at A0 untill it hits Γ at a point, say, B 0 . The point A0 is chosen so that B 0 ∈ [0, B]. Then B 0 ∈ [B, P (B)], or else P −1 (B 0 ) will be well defined which is impossible. Hence, the distance |B − B 0 | provides a lower estimate for ρ. For A and A0 specially chosen, this distance is estimated from below in 3.5. To estimate the width of the neighborhood of D to which the Poincar´e map of (1.4) may be analytically extended from D, we use the Gronwall inequality. To apply it, we need to estimate from above the time of the first return to Γ. This is done in Section 4. In Section 5 the main theorem is deduced from Theorem 3 stated in the next subsection. This theorem provides an estimate of the width of the domain of the Poincar´e map, and is proved in 2.2. 1.4. Refinement of Theorem 1. The parameter ε in Theorem 1 may be estimated from below through the right-hand side of (1.2) in the following way. Let D, D0 and Γ be the same as in Theorem 1.
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Theorem 3. Let P : D → D0 be the Poincar´e map of (1.2). For any x ∈ D denote by ϕx,P (x) the arc of the phase curve of (1.1) starting at x and ending at P (x). Let [ Ω(D) = ϕx,P (x) , x∈D
and 1 ≤ µ = max |v|, 2
L = 2µ.
(1.5)
U (Ω)
Let t(x) be the time length of the arc ϕx,P (x) , and Tmax = maxx∈D t(x), Let δ ≤ e−LT ,
λ=
T = Tmax + 1. √ 2
ε = δ2 .
δ,
(1.6)
Suppose that (z1 , z2 ) are coordinates in C , Γ = {z1 = 0}, v = (v1 , v2 ). Let K ⊂ D be a segment, K 0 = P (K), Πδ = U δ (0) × U λ (K 0 ) ⊂ C2 . Suppose that v 2 (1.7) ≤ µ in Πδ . v1 C
Then
(1) The Poincar´e map P : K → K 0 of (1.2) may be analytically extended to U ε (K) ⊂ C Γ, and P (U ε (K)) ⊂ U 1 (K 0 ). (2) For system (1.2), we have #LC(K) ≤ e2|D|e
2LT
|D0 | + 2 . ρ(K, ∂K 0 )
log
The S same is true for P replaced by P −1 . In this case P −1 (D) = D0 , Ω = x∈D0 ϕx,P (x) . Theorems 1, 3 are proved in Section 2, Theorem 2 is proved in Sections 3–5. 2. Estimates of the number of limit cycles 2.1. An estimate using the hidden data. In this subsection Theorem 1 is proved. It uses the width of the complex domain U ε to which the Poincar´e map may be extended. The parameter ε is not easy to find by straightforward calculation from equation (1.2), see Theorem 3. This parameter is called “hidden data” in the title. Proof of Theorem 1. In [IYa], [I] a Growth-and-Zeros theorem for holomorphic functions was proved; we quote the corollary that we need below: Theorem 4 [I]. Let D be a real segment, U = U ε (D) be its ε-neighborhood in C, |D| be the length of the segment. Let f be a holomorphic function in U continuous up to the boundary. Then −1
#{z ∈ D : f (z) = 0} ≤ e2|D|ε
log
maxU¯ |f | . maxD |f |
(2.1)
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To prove Theorem 1, we apply this result to the increment of the Poincar´e map P in U : f (z) = P (z) − z. The diameter of the domain U 1 (D0 ) is equal to |D0 | + 2. Inclusions D ⊂ D0 , P (U ) ⊂ U 1 (D0 ) imply: max |f | ≤ |D0 | + 2. ¯ U
On the other hand, P moves one of the endpoints of D to an endpoint of D0 located at the distance ρ from its source. Hence max |f | ≥ ρ. D
Now (2.1) implies the estimate of Theorem 1.
2.2. An estimate using available data. A lower estimate of the parameter ε in Theorem 1 is given below through the data that often may be computed by equation (1.2) in a straightforward manner. This will prove the first statement of Theorem 3. The second statement is an immediate consequence of the first one and Theorem 1. Proof of Statement 1 of Theorem 3. We need to prove that for any x ∈ K the Poincar´e map of (1.2) may be extended to an ε-neighborhood of x in C. Let, as before, ϕx,P (x) = γx be the arc of the time oriented orbit of (1.2) from x to P (x), t(x) be the time length of γx . Let x = Re z1 , y = Re z2 , where z1 , z2 are the coordinates in C2 . By (1.5) and the Cauchy inequality, ∂vi , i = 1, 2; j = 1, 2 ≤ µ. max U 1 (Ω) ∂zj Hence, in the norm |(z1 , z2 )| = max(|z1 |, |z2 |), we have:
∂v
≤ L. max
U 1 (Ω) ∂(z1 , z2 ) By (1.6),
εeLT ≤ δ.
Let z ∈ U ε (K); hence, |z − x| ≤ ε for some x ∈ K. Let ϕz be a solution of the complexified equation (1.2) with complex time, passing through z for t = 0: ϕz (0) = z. By (1.5), (1.6) and the definition of T , we have εeLT < 1. Then, by the Gronwall inequality, |ϕz (t) − ϕx (t)| ≤ εeLt . Hence, |ϕz (t(x)) − ϕx (t(x))| ≤ δ. Note that ϕx (t(x)) = P (x) ∈ K 0 , because x ∈ K. Therefore, a := ϕz (t(x)) ∈ Πδ . In general a 6∈ C Γ. Let a = (a1 , a2 ), P (x) = (0, b). Then |a2 − b| ≤ δ.
(2.2)
The leaf ϕa in some neighborhood of a is a solution z2 = w(z1 ) of the equation dz2 v2 = . dz1 v1
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The function w may be extended over any segment in U δ (0) until its graph reaches ∂Πδ . Over the segment [a1 , 0] this graph will not reach ∂Πδ , because, by (1.7) and (2.2), at any point z1 ∈ [a1 , 0] where w is well defined, |w(z1 ) − b| ≤ |w(z1 ) − a2 | + |a2 − b| ≤ µδ + δ. √ But (µ+1)δ ≤ λ = δ, because for T ≥ 1, µ > 0, we have eµT > µ+1. Hence, w(0) is well defined. Let P (z) = w(0). This is the desired analytic extension of P from K to U ε (K); moreover, P (U ε (K)) ⊂ U λ (K 0 ) ⊂ U 1 (K 0 ). This proves Statement 1 of Theorem 2. ´ map for the Lie ´nard equation 3. Domain of the inverse Poincare In this and next two sections, Theorem 2 is proved. The proof is based on Theorems 1, 3 applied to the inverse Poincar´e map. Hence, in what follows, D and D0 belong to the domain and the image of the inverse Poincar´e map respectively. We mainly study equation (1.4) in the right halfplane x > 0. The conclusions may be easily reformulated for the halfplane x < 0, because the symmetry (x, y) 7→ (−x, −y) brings (1.4) to an equation of the same type. All the estimates below hold true for C ≥ 2, except for the very end of 5.4, where the inequality C ≥ 4 is needed. 3.1. C-monic polynomials. The following elementary estimates will be used all over this and the next sections. Definition 3.1. A C-monic polynomial is a real polynomial in one variable with the highest coefficient one and other coefficients no greater than C in absolute value, with zero constant term. Proposition 3.1 (Properties of C-monic polynomials). Let F be a C-monic polynomial of degree n, C ≥ 2. Then 1. 2.
F (x) > 0
n
for x ≥ C + 1,
min F (x) ≥ −(C + 1) , x>0
max |F (x)| ≤ 2An
for A ≥ C + 1,
x∈[0,A]
3. 4.
|F (x)| ≥
1 n |x| 2
for |x| ≥ 2C + 1,
max |F 0 (x)| ≤ Cn2 X n−1 [0,X]
for X ≥ 1,
(3.1) (3.2) (3.3) (3.4)
1 . (3.5) 2 n Remark. (3.1) implies that F (x) < 0 for x ≤ −(C + 1), maxx 0. This yields (3.1). 2. We have for x ≥ 0 |F (x)| ≤ xn + C
n−1 X 1
xj = G(x).
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589
The polynomial G is monotonious in x for x > 0. Hence, for A ≥ C + 1: max G(x) = G(A) = An + C [0,A]
An − 1 ≤ 2An . A−1
3. For x ≥ 2C + 1, |F1 (x)| < xn /2, see the first inequality in (3.6). This implies (3.3). 4. Estimate (3.4) is straightforward. 5. For |z| ≤ 1/2, z ∈ C, |F (z)| =
|a1 | +
n−1 X
j−1
aj |z|
|z| ≤
|a1 | + C|z|
1
n
1 − |z| 1 − |z|
|z|
≤ (|a1 | + 2C|z|)|z| ≤ 2C|z|. 3.2. Diacritical nodes at infinity. Behavior of the solution of the Li´enard equation near infinity is well known, see [LMP]. We need only a part of this information. Proposition 3.2. Equation (1.4) has a diacritical node at infinity; in the projective chart (u, v) = (1/x, y/x) (3.7) it is (u, v) = (0, 0). Proof. In the chart (u, v), after change of time, (1.4) takes the form: u˙ = u(F˜ (u) − un−1 v),
v˙ = v F˜ (u) − un−1 (1 + v 2 ),
(3.8)
where F˜ (u) = u F (1/u). Note that the free term of F˜ (u) is 1 because F is monic. Hence, the linear part of (3.8) at zero is u˙ = u, v˙ = v. This proves the proposition. n
3.3. Overflowing domains near infinity. The infinite line in the chart (x, y) becomes u = 0 in the chart (u, v). In the neighborhood of zero the field (3.8) is close to its linear part. Hence, parabolic sectors v 2 ≤ |u| ≤ a,
a small,
(3.9)
are overflowing domains of (3.8): the vector field on the boundary is directed outwards everywhere except for the origin, see Fig. 1. Proposition 3.3. Domains + DC = {(x, y) ∈ R2 : x ≥ 2C, |y| ≤ − DC
2
= {(x, y) ∈ R : (−x, y) ∈
+ DC }
√
x},
(3.10)
are overflowing for system (1.4) with n odd. Proof. For n odd, the image of the vector field (1.4) under the map (3.7) equals to the vector field (3.8) multiplied by a positive function. Domains (3.10) are transformed by (3.7) to (3.9) with a = 1/(2C). Hence, it is sufficient to prove that the domain (3.9) with this a is overflowing for (3.8).
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YU. ILYASHENKO AND A. PANOV
u
y − DC
+ DC
v
x
(a)
(b)
Figure 1. Overflowing domains (a) in (u, v)-chart, (b) in (x, y)-chart. On the segment u = a, |v| ≤ u˙ = F˜ (u) − un−1 v ≥ 1 − u
√
a, a = 1/(2C) we have:
n X
Caj − an−1/2 ≥ 1 − Ca
1
1 − an−1 − a > 0, 1−a
because C ≥ 2, hence a ≤ 1/4. √ On the arc of the parabola u = v 2 , |v| ≤ a, d 2 (v − u) = v 2 (F˜ (v 2 ) − 2v 2n−3 − v 2n−1 ). dt For 0 ≤ u ≤ a we have: n−1
1−u F˜ (u) ≥ 1 − Cu ≥ 1−u √ On the other hand, for |v| ≤ a, where a is the same
1 . 3 as before, n ≥ 3, we have:
|2v 2n−3 | + |v 2n−1 | ≤ 2an−3/2 + an−1/2 ≤
1 1 1 + < . 4 32 3
d (v 2 − u) > 0 on the arc considered. In the same way, the arc u = −v 2 , Hence, dt √ √ |v| ≤ a and the segment u = −a, |v| ≤ a are treated.
3.4. Domain of the inverse Poincar´ e map. The line Γ : x = 0 is the horizontal isocline of (1.4), the graph Λ : y = F (x) is the vertical one. For n odd, the graph of a monic polynomial F has two branches going to infinity in the first and third quadrants. The isoclines Λ and Γ split R2 into four regions where the direction of v in terms of “up-down”,“right-left” is the same, see Figure 2. In G1 : x > 0, y > F (x), the field v points right-down; in G2 : x > 0, y < F (x), left-down; in G3 : x < 0, y < F (x), left-up; in G4 : x < 0, y > F (x), right-up. Note that |y| ˙ is bounded in any strip |x| ≤ α. Hence, any orbit starting at y ∈ Γ, returns to Γ. It may return to the singular point zero, where Γ is not transversal to the field (1.4). Otherwise, the Poincar´e map is well defined at y and near y. In the backward time, not any orbit starting from Γ returns to Γ: it may be cap+ − tured by one of the domains DC , DC which become absorbing after time reversal. We will now “estimate from above” the domain D0 of the inverse Poincar´e map.
´ NUMBER OF LIMIT CYCLES OF LIENARD EQUATIONS
G4
591
G1
A−
− DC
D
+ DC
P (B) D0 A0 G3
A
G2 ψ B0
0 F (x) by (3.1). Take a point A = (4k 2 , −2k) ∈ ∂DC . The positive semiorbit ϕ of (1.4) starting at A will go left and down until it crosses either Γ or Λ. But the second case is impossible because in the right halfplane the field v points downwards on Λ. Denote by B the first intersection of ϕ and Γ, and let Y0 = |y(B)|, see Figure 2. + Any orbit of the reversed system (1.4) starting on Γ below B enters DC and is absorbed. Hence, the inverse Poincar´e map is not defined on Γ for y < −Y0 . Therefore, any closed orbit of (1.4) crosses [0, −Y0 ].
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YU. ILYASHENKO AND A. PANOV
We will now estimate Y0 from above. By (3.1), x˙ = y − F (x) < −k
for (x, y) ∈ G2 ∩ {y < −2k}.
Hence, the time length of the arc ϕA,B is no greater than x(A)/k = 4k. On the other hand, in the strip 0 ≤ x ≤ 4k 2 we have: y˙ = −x ≥ −4k 2 ,
|y| ˙ ≤ 4k 2 .
Hence, Y0 = |y(B)| ≤ | − 2k − 16k 3 | < 17k 3 = Y. The arguments for G4 and y > 0 are obtained from the previous ones by symmetry; they show that P −1 is not defined for y ≥ Y on Γ. This proves Lemma 1, together with a stronger statement: every periodic orbit crosses [0, B]. Remarks. 1. The estimate for Y0 above is very robust. We proved at the same time that Y0 + 2 < Y . 2. The arguments above prove as well that an orbit starting at −A = (−4k 2 , 2k) hits the y axis at a point lower than (0, Y ). Therefore, any orbit that starts on Γ crosses the segment [0, Y ] after a half curcuit around the origin. 3.5. Increment of the inverse Poincar´ e map. In this subsection we construct a segment D that belongs to the domain of the inverse Poincar´e map and that is crossed by any periodic orbit of (1.4). Let E ∈ [0, −Y ] be the fixed point of the Poincar´e map P of (1.4) with the maximal y-coordinate. Let B be the same as in 3.4, see Figure 2. Let ( [0, B] in case |a1 | ≤ 1/4, D0 = D = P (D0 ). (3.12) [E, B] in case |a1 | > 1/4, Any periodic orbit of (1.4) crosses D0 , hence D; see the statement at the end of the proof of Lemma 1. The splitting in two cases above is motivated in the following way. In the first case, for |a1 | ≤ 1/4, the singular point 0 is a focus. The domain of the Poincar´e map and of its complex extension is easily studied near zero. The time length of corresponding arcs of the phase curves of (1.4) is estimated from above, see 4.1 below. Periodic orbits may occur in an arbitrary neighborhood of 0. In the second case, 0 has a well controlled basin of attraction in forward or backward time. Periodic orbits of (1.4) cannot touch this basin. Therefore in case 2 we consider the Poincar´e map on a segment separated from 0. Note that in both cases the upper end of D (0 or E) is mapped by P into itself. Hence, in both cases ρ = |B − P (B)|. In what follows, the lower estimate for ρ, hence, for the increment of the Poincar´e map is given. Let A0 = ( 94 k 2 , − 32 k), ψ be the positive semiorbit of (1.4) starting at A0 , see Fig. 2. The same arguments as before show that ψ crosses Γ; let B 0 ∈ [0, B] be the first intersection point of ψ and Γ.
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593
The inverse Poincar´e map is not defined at any y ∈ Γ below B 0 . Indeed, any + negative semiorbit starting below B 0 , hits DC and is absorbed, thus never returns to Γ. Hence, the image P (B) lies above B 0 , and therefore Y1 = |y(B 0 )|, Y0 = |y(B)|
ρ ≥ Y0 − Y1 ,
as before. We will prove that ρ is no less than a constant of order k, see Corollary 1 below. The reason is that in G2 ∩ {y ≤ −3k/2} the vector field of (1.4) is almost horizontal. Proposition 3.4. Let y1 (x) and y2 (x) be solutions of the equation dy x = dx F (x) − y
(3.13)
that corresponds to system (1.4). Let x0 ≤ 3k 2 , yj (x0 ) ≤ −3k/2, j = 1, 2. Then 1 |y1 (0) − y2 (0)| ≥ |y1 (x0 ) − y2 (x0 )|. 2 Corollary 1. For system (1.4), we have ρ = d(D, ∂D0 ) ≥ k/4. Proof. Let y = y1 (x), y = y2 (x) be the orbits of (1.4) passing through A and A0 and considered over [0, 4k 2 ] and [0, 49 k 2 ] respectively. Let x0 = 94 k 2 . Then y1 (x0 ) < −2k, y2 (x0 ) = −3k/2. By Proposition 3.4, 1 k Y1 − Y0 = y2 (0) − y1 (0) ≥ (y2 (x0 ) − y1 (x0 )) ≥ . 2 4 Proof of Proposition 3.4. Let y(x, y0 ) be the solution of (3.13) with the initial condition y(x0 , y0 ) = y0 . Consider the variational equation for (3.13) with respect to the initial conditions: x ∂y(x, y0 ) dξ = ξ, ξ= . (3.14) dx (F (x) − y)2 ∂y0 The solution y(x, y0 ) of (3.13) with y0 < −k is monotonously increasing on [0, x0 ], because F (x) − y > 0 in x > 0, y < −k. Hence, in (3.14), y = y(x, y0 ) ≤ −k for x ∈ [0, x0 ]. Let x a(x, y0 ) = . (F (x) − y(x, y0 ))2 Rx Then ξ(x, y0 ) = exp x0 a(x, y0 ) dx := exp I(x, y0 ). We will prove that I(0, y0 ) > − log 2, provided that y0 ≤ −3k/2. This will prove Proposition 3.4. By (3.3), the inequality F (x) − y > xn /2 holds in the domain x ∈ [2C + 2, x0 ], y < −k. Hence, for (x, y(x, y0 )) lying in this domain, a(x, y0 ) ≤ 4/x2n−1 . For these x, the inequalities C ≥ 2, n ≥ 3 imply: 4 1 I(x, y0 ) ≥ − ≥− . 2n−2 (2n − 2)(2C + 2) 8 By (3.1), the inequality F (x) − y > k/2 holds in the domain x ∈ [0, 2C + 2], y < −3k/2. Hence, for x ∈ [0, 2C + 2], y ≤ −3k/2, a(x, y0 ) ≤ 4x/k 2 , I(0, y0 ) ≥
Z
0
2C+2
a(x) dx −
1 8(C + 1)2 1 1 ≥− − ≥ − > − log 2. 8 (C + 1)2n 8 4
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YU. ILYASHENKO AND A. PANOV
3.6. Suspension over the Poincar´ e map. Let D be the same as in (3.12). Denote by γy the arc ϕy,P −1 (y) of the orbit of (1.4) with time reversed, ran from y to P −1 (y), y ∈ D. Let [ Ω= γy . (3.15) y∈D
Proposition 3.5. The domain Ω defined in (3.15) belongs to a rectangle Π = S × Σ,
S ⊂ Ox ,
Σ ⊂ Oy = Γ,
Σ = {|y| ≤ Y = 17k 3 },
(3.16) 3
S = {|x| ≤ X := 4(C + 1) }. Proof. The maximal value x(y) of x on γy is achieved at a point a(y) of intersection of γy with the graph Λ in the right halfplane: a(y) = (x(y), F (x(y))). We need to prove that x(y) ≤ X. By Remark 2 of 3.4, the point a(y) is lower than Y . Hence, F (x(y)) ≤ Y. Suppose that x(y) ≥ 2C + 1; if not, the proposition is proved. Then, by (3.3), F (x(y)) ≥ 21 xn (y). Hence, 1 n 1/n 3 x (y) ≤ Y, x(y) ≤ (2Y ) = (34k 3 )1/n ≤ 4(C + 1) = X. (3.17) 2 Together with the symmetry arguments at the beginning of the section, this proves the estimate x|γy ≥ −X as well. 4. Time of the first return 4.1. Neighborhood of the singular point. The singular point (0, 0) is a focus if |a1 | < 2 and a node if a1 ≥ 2. We will use another distinction: |a1 | ≤ 1/4, the polar angle near 0 varies monotonously in a fast way; |a1 | > 1/4, the polar angle varies slowly monotonously or nonmonotonously. Let ϕ, r be polar coordinates on R2 , ϕ, ˙ r˙ be derivatives with respect to (1.4). Proposition 4.1. Let |a1 | ≤ 1/4. Then ϕ˙ ≤ −1/2 in the disk 1 . (4.1) 8C The Poincar´e map, together with its inverse, is well defined near zero and admits a complex extension to a disk |z| ≤ βe−2π . Moreover, in this disk, Dβ : r ≤ β =
|P (z)/z| ≤ e2π ,
|P −1 (z)/z| ≤ e2π .
(4.2)
Proof. A system x˙ = f , y˙ = g in polar coordinates has the form: xf + yg xg − yf := R, ϕ˙ = := Φ, (4.3) r r2 ϕ ∈ S 1 , r ∈ R+ . Let us complexify r, replacing it by z ∈ C. Then, for (1.4), in the domain Uβ : |z| ≤ β, ϕ ∈ S 1 , r˙ =
´ NUMBER OF LIMIT CYCLES OF LIENARD EQUATIONS
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Φ(z, ϕ) = −1 + sin ϕ cos ϕ F1 (z cos ϕ),
(4.4)
we have: ϕ˙ = Re Φ,
1
where F1 (x) = F (x)/x. By (3.5), for |z| ≤ β, ϕ ∈ S , 1 1 + 2Cβ ≤ . (4.5) 4 2 Hence, |F1 (z cos ϕ)| ≤ 1/2, Re Φ < −1/2 in Uβ . This proves the first statement of the proposition. Let us prove the second statement. By (4.3), |F1 (z cos ϕ)| ≤
z˙ = R,
R(z, ϕ) = − cos2 ϕ F1 (z cos ϕ)z.
(4.6)
By (4.5), in Uβ , |z/z| ˙ ≤ 1/2. For the system (4.3) with r complexified, that is, for the system (4.4), (4.6), the absolute value of the time of the first return for the orbits that remain in Uβ is no greater than 4π. This follows from (4.4) and the inequality Re Φ < −1/2. Hence the orbits with the initial condition: |z| ≤ βe−2π ,
ϕ=0
remain in Uβ during at least one circuit along S 1 and come back to the disk |z| ≤ β, ϕ = 0. This proves that the Poincar´e map of (4.3) may be analytically extended to the disk |z| ≤ βe−2π . The same arguments prove this statement for P −1 . Statement (4.2) follows from the inequalities |z/z| ˙ ≤ 1/2, |ϕ| ˙ ≥ 1/2. Proposition 4.2. If |a1 | > 1/4, then r˙ ≥ 0 or r˙ ≤ 0 in the disk Dβ , see (4.1). Proof. By (4.5), in Dβ , r˙ = − cos2 ϕ F1 (r cos ϕ)r, F1 = F/x. Moreover, F1 (x) = (a1 + F2 (x)),
|F2 (x)| ≤ 2C|x|;
the latter inequality is contained in the proof of (3.5). If |x| ≤ β, then 2C|x| ≤ 1/4. Hence, sgn F1 = sgn a1 in Dβ . This proves Proposition 4.2. Remark. Equation (1.4) has no invariant circles. So Proposition 4.2 implies that for |a1 | > 1/4, 0 is either attracting or repelling fixed point with the basin that contains Dβ . Hence, no periodic orbit touches Dβ . Therefore, in case |a1 | > 1/4, by (3.12), we have D ∩ Dβ = ∅, D0 ∩ Dβ = ∅. 4.2. Strip around the vertical isocline. Lemma 2. Let, as before, γy be the arc ϕy,P −1 (y) of the phase curve of (1.4), D and X be the same as in (3.12) and (3.17), y ∈ D. Then t(y), the time length of γy , admits an estimate Tmax = max t(y) ≤ 200 C 3 n2 X n . y∈D
(4.7)
Proof. The arcs γy , y ∈ D, belong to the strip {|x| ≤ X}, see Proposition 3.5. On the other hand, the time length of any arc of the orbit of (1.4) located in Dβ and making no more than 1 circuit around zero is no greater than 4π for |a1 | ≤ 1/4, see Proposition 4.1. For |a1 | > 1/4, no closed orbit touches Dβ by Proposition 4.2. By definition of D, see (3.12), no arc γy , y ∈ D, touches Dβ provided that |a1 | > 1/4.
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Let us now estimate the time length of the part of γy located in G = {|x| ≤ X} \ Dβ . We will split G into two domains: |x| ˙ ≤ α and |x| ˙ > α for α small to be chosen later. The second domain contains two parts of γy : one with x˙ < −α, the other with x˙ > α. The time length of any of them is no greater than 2X/α. We will now choose α so small that the curvilinear strip Sα = {(x, y) ∈ R2 : |y − F (x)| ≤ α, |x| ≤ X, (x, y) ∈ / Dβ } is crossed by the orbits of (1.4) in the time no greater than 1. Proposition 4.3. Let 1 ω 3 , X = 4(C + 1) , α = . (4.8) 24C 2 2Cn2 X n−1 Then the time length of any arc of the orbit of (1.4) located in Sα is no greater than 1. ω=
Proof. By the symmetry arguments it is sufficient to prove that in Sα ∩ {x > 0} d (y − F (x)) ≤ −2α. dt Let us first prove that in Sα we have: |x| > ω. Namely, let |x| ≤ ω, |y − F (x)| ≤ α. Then (x, y) ∈ Dβ . Indeed, |x| + |y| ≤ ω + max[0,ω] |F | + α. By (4.8), α < ω < 1/2. By (3.5), |F (x)| ≤ 2Cω. Hence, for x ∈ [0, ω], |x| + |y| ≤ (2C + 2)ω ≤ 3Cω =
1 = β. 8C
By (3.4), |F 0 (x)| ≤ Cn2 X n−1 in G. Therefore, for x > 0, (x, y) ∈ Sα we have: x > ω, and d (y − F (x)) = −x − (y − F (x))F 0 (x) ≤ −ω + αCn2 X n−1 ≤ −2α, dt because α=
ω 2Cn2 X n−1
100 for C ≥ 2.
(5.3)
Take n
L = 2µ = 6(X + 2) . Now, let T = Tmax + 1, see (4.7). Then, by Lemma 2 and Remark in 4.3, T = Tmax + 1 ≤ 200 C 3 n2 X n .
(5.4) (5.5)
Let
1 2n+3 δ = exp − n2 (X + 2) , ε = δ2 . (5.6) 2 This is the same ε as in (5.1). Let us check that, for L and T from (5.4), (5.5), δ from (5.6), the inequality δ ≤ e−LT holds. Indeed, 1 n 2n+3 LT ≤ 1200 C 3 n2 X n (X + 2) ≤ n2 (X + 2) = − log δ, 2 because X > 100. This checks assumption (1.6) of Theorem 3. 5.3. Application of Theorem 3. Let us first carry on the proof of Lemma 3 in Case I: |a1 | ≤ 1/4. By (3.12), in this case D0 = [0, B], D = P (D0 ). The set D will be split into two parts: a short segment [0, B1 ] and a longer segment [B1 , B]. The map P −1 will be extended to U ε ([0, B1 ]) by Proposition 4.1, and to U ε ([B1 , B]) by Theorem 3. In more detail, let β be the same as in (4.1), 1 β0 = βe−4π , B0 = (0, −β0 ), B1 = P (B0 ), 2 (5.7) K 0 = [B0 , B], K = P (K 0 ) = [B1 , P (B)].
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The map P −1 can be analytically extended to U ε ([0, B1 ]). Indeed, by (4.2), 1 −6π 1 βe ≤ |y(B1 )| ≤ βe−2π . (5.8) 2 2 Hence, U ε ([0, B1 ]) ⊂ {|z| ≤ βe−2π }, because ε 12 βe−2π by (4.1), (5.1). By Proposition 4.1, P −1 can be analytically extended to U ε ([0, B1 ]). From now on, the proof of Lemma 3 is completed in both cases simultaneously. In Case I (|a1 | ≤ 1/4), K is defined by (5.7). In Case II (|a1 | > 1/4), let K = D, K 0 = D0 = P −1 (D), D is from (3.12). In this case K and K 0 do not touch the disk Dβ , by the remark in 4.1. Hence, K 0 ⊂ [B0 , B] in Case II. It is sufficient to prove that P −1 can be analytically extended to U ε (K). Indeed, in Case I, D = [0, B1 ] ∪ K, and the extension of P −1 to U ε ([0, B1 ]) is already considered. In Case II, let D be from (3.12), D = K. Let us check assumptions of Theorem 3 for system (1.4), the map P −1 and K defined above. Assumption (1.5) follows from (5.2), (5.4). Assumption (1.6) is verified in 5.2. Let us prove (1.7) for Πδ replaced by a larger domain √ Π0δ = U δ (0) × U λ ([B0 , B]), λ = δ. More precisely, Πδ = Π0δ in Case I; Πδ ⊂ Π0δ in Case II. By(3.5), in Π0δ |v1 (z)| ≥ |z2 | − F (z1 ) ≥ |y(B1 )| − λ − 2Cδ. By (5.8), (4.1), (5.1), in Π0δ
1 −6π βe − λ − 2Cδ > δ. 2 On the other hand, v2 = −x. In Π0δ , |v2 | ≤ δ. Hence, v 2 ≤ 1 < µ. v1 This verifies condition (1.7) of Theorem 3. Theorem 3 may be now applied. It implies that the inverse Poincar´e map of (1.4) may be analytically extended to U ε (K), and P −1 (U ε (K)) ⊂ U 1 (K 0 ). This proves Lemma 3. |v1 | ≥
5.4. Final estimate and simplifications. The estimate of Theorem 1 implies: |D0 | + 2 , ρ where ε is from Lemma 3. By Lemma 1, and Remark in 3.4, −1
L(n, C) ≤ e2|D|ε
|D0 | + 2 ≤ Y = 17k 3 ,
log
n
k = (C + 1) .
By Corollary 2 of Proposition 3.4, ρ ≥ k/4. Hence, log
Y ≤ log 68 + 2n log(C + 1) := m. ρ
Therefore, for ε given by (5.1), L(n, C) ≤ m exp 2Y ε−1 . Elementary estimates show that for C ≥ 4, n ≥ 5, L(n, C) ≤ exp(exp C 14n ). This proves Theorem 2.
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References [I]
Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity 13 (2000), no. 4, 1337–1342. MR 2001h:34047 [IYa] Yu. Ilyashenko, S. Yakovenko, Counting real zeros of analytic functions satisfying linear ordinary differential equations, J. Differential Equations 126 (1996), no. 1, 87–105. MR 97a:34010 [LMP] A. Lins, W. de Melo, C. C. Pugh, On Li´ enard’s equation, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., vol. 597, Springer, Berlin, 1977, pp. 335–357. MR 56 #6730 [LL] N. G. Lloyd, S. Lynch, Small-amplitude limit cycles of certain Li´ enard systems, Proc. Roy. Soc. London Ser. A 418 (1988), no. 1854, 199–208. MR 89g:34040 [S] S. Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 99h:01033 Cornell University, US; Moscow State and Independent Universities, Steklov Math. Institute, Moscow (MIAN) E-mail address: [email protected], [email protected] Moscow State University and Independent University of Moscow E-mail address: andrei [email protected]
SOME UPPER ESTIMATES OF THE NUMBER OF LIMIT CYCLES OF PLANAR VECTOR FIELDS ´ WITH APPLICATIONS TO LIENARD EQUATIONS YU. ILYASHENKO AND A. PANOV To the memory of Ivan Georgievich Petrovskii, great personality and great mathematician
Abstract. We estimate the number of limit cycles of planar vector fields through the size of the domain of the Poincar´e map, the increment of this map, and the width of the complex domain to which the Poincar´e map may be analytically extended. The estimate is based on the relationship between the growth and zeros of holomorphic functions [IYa], [I]. This estimate is then applied to getting the upper bound of the number of limit cycles of the Li´enard equation x˙ = y − F (x), y˙ = −x through the (odd) power of the monic polynomial F and magnitudes of its coefficients. 2000 Math. Subj. Class. 34Cxx, 34Mxx. Key words and phrases. Limit cycles, Poincar´ e map, Li´ enard equation.
1. Introduction 1.1. Hilbert’s 16th Problem and Li´ enard equations. Questions related to bounds of the number of limit cycles of planar vector fields go back to the second half of Hilbert’s 16th Problem. Smale suggested to study a special class, namely, Li´enard equations with polynomials of odd degree, where the Poincar´e map is globally defined and nonidentical [S]. These equations were studied by many authors; in [LMP] small perturbations of linear center are examined; small amplitude limit cycles are discussed in [LL]. All these approaches are based on different kinds of perturbations techniques. Below we study equations far from integrable ones and limit cycles distant from equilibrium points. We use the method developed in [IYa] for linear equations and based on the growth-and-zeros theorem for holomorphic functions, see 2.1 below. Using this theorem, we obtain upper bounds for the number of limit cycles of planar vector fields (Theorems 1 and 3 below). The main result is Theorem 2, which gives a bound for the number of limit cycles the Li´enard Received October 30, 2001; in revised form December 19, 2001. The authors were supported by part by the grants RFBR 95-00-0455, CRDF RM1-2086; the first author was also supported by the grants NSF DMS 997-0372, NSF 0010404. c
2001 Independent University of Moscow
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YU. ILYASHENKO AND A. PANOV
equation x˙ = y − F (x), y˙ = −x (1.1) provided that the polynomial F is monic, and the magnitude of the coefficients is no greater than C. The estimate depends on this C as well as on the degree n of the polynomial. The estimate of Theorem 2 is triple exponential, and does not pretend to be realistic (Smale conjectures that it is polynomial in n and does not depend on C [S]). Yet it is the only known estimate of the number of limit cycles of Li´enard equations with an arbitrary odd degree of the polynomial. For n ≤ 3, the Li´enard equation has no more than one limit cycle [LMP]. In [LMP] the authors suggested that the number of limit cycles of (1.1) is no greater than [n/2] + 1. This estimate is proved for small perturbations of a linear center of the form: x˙ = y − εF (x), y˙ = −x/ε 1. For almost 25 years this conjecture remains neither proved nor disproved. But there is no other result that confirms this conjecture up to now. Recently the same method was applied by the first author to bound the number of limit cycles of the Abel equation [I]. Let v be an analytic vector field in the real plane, that may be extended to C2 . For any set D in a metric space denote by U ε (D) the ε-neighborhood of D. The metrics in C and C2 are given by: ρ(z, w) = |z − w|,
z, w ∈ C;
ρ(z, w) = max(|z1 − w1 |, |z2 − w2 |),
z, w ∈ C2 .
Denote by |D| the length of the segment D. For any larger segment D0 ⊃ D, let ρ(D, ∂D0 ) be the Hausdorff distance between D and ∂D0 . 1.2. Upper bounds of the number of limit cycles. Consider the system x ∈ R2 .
x˙ = v(x),
(1.2)
Theorem 1. Let Γ be a cross-section of the vector field v, D ⊂ Γ a segment. Let P be the Poincar´e map of (1.2) defined on D, and D ⊂ D0 = P (D). Suppose that P may be analytically extended to U = U ε (D) ⊂ C, ε < 1, and P (U ) ⊂ U 1 (D0 ) ⊂ C.
(1.3)
Then the number #LC(D) of limit cycles that cross D admits an upper estimate: −1
#LC(D) ≤ e2|D|ε
log
|D0 | + 2 . ρ(D, ∂D0 )
The same is true for P replaced by P −1 . Theorem 1 is derived in §2 from the Growth-and-Zeros theorem [IYa], [I]. An application of Theorem 1 to the Li´enard equation provides the following Theorem 2. Consider the Li´enard equation x˙ = y − F (x), y˙ = −x, F (x) = xn +
n−1 X 1
aj xj , |aj | ≤ C, C ≥ 4, n ≥ 5, (1.4)
´ NUMBER OF LIMIT CYCLES OF LIENARD EQUATIONS
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and suppose that n is odd. Then the number L(n, C) of limit cycles of (1.4) admits the following upper estimate: L(n, C) ≤ exp(exp C 14n ). Remark. The assumption F (0) = 0 does not reduce the generality; it may be fulfilled by a shift y 7→ y + a. The assumption that F is monic may be fulfilled by rescaling in x, y and reversing time if necessary. Assumption: n ≥ 5 does not reduce the generality. Indeed, for n = 3, L(n, C) = 1, and n is odd. 1.3. Plan of the proof of Theorem 2. In order to deduce Theorem 2 from Theorem 1, we need the following data: • D, a subdomain of the inverse Poincar´e map that is crossed by all the limit cycles of (1.4); • ρ(D, ∂D0 ), where D0 = P −1 (D) ⊂ D; • ε, the radius of the neighborhood of D to which the inverse Poincar´e map P −1 may be extended with restriction (1.3). The main step is the description of D. Suppose that (1.4) has at least one limit cycle, or else there is nothing to discuss. The cycles of (1.4) form a nest around the origin. Denote by E the most interior cycle of this nest. Let Γ be a section x = 0, y ≤ 0. It is transversal to (1.4) everywhere except for 0. Let E = E ∩ Γ. In order to specify D, we analyze infinite singular points of (1.4), see [LMP]. It appears that (1.4) has two unbounded repelling domains D+ , D− shown in Figure 2 below. Consider any orbit starting from a point of D+ , say A. It hits Γ at a point B. After that the orbit winds around zero, never crossing the cycle E. All the periodic solutions of (1.4) cross Γ inside the segment [0, P (B)] that belongs to [0, P (B)]. Indeed, orbits with initial condition below P (B) on Γ are absorbed by D+ in the reversed time after one circuit around zero. We will choose D ⊂ [0, P (B)]. This will give an upper estimate for the length of D as soon as |P (B)| is estimated through C. This is done in 3.4. The gap ρ(D, ∂D0 ) is estimated from below by |B − P (B)|, which is, in turn, estimated like follows. Take a point A0 ∈ D+ , see Fig. 2, and consider an arc of the orbit from the moment when it starts at A0 untill it hits Γ at a point, say, B 0 . The point A0 is chosen so that B 0 ∈ [0, B]. Then B 0 ∈ [B, P (B)], or else P −1 (B 0 ) will be well defined which is impossible. Hence, the distance |B − B 0 | provides a lower estimate for ρ. For A and A0 specially chosen, this distance is estimated from below in 3.5. To estimate the width of the neighborhood of D to which the Poincar´e map of (1.4) may be analytically extended from D, we use the Gronwall inequality. To apply it, we need to estimate from above the time of the first return to Γ. This is done in Section 4. In Section 5 the main theorem is deduced from Theorem 3 stated in the next subsection. This theorem provides an estimate of the width of the domain of the Poincar´e map, and is proved in 2.2. 1.4. Refinement of Theorem 1. The parameter ε in Theorem 1 may be estimated from below through the right-hand side of (1.2) in the following way. Let D, D0 and Γ be the same as in Theorem 1.
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Theorem 3. Let P : D → D0 be the Poincar´e map of (1.2). For any x ∈ D denote by ϕx,P (x) the arc of the phase curve of (1.1) starting at x and ending at P (x). Let [ Ω(D) = ϕx,P (x) , x∈D
and 1 ≤ µ = max |v|, 2
L = 2µ.
(1.5)
U (Ω)
Let t(x) be the time length of the arc ϕx,P (x) , and Tmax = maxx∈D t(x), Let δ ≤ e−LT ,
λ=
T = Tmax + 1. √ 2
ε = δ2 .
δ,
(1.6)
Suppose that (z1 , z2 ) are coordinates in C , Γ = {z1 = 0}, v = (v1 , v2 ). Let K ⊂ D be a segment, K 0 = P (K), Πδ = U δ (0) × U λ (K 0 ) ⊂ C2 . Suppose that v 2 (1.7) ≤ µ in Πδ . v1 C
Then
(1) The Poincar´e map P : K → K 0 of (1.2) may be analytically extended to U ε (K) ⊂ C Γ, and P (U ε (K)) ⊂ U 1 (K 0 ). (2) For system (1.2), we have #LC(K) ≤ e2|D|e
2LT
|D0 | + 2 . ρ(K, ∂K 0 )
log
The S same is true for P replaced by P −1 . In this case P −1 (D) = D0 , Ω = x∈D0 ϕx,P (x) . Theorems 1, 3 are proved in Section 2, Theorem 2 is proved in Sections 3–5. 2. Estimates of the number of limit cycles 2.1. An estimate using the hidden data. In this subsection Theorem 1 is proved. It uses the width of the complex domain U ε to which the Poincar´e map may be extended. The parameter ε is not easy to find by straightforward calculation from equation (1.2), see Theorem 3. This parameter is called “hidden data” in the title. Proof of Theorem 1. In [IYa], [I] a Growth-and-Zeros theorem for holomorphic functions was proved; we quote the corollary that we need below: Theorem 4 [I]. Let D be a real segment, U = U ε (D) be its ε-neighborhood in C, |D| be the length of the segment. Let f be a holomorphic function in U continuous up to the boundary. Then −1
#{z ∈ D : f (z) = 0} ≤ e2|D|ε
log
maxU¯ |f | . maxD |f |
(2.1)
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To prove Theorem 1, we apply this result to the increment of the Poincar´e map P in U : f (z) = P (z) − z. The diameter of the domain U 1 (D0 ) is equal to |D0 | + 2. Inclusions D ⊂ D0 , P (U ) ⊂ U 1 (D0 ) imply: max |f | ≤ |D0 | + 2. ¯ U
On the other hand, P moves one of the endpoints of D to an endpoint of D0 located at the distance ρ from its source. Hence max |f | ≥ ρ. D
Now (2.1) implies the estimate of Theorem 1.
2.2. An estimate using available data. A lower estimate of the parameter ε in Theorem 1 is given below through the data that often may be computed by equation (1.2) in a straightforward manner. This will prove the first statement of Theorem 3. The second statement is an immediate consequence of the first one and Theorem 1. Proof of Statement 1 of Theorem 3. We need to prove that for any x ∈ K the Poincar´e map of (1.2) may be extended to an ε-neighborhood of x in C. Let, as before, ϕx,P (x) = γx be the arc of the time oriented orbit of (1.2) from x to P (x), t(x) be the time length of γx . Let x = Re z1 , y = Re z2 , where z1 , z2 are the coordinates in C2 . By (1.5) and the Cauchy inequality, ∂vi , i = 1, 2; j = 1, 2 ≤ µ. max U 1 (Ω) ∂zj Hence, in the norm |(z1 , z2 )| = max(|z1 |, |z2 |), we have:
∂v
≤ L. max
U 1 (Ω) ∂(z1 , z2 ) By (1.6),
εeLT ≤ δ.
Let z ∈ U ε (K); hence, |z − x| ≤ ε for some x ∈ K. Let ϕz be a solution of the complexified equation (1.2) with complex time, passing through z for t = 0: ϕz (0) = z. By (1.5), (1.6) and the definition of T , we have εeLT < 1. Then, by the Gronwall inequality, |ϕz (t) − ϕx (t)| ≤ εeLt . Hence, |ϕz (t(x)) − ϕx (t(x))| ≤ δ. Note that ϕx (t(x)) = P (x) ∈ K 0 , because x ∈ K. Therefore, a := ϕz (t(x)) ∈ Πδ . In general a 6∈ C Γ. Let a = (a1 , a2 ), P (x) = (0, b). Then |a2 − b| ≤ δ.
(2.2)
The leaf ϕa in some neighborhood of a is a solution z2 = w(z1 ) of the equation dz2 v2 = . dz1 v1
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The function w may be extended over any segment in U δ (0) until its graph reaches ∂Πδ . Over the segment [a1 , 0] this graph will not reach ∂Πδ , because, by (1.7) and (2.2), at any point z1 ∈ [a1 , 0] where w is well defined, |w(z1 ) − b| ≤ |w(z1 ) − a2 | + |a2 − b| ≤ µδ + δ. √ But (µ+1)δ ≤ λ = δ, because for T ≥ 1, µ > 0, we have eµT > µ+1. Hence, w(0) is well defined. Let P (z) = w(0). This is the desired analytic extension of P from K to U ε (K); moreover, P (U ε (K)) ⊂ U λ (K 0 ) ⊂ U 1 (K 0 ). This proves Statement 1 of Theorem 2. ´ map for the Lie ´nard equation 3. Domain of the inverse Poincare In this and next two sections, Theorem 2 is proved. The proof is based on Theorems 1, 3 applied to the inverse Poincar´e map. Hence, in what follows, D and D0 belong to the domain and the image of the inverse Poincar´e map respectively. We mainly study equation (1.4) in the right halfplane x > 0. The conclusions may be easily reformulated for the halfplane x < 0, because the symmetry (x, y) 7→ (−x, −y) brings (1.4) to an equation of the same type. All the estimates below hold true for C ≥ 2, except for the very end of 5.4, where the inequality C ≥ 4 is needed. 3.1. C-monic polynomials. The following elementary estimates will be used all over this and the next sections. Definition 3.1. A C-monic polynomial is a real polynomial in one variable with the highest coefficient one and other coefficients no greater than C in absolute value, with zero constant term. Proposition 3.1 (Properties of C-monic polynomials). Let F be a C-monic polynomial of degree n, C ≥ 2. Then 1. 2.
F (x) > 0
n
for x ≥ C + 1,
min F (x) ≥ −(C + 1) , x>0
max |F (x)| ≤ 2An
for A ≥ C + 1,
x∈[0,A]
3. 4.
|F (x)| ≥
1 n |x| 2
for |x| ≥ 2C + 1,
max |F 0 (x)| ≤ Cn2 X n−1 [0,X]
for X ≥ 1,
(3.1) (3.2) (3.3) (3.4)
1 . (3.5) 2 n Remark. (3.1) implies that F (x) < 0 for x ≤ −(C + 1), maxx 0. This yields (3.1). 2. We have for x ≥ 0 |F (x)| ≤ xn + C
n−1 X 1
xj = G(x).
´ NUMBER OF LIMIT CYCLES OF LIENARD EQUATIONS
589
The polynomial G is monotonious in x for x > 0. Hence, for A ≥ C + 1: max G(x) = G(A) = An + C [0,A]
An − 1 ≤ 2An . A−1
3. For x ≥ 2C + 1, |F1 (x)| < xn /2, see the first inequality in (3.6). This implies (3.3). 4. Estimate (3.4) is straightforward. 5. For |z| ≤ 1/2, z ∈ C, |F (z)| =
|a1 | +
n−1 X
j−1
aj |z|
|z| ≤
|a1 | + C|z|
1
n
1 − |z| 1 − |z|
|z|
≤ (|a1 | + 2C|z|)|z| ≤ 2C|z|. 3.2. Diacritical nodes at infinity. Behavior of the solution of the Li´enard equation near infinity is well known, see [LMP]. We need only a part of this information. Proposition 3.2. Equation (1.4) has a diacritical node at infinity; in the projective chart (u, v) = (1/x, y/x) (3.7) it is (u, v) = (0, 0). Proof. In the chart (u, v), after change of time, (1.4) takes the form: u˙ = u(F˜ (u) − un−1 v),
v˙ = v F˜ (u) − un−1 (1 + v 2 ),
(3.8)
where F˜ (u) = u F (1/u). Note that the free term of F˜ (u) is 1 because F is monic. Hence, the linear part of (3.8) at zero is u˙ = u, v˙ = v. This proves the proposition. n
3.3. Overflowing domains near infinity. The infinite line in the chart (x, y) becomes u = 0 in the chart (u, v). In the neighborhood of zero the field (3.8) is close to its linear part. Hence, parabolic sectors v 2 ≤ |u| ≤ a,
a small,
(3.9)
are overflowing domains of (3.8): the vector field on the boundary is directed outwards everywhere except for the origin, see Fig. 1. Proposition 3.3. Domains + DC = {(x, y) ∈ R2 : x ≥ 2C, |y| ≤ − DC
2
= {(x, y) ∈ R : (−x, y) ∈
+ DC }
√
x},
(3.10)
are overflowing for system (1.4) with n odd. Proof. For n odd, the image of the vector field (1.4) under the map (3.7) equals to the vector field (3.8) multiplied by a positive function. Domains (3.10) are transformed by (3.7) to (3.9) with a = 1/(2C). Hence, it is sufficient to prove that the domain (3.9) with this a is overflowing for (3.8).
590
YU. ILYASHENKO AND A. PANOV
u
y − DC
+ DC
v
x
(a)
(b)
Figure 1. Overflowing domains (a) in (u, v)-chart, (b) in (x, y)-chart. On the segment u = a, |v| ≤ u˙ = F˜ (u) − un−1 v ≥ 1 − u
√
a, a = 1/(2C) we have:
n X
Caj − an−1/2 ≥ 1 − Ca
1
1 − an−1 − a > 0, 1−a
because C ≥ 2, hence a ≤ 1/4. √ On the arc of the parabola u = v 2 , |v| ≤ a, d 2 (v − u) = v 2 (F˜ (v 2 ) − 2v 2n−3 − v 2n−1 ). dt For 0 ≤ u ≤ a we have: n−1
1−u F˜ (u) ≥ 1 − Cu ≥ 1−u √ On the other hand, for |v| ≤ a, where a is the same
1 . 3 as before, n ≥ 3, we have:
|2v 2n−3 | + |v 2n−1 | ≤ 2an−3/2 + an−1/2 ≤
1 1 1 + < . 4 32 3
d (v 2 − u) > 0 on the arc considered. In the same way, the arc u = −v 2 , Hence, dt √ √ |v| ≤ a and the segment u = −a, |v| ≤ a are treated.
3.4. Domain of the inverse Poincar´ e map. The line Γ : x = 0 is the horizontal isocline of (1.4), the graph Λ : y = F (x) is the vertical one. For n odd, the graph of a monic polynomial F has two branches going to infinity in the first and third quadrants. The isoclines Λ and Γ split R2 into four regions where the direction of v in terms of “up-down”,“right-left” is the same, see Figure 2. In G1 : x > 0, y > F (x), the field v points right-down; in G2 : x > 0, y < F (x), left-down; in G3 : x < 0, y < F (x), left-up; in G4 : x < 0, y > F (x), right-up. Note that |y| ˙ is bounded in any strip |x| ≤ α. Hence, any orbit starting at y ∈ Γ, returns to Γ. It may return to the singular point zero, where Γ is not transversal to the field (1.4). Otherwise, the Poincar´e map is well defined at y and near y. In the backward time, not any orbit starting from Γ returns to Γ: it may be cap+ − tured by one of the domains DC , DC which become absorbing after time reversal. We will now “estimate from above” the domain D0 of the inverse Poincar´e map.
´ NUMBER OF LIMIT CYCLES OF LIENARD EQUATIONS
G4
591
G1
A−
− DC
D
+ DC
P (B) D0 A0 G3
A
G2 ψ B0
0 F (x) by (3.1). Take a point A = (4k 2 , −2k) ∈ ∂DC . The positive semiorbit ϕ of (1.4) starting at A will go left and down until it crosses either Γ or Λ. But the second case is impossible because in the right halfplane the field v points downwards on Λ. Denote by B the first intersection of ϕ and Γ, and let Y0 = |y(B)|, see Figure 2. + Any orbit of the reversed system (1.4) starting on Γ below B enters DC and is absorbed. Hence, the inverse Poincar´e map is not defined on Γ for y < −Y0 . Therefore, any closed orbit of (1.4) crosses [0, −Y0 ].
592
YU. ILYASHENKO AND A. PANOV
We will now estimate Y0 from above. By (3.1), x˙ = y − F (x) < −k
for (x, y) ∈ G2 ∩ {y < −2k}.
Hence, the time length of the arc ϕA,B is no greater than x(A)/k = 4k. On the other hand, in the strip 0 ≤ x ≤ 4k 2 we have: y˙ = −x ≥ −4k 2 ,
|y| ˙ ≤ 4k 2 .
Hence, Y0 = |y(B)| ≤ | − 2k − 16k 3 | < 17k 3 = Y. The arguments for G4 and y > 0 are obtained from the previous ones by symmetry; they show that P −1 is not defined for y ≥ Y on Γ. This proves Lemma 1, together with a stronger statement: every periodic orbit crosses [0, B]. Remarks. 1. The estimate for Y0 above is very robust. We proved at the same time that Y0 + 2 < Y . 2. The arguments above prove as well that an orbit starting at −A = (−4k 2 , 2k) hits the y axis at a point lower than (0, Y ). Therefore, any orbit that starts on Γ crosses the segment [0, Y ] after a half curcuit around the origin. 3.5. Increment of the inverse Poincar´ e map. In this subsection we construct a segment D that belongs to the domain of the inverse Poincar´e map and that is crossed by any periodic orbit of (1.4). Let E ∈ [0, −Y ] be the fixed point of the Poincar´e map P of (1.4) with the maximal y-coordinate. Let B be the same as in 3.4, see Figure 2. Let ( [0, B] in case |a1 | ≤ 1/4, D0 = D = P (D0 ). (3.12) [E, B] in case |a1 | > 1/4, Any periodic orbit of (1.4) crosses D0 , hence D; see the statement at the end of the proof of Lemma 1. The splitting in two cases above is motivated in the following way. In the first case, for |a1 | ≤ 1/4, the singular point 0 is a focus. The domain of the Poincar´e map and of its complex extension is easily studied near zero. The time length of corresponding arcs of the phase curves of (1.4) is estimated from above, see 4.1 below. Periodic orbits may occur in an arbitrary neighborhood of 0. In the second case, 0 has a well controlled basin of attraction in forward or backward time. Periodic orbits of (1.4) cannot touch this basin. Therefore in case 2 we consider the Poincar´e map on a segment separated from 0. Note that in both cases the upper end of D (0 or E) is mapped by P into itself. Hence, in both cases ρ = |B − P (B)|. In what follows, the lower estimate for ρ, hence, for the increment of the Poincar´e map is given. Let A0 = ( 94 k 2 , − 32 k), ψ be the positive semiorbit of (1.4) starting at A0 , see Fig. 2. The same arguments as before show that ψ crosses Γ; let B 0 ∈ [0, B] be the first intersection point of ψ and Γ.
´ NUMBER OF LIMIT CYCLES OF LIENARD EQUATIONS
593
The inverse Poincar´e map is not defined at any y ∈ Γ below B 0 . Indeed, any + negative semiorbit starting below B 0 , hits DC and is absorbed, thus never returns to Γ. Hence, the image P (B) lies above B 0 , and therefore Y1 = |y(B 0 )|, Y0 = |y(B)|
ρ ≥ Y0 − Y1 ,
as before. We will prove that ρ is no less than a constant of order k, see Corollary 1 below. The reason is that in G2 ∩ {y ≤ −3k/2} the vector field of (1.4) is almost horizontal. Proposition 3.4. Let y1 (x) and y2 (x) be solutions of the equation dy x = dx F (x) − y
(3.13)
that corresponds to system (1.4). Let x0 ≤ 3k 2 , yj (x0 ) ≤ −3k/2, j = 1, 2. Then 1 |y1 (0) − y2 (0)| ≥ |y1 (x0 ) − y2 (x0 )|. 2 Corollary 1. For system (1.4), we have ρ = d(D, ∂D0 ) ≥ k/4. Proof. Let y = y1 (x), y = y2 (x) be the orbits of (1.4) passing through A and A0 and considered over [0, 4k 2 ] and [0, 49 k 2 ] respectively. Let x0 = 94 k 2 . Then y1 (x0 ) < −2k, y2 (x0 ) = −3k/2. By Proposition 3.4, 1 k Y1 − Y0 = y2 (0) − y1 (0) ≥ (y2 (x0 ) − y1 (x0 )) ≥ . 2 4 Proof of Proposition 3.4. Let y(x, y0 ) be the solution of (3.13) with the initial condition y(x0 , y0 ) = y0 . Consider the variational equation for (3.13) with respect to the initial conditions: x ∂y(x, y0 ) dξ = ξ, ξ= . (3.14) dx (F (x) − y)2 ∂y0 The solution y(x, y0 ) of (3.13) with y0 < −k is monotonously increasing on [0, x0 ], because F (x) − y > 0 in x > 0, y < −k. Hence, in (3.14), y = y(x, y0 ) ≤ −k for x ∈ [0, x0 ]. Let x a(x, y0 ) = . (F (x) − y(x, y0 ))2 Rx Then ξ(x, y0 ) = exp x0 a(x, y0 ) dx := exp I(x, y0 ). We will prove that I(0, y0 ) > − log 2, provided that y0 ≤ −3k/2. This will prove Proposition 3.4. By (3.3), the inequality F (x) − y > xn /2 holds in the domain x ∈ [2C + 2, x0 ], y < −k. Hence, for (x, y(x, y0 )) lying in this domain, a(x, y0 ) ≤ 4/x2n−1 . For these x, the inequalities C ≥ 2, n ≥ 3 imply: 4 1 I(x, y0 ) ≥ − ≥− . 2n−2 (2n − 2)(2C + 2) 8 By (3.1), the inequality F (x) − y > k/2 holds in the domain x ∈ [0, 2C + 2], y < −3k/2. Hence, for x ∈ [0, 2C + 2], y ≤ −3k/2, a(x, y0 ) ≤ 4x/k 2 , I(0, y0 ) ≥
Z
0
2C+2
a(x) dx −
1 8(C + 1)2 1 1 ≥− − ≥ − > − log 2. 8 (C + 1)2n 8 4
594
YU. ILYASHENKO AND A. PANOV
3.6. Suspension over the Poincar´ e map. Let D be the same as in (3.12). Denote by γy the arc ϕy,P −1 (y) of the orbit of (1.4) with time reversed, ran from y to P −1 (y), y ∈ D. Let [ Ω= γy . (3.15) y∈D
Proposition 3.5. The domain Ω defined in (3.15) belongs to a rectangle Π = S × Σ,
S ⊂ Ox ,
Σ ⊂ Oy = Γ,
Σ = {|y| ≤ Y = 17k 3 },
(3.16) 3
S = {|x| ≤ X := 4(C + 1) }. Proof. The maximal value x(y) of x on γy is achieved at a point a(y) of intersection of γy with the graph Λ in the right halfplane: a(y) = (x(y), F (x(y))). We need to prove that x(y) ≤ X. By Remark 2 of 3.4, the point a(y) is lower than Y . Hence, F (x(y)) ≤ Y. Suppose that x(y) ≥ 2C + 1; if not, the proposition is proved. Then, by (3.3), F (x(y)) ≥ 21 xn (y). Hence, 1 n 1/n 3 x (y) ≤ Y, x(y) ≤ (2Y ) = (34k 3 )1/n ≤ 4(C + 1) = X. (3.17) 2 Together with the symmetry arguments at the beginning of the section, this proves the estimate x|γy ≥ −X as well. 4. Time of the first return 4.1. Neighborhood of the singular point. The singular point (0, 0) is a focus if |a1 | < 2 and a node if a1 ≥ 2. We will use another distinction: |a1 | ≤ 1/4, the polar angle near 0 varies monotonously in a fast way; |a1 | > 1/4, the polar angle varies slowly monotonously or nonmonotonously. Let ϕ, r be polar coordinates on R2 , ϕ, ˙ r˙ be derivatives with respect to (1.4). Proposition 4.1. Let |a1 | ≤ 1/4. Then ϕ˙ ≤ −1/2 in the disk 1 . (4.1) 8C The Poincar´e map, together with its inverse, is well defined near zero and admits a complex extension to a disk |z| ≤ βe−2π . Moreover, in this disk, Dβ : r ≤ β =
|P (z)/z| ≤ e2π ,
|P −1 (z)/z| ≤ e2π .
(4.2)
Proof. A system x˙ = f , y˙ = g in polar coordinates has the form: xf + yg xg − yf := R, ϕ˙ = := Φ, (4.3) r r2 ϕ ∈ S 1 , r ∈ R+ . Let us complexify r, replacing it by z ∈ C. Then, for (1.4), in the domain Uβ : |z| ≤ β, ϕ ∈ S 1 , r˙ =
´ NUMBER OF LIMIT CYCLES OF LIENARD EQUATIONS
595
Φ(z, ϕ) = −1 + sin ϕ cos ϕ F1 (z cos ϕ),
(4.4)
we have: ϕ˙ = Re Φ,
1
where F1 (x) = F (x)/x. By (3.5), for |z| ≤ β, ϕ ∈ S , 1 1 + 2Cβ ≤ . (4.5) 4 2 Hence, |F1 (z cos ϕ)| ≤ 1/2, Re Φ < −1/2 in Uβ . This proves the first statement of the proposition. Let us prove the second statement. By (4.3), |F1 (z cos ϕ)| ≤
z˙ = R,
R(z, ϕ) = − cos2 ϕ F1 (z cos ϕ)z.
(4.6)
By (4.5), in Uβ , |z/z| ˙ ≤ 1/2. For the system (4.3) with r complexified, that is, for the system (4.4), (4.6), the absolute value of the time of the first return for the orbits that remain in Uβ is no greater than 4π. This follows from (4.4) and the inequality Re Φ < −1/2. Hence the orbits with the initial condition: |z| ≤ βe−2π ,
ϕ=0
remain in Uβ during at least one circuit along S 1 and come back to the disk |z| ≤ β, ϕ = 0. This proves that the Poincar´e map of (4.3) may be analytically extended to the disk |z| ≤ βe−2π . The same arguments prove this statement for P −1 . Statement (4.2) follows from the inequalities |z/z| ˙ ≤ 1/2, |ϕ| ˙ ≥ 1/2. Proposition 4.2. If |a1 | > 1/4, then r˙ ≥ 0 or r˙ ≤ 0 in the disk Dβ , see (4.1). Proof. By (4.5), in Dβ , r˙ = − cos2 ϕ F1 (r cos ϕ)r, F1 = F/x. Moreover, F1 (x) = (a1 + F2 (x)),
|F2 (x)| ≤ 2C|x|;
the latter inequality is contained in the proof of (3.5). If |x| ≤ β, then 2C|x| ≤ 1/4. Hence, sgn F1 = sgn a1 in Dβ . This proves Proposition 4.2. Remark. Equation (1.4) has no invariant circles. So Proposition 4.2 implies that for |a1 | > 1/4, 0 is either attracting or repelling fixed point with the basin that contains Dβ . Hence, no periodic orbit touches Dβ . Therefore, in case |a1 | > 1/4, by (3.12), we have D ∩ Dβ = ∅, D0 ∩ Dβ = ∅. 4.2. Strip around the vertical isocline. Lemma 2. Let, as before, γy be the arc ϕy,P −1 (y) of the phase curve of (1.4), D and X be the same as in (3.12) and (3.17), y ∈ D. Then t(y), the time length of γy , admits an estimate Tmax = max t(y) ≤ 200 C 3 n2 X n . y∈D
(4.7)
Proof. The arcs γy , y ∈ D, belong to the strip {|x| ≤ X}, see Proposition 3.5. On the other hand, the time length of any arc of the orbit of (1.4) located in Dβ and making no more than 1 circuit around zero is no greater than 4π for |a1 | ≤ 1/4, see Proposition 4.1. For |a1 | > 1/4, no closed orbit touches Dβ by Proposition 4.2. By definition of D, see (3.12), no arc γy , y ∈ D, touches Dβ provided that |a1 | > 1/4.
596
YU. ILYASHENKO AND A. PANOV
Let us now estimate the time length of the part of γy located in G = {|x| ≤ X} \ Dβ . We will split G into two domains: |x| ˙ ≤ α and |x| ˙ > α for α small to be chosen later. The second domain contains two parts of γy : one with x˙ < −α, the other with x˙ > α. The time length of any of them is no greater than 2X/α. We will now choose α so small that the curvilinear strip Sα = {(x, y) ∈ R2 : |y − F (x)| ≤ α, |x| ≤ X, (x, y) ∈ / Dβ } is crossed by the orbits of (1.4) in the time no greater than 1. Proposition 4.3. Let 1 ω 3 , X = 4(C + 1) , α = . (4.8) 24C 2 2Cn2 X n−1 Then the time length of any arc of the orbit of (1.4) located in Sα is no greater than 1. ω=
Proof. By the symmetry arguments it is sufficient to prove that in Sα ∩ {x > 0} d (y − F (x)) ≤ −2α. dt Let us first prove that in Sα we have: |x| > ω. Namely, let |x| ≤ ω, |y − F (x)| ≤ α. Then (x, y) ∈ Dβ . Indeed, |x| + |y| ≤ ω + max[0,ω] |F | + α. By (4.8), α < ω < 1/2. By (3.5), |F (x)| ≤ 2Cω. Hence, for x ∈ [0, ω], |x| + |y| ≤ (2C + 2)ω ≤ 3Cω =
1 = β. 8C
By (3.4), |F 0 (x)| ≤ Cn2 X n−1 in G. Therefore, for x > 0, (x, y) ∈ Sα we have: x > ω, and d (y − F (x)) = −x − (y − F (x))F 0 (x) ≤ −ω + αCn2 X n−1 ≤ −2α, dt because α=
ω 2Cn2 X n−1
100 for C ≥ 2.
(5.3)
Take n
L = 2µ = 6(X + 2) . Now, let T = Tmax + 1, see (4.7). Then, by Lemma 2 and Remark in 4.3, T = Tmax + 1 ≤ 200 C 3 n2 X n .
(5.4) (5.5)
Let
1 2n+3 δ = exp − n2 (X + 2) , ε = δ2 . (5.6) 2 This is the same ε as in (5.1). Let us check that, for L and T from (5.4), (5.5), δ from (5.6), the inequality δ ≤ e−LT holds. Indeed, 1 n 2n+3 LT ≤ 1200 C 3 n2 X n (X + 2) ≤ n2 (X + 2) = − log δ, 2 because X > 100. This checks assumption (1.6) of Theorem 3. 5.3. Application of Theorem 3. Let us first carry on the proof of Lemma 3 in Case I: |a1 | ≤ 1/4. By (3.12), in this case D0 = [0, B], D = P (D0 ). The set D will be split into two parts: a short segment [0, B1 ] and a longer segment [B1 , B]. The map P −1 will be extended to U ε ([0, B1 ]) by Proposition 4.1, and to U ε ([B1 , B]) by Theorem 3. In more detail, let β be the same as in (4.1), 1 β0 = βe−4π , B0 = (0, −β0 ), B1 = P (B0 ), 2 (5.7) K 0 = [B0 , B], K = P (K 0 ) = [B1 , P (B)].
598
YU. ILYASHENKO AND A. PANOV
The map P −1 can be analytically extended to U ε ([0, B1 ]). Indeed, by (4.2), 1 −6π 1 βe ≤ |y(B1 )| ≤ βe−2π . (5.8) 2 2 Hence, U ε ([0, B1 ]) ⊂ {|z| ≤ βe−2π }, because ε 12 βe−2π by (4.1), (5.1). By Proposition 4.1, P −1 can be analytically extended to U ε ([0, B1 ]). From now on, the proof of Lemma 3 is completed in both cases simultaneously. In Case I (|a1 | ≤ 1/4), K is defined by (5.7). In Case II (|a1 | > 1/4), let K = D, K 0 = D0 = P −1 (D), D is from (3.12). In this case K and K 0 do not touch the disk Dβ , by the remark in 4.1. Hence, K 0 ⊂ [B0 , B] in Case II. It is sufficient to prove that P −1 can be analytically extended to U ε (K). Indeed, in Case I, D = [0, B1 ] ∪ K, and the extension of P −1 to U ε ([0, B1 ]) is already considered. In Case II, let D be from (3.12), D = K. Let us check assumptions of Theorem 3 for system (1.4), the map P −1 and K defined above. Assumption (1.5) follows from (5.2), (5.4). Assumption (1.6) is verified in 5.2. Let us prove (1.7) for Πδ replaced by a larger domain √ Π0δ = U δ (0) × U λ ([B0 , B]), λ = δ. More precisely, Πδ = Π0δ in Case I; Πδ ⊂ Π0δ in Case II. By(3.5), in Π0δ |v1 (z)| ≥ |z2 | − F (z1 ) ≥ |y(B1 )| − λ − 2Cδ. By (5.8), (4.1), (5.1), in Π0δ
1 −6π βe − λ − 2Cδ > δ. 2 On the other hand, v2 = −x. In Π0δ , |v2 | ≤ δ. Hence, v 2 ≤ 1 < µ. v1 This verifies condition (1.7) of Theorem 3. Theorem 3 may be now applied. It implies that the inverse Poincar´e map of (1.4) may be analytically extended to U ε (K), and P −1 (U ε (K)) ⊂ U 1 (K 0 ). This proves Lemma 3. |v1 | ≥
5.4. Final estimate and simplifications. The estimate of Theorem 1 implies: |D0 | + 2 , ρ where ε is from Lemma 3. By Lemma 1, and Remark in 3.4, −1
L(n, C) ≤ e2|D|ε
|D0 | + 2 ≤ Y = 17k 3 ,
log
n
k = (C + 1) .
By Corollary 2 of Proposition 3.4, ρ ≥ k/4. Hence, log
Y ≤ log 68 + 2n log(C + 1) := m. ρ
Therefore, for ε given by (5.1), L(n, C) ≤ m exp 2Y ε−1 . Elementary estimates show that for C ≥ 4, n ≥ 5, L(n, C) ≤ exp(exp C 14n ). This proves Theorem 2.
´ NUMBER OF LIMIT CYCLES OF LIENARD EQUATIONS
599
References [I]
Yu. Ilyashenko, Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions, Nonlinearity 13 (2000), no. 4, 1337–1342. MR 2001h:34047 [IYa] Yu. Ilyashenko, S. Yakovenko, Counting real zeros of analytic functions satisfying linear ordinary differential equations, J. Differential Equations 126 (1996), no. 1, 87–105. MR 97a:34010 [LMP] A. Lins, W. de Melo, C. C. Pugh, On Li´ enard’s equation, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lecture Notes in Math., vol. 597, Springer, Berlin, 1977, pp. 335–357. MR 56 #6730 [LL] N. G. Lloyd, S. Lynch, Small-amplitude limit cycles of certain Li´ enard systems, Proc. Roy. Soc. London Ser. A 418 (1988), no. 1854, 199–208. MR 89g:34040 [S] S. Smale, Mathematical problems for the next century, Math. Intelligencer 20 (1998), no. 2, 7–15. MR 99h:01033 Cornell University, US; Moscow State and Independent Universities, Steklov Math. Institute, Moscow (MIAN) E-mail address: [email protected], [email protected] Moscow State University and Independent University of Moscow E-mail address: andrei [email protected]
