Dulac-Cherkas function and its application to estimation of limit cycles number for some classes of planar autonomous systems

Dulac-Cherkas function and its application to estimation of limit cycles number for some classes of planar autonomous systems Castro Urdiales / September 12-16, 2011 Alexander Grin Yanka Kupala State University of Grodno a joint work with Leonid A. Cherkas Belarusian State University of Informatics and Radioelectronics Klaus R. Schneider Weierstrass Institute for Applied Analysis and Stochastics 1

Abstract The talk deals with the estimation of limit cycles number and their localization for real planar autonomous system dx dt = P(x,y), dy dt = Q(x,y), Ω R2, P,Q C 1 (Ω). (1) It is based on the method suggested by L.A. Cherkas (1997) with the use of a Dulac function in the form B = Ψ(x,y) 1/k, 0 k R, Ψ(x,y) C 1 (Ω), satisfying the inequality Φ kψdiv X + Ψ x P + Ψ Q > 0(< 0), (x,y) Ω, X = (P,Q). (2) y In this case Ψ is called a Dulac-Cherkas (DC) function for system (1) in the region Ω. The main subject is limit cycles of systems (1). Main goal is an application DC function to estimation of limit cycles number for some classes of planar autonomous systems. 2

Outline 1. DC function, its relationship to Dulac criterion and main properties. 2. Numerical aspects for the construction of DC function. 3. Application of DC function to some classes of system (1). 4. An extension of DC function to system (1) with cylindrical phase space. 5. Conclusions 3

DC function, its relationship to Dulac criterion and main properties The Dulac criterion for the nonexistence of a limit cycle: Theorem 0.1 Let D Ω be a simply-connected region. If there exists a function B C 1 (D) such that div(bx) does not change sign in D and is not identically zero on any subregion of D, then (1) has no closed orbit lying entirely in D. Remark 0.2 The proof of Theorem 0.1 is based on Green s Theorem. For B 1, Theorem 0.1 represents the well-known criterion of Bendixson. Definition 0.3 A function B C 1 (D) such that div(bx) has the same sign in some connected region D and is not identically zero on any subregion of D is called a Dulac function to system (1) in D. 4

In other words: A function B(x,y) C 1 (Ω) such that the expression F(x,y) = div(bx) = B x P + B y Q + B( P x + Q y ) does not change sign in Ω and such that any curve F(x,y) = 0 does not contain a limit cycle, is called a Dulac function for system (1). The existence of Dulac function can be used to estimate the number of limit cycles of system (1) in some regions. Namely, system (1) has at most p 1 limit cycles in a p connected region Ω if a Dulac function exists. The condition imposed by the criterion on the function F(x,y) implies that the curve B(x,y) = 0 is transversal to the vector field X. Consequently, the limit cycles do not cross it, and their analysis can be performed separately in each of the regions B > 0 and B < 0. However, the criterion itself provides no method for the construction of the function B and for the localization of limit cycles lying in the region Ω. The most widespread forms of B were x a y b, a,b R and e xa y b. 5

There is a connection between a DC function and a Dulac function. Lemma 0.4 Let Ω D be connected, let Ψ be a DC function of system (1) in Ω. Then B := Ψ 1/k is a Dulac function in each subregion of Ω where Ψ is positive or negative. The proof is based on the fact that in a subregion of Ω with Ψ > 0 we have div ( Ψ 1/k X) = sign Ψ 1 k Ψ 1 1 Ψ k x P + sign Ψ 1 k Ψ 1 1 Ψ k y Q + Ψ 1 k div X = sign Ψ 1 k Ψ 1 k 1 Φ. (3) 6

The curve W defined by plays an important role. W := {(x,y) Ω : Ψ(x,y) = 0} Lemma 0.5 Any trajectory of system (1) meeting the curve W intersects W transversally. Lemma 0.6 The curve W does not contain any equilibrium of system (1). Lemma 0.7 Let W 1 and W 2 be two different smooth local open branches of the curve W such that W 1 W 2 is not connected, that is, W 1 W 2 is empty. Then W 1 and W 2 do not meet. Lemma 0.8 The curve W decomposes the region D in subregions on which Ψ is definite and where the transition from one subregion to an adjacent subregion is connected with a sign change of Ψ. Theorem 0.9 Let Ψ be a DC function of system (1) in D Ω. Then any limit cycle of system (1) which is entirely located in D does not intersect the curve W. [L.A. Cherkas, 1997]; [A. Grin, K. Schneider, 2007] 7

The following theorem represents a generalization of Dulac s criterion on the nonexistence of a limit cycle. Theorem 0.10 Let Ω be a connected region and let Ψ be a DC function for system (1) in Ω. Furthermore, we suppose that W decomposes Ω in simply connected subregions Ω i,i = 1,...,l. Then system (1) has no limit cycle in Ω. We note that in Theorem 0.10 the sign of k in the Dulac function plays no role. This is due to the property that the subregions are simply connected and Ψ does not vanish in these regions. The following generalization of Dulac s criterion admits that Ψ can change its sign in simply connected region, but in that case, the sign of k is essential. 8

Theorem 0.11 Let Ω be a simply-connected region and let Ψ be a DC function for system (1) in Ω, where k is positive. Then system (1) has no limit cycle in Ω. Idea of the proof: We assume Φ > 0 in Ω. Then we suppose that the decomposition of Ω by W contains a multiply connected region Ω i and that Ω i contains a limit cycle Γ of system (1) surrounding in its interior a closed component W 0 of W. We use the fact that dψ is positive on W dt 0 and consider the region Ψ > 0 or Ψ < 0. In the case Ψ > 0 from (2) integration along the limit cycle Γ yields Γ div f dt = T 0 1 ( kψ Φ dψ dt ) T dt = 0 Φ kψ dt > 0, hence, Γ is orbitally unstable. By the same way we obtain that in any equilibrium point in the interior of Γ the inequality divf = Φ dψ dt kψ = Φ kψ > 0 holds. It leads to the contradiction which proves that there is no limit cycle in the domain Ψ > 0. The case Ψ < 0 can be treated analogously. We get that any trajectory starting near Γ has to cross W 0 but this contradicts the fact that dψ dt > 0. 9

The proof of Theorem 0.11 does not work in case k < 0. The following result include the case k < 0. Theorem 0.12 Let Ω D be a connected subset of R 2, and let Ψ be a DC function for system (1) in Ω. Then in each p-connected subdomain Ω of Ω with Ω W Ω, that means any point of the boundary of Ω belongs to W or to Ω, the number of limit cycles is at most p 1 and any limit cycle in Ω is stable (unstable) if k sign (ΨΦ) < 0 (k sign (ΨΦ) > 0). The proof of this theorem follows the same line as in the proof of Theorem 0.11. Theorem 0.13 Let system (1) has an unique equilibrium point an antisaddle O, divx(o) 0, in a symply-connected region Ω and there is a function Ψ satisfying the condition (2). If in Ω the equation Ψ(x,y) = 0 defines a nest of q embedded ovals then in each of the q 1 doubly-connected subregions Ω i bounded by neighboring ovals system (1) has exactly one limit cycle; and in the Ω in the whole it has at most q limit cycles. 10

Remark 0.14 The substantial difference of such approach to estimate the number of limit cycles from the classical one is that we must not localize limit cycles because this localization follows from the topological analysis of the curve Ψ(x,y) = 0. Furthermore, the inequality (2) is linear in Ψ and is considered not on a part of Ω but on the entire region. This approach was exploited by us to estimate the number of limit cycles for some classes of polynomial systems with the help of construction of polynomial DC functions. This approach was exploited also by Gasull A., Giacomini H. A new criterion for controlling the number of limit cycles of some generalized Liénard equations (2002), Gasull A., Giacomini H. Upper bounds for the number of limit cycles through linear differential equations (2006), Gasull A., Giacomini H., Llibre J. New criteria for the existence and non-existence of limit cycles in Liénard differential systems (2008), Gasull A., Giacomini H. Upper bounds for the number of limit cycles of some planar polynomial differential systems (2008). 11

Numerical aspects for the construction of DC function We look for a function Ψ(x,y) in the form of a linear combination of given basic functions Ψ = Ψ(x,y,C) = n C j Ψ j (x,y), C j = const, C = (C 1,..., C n ). (4) j=1 Then Φ = Φ(x,y,C) = n C j Φ j (x,y), (5) j=1 and the existence of a linear combination (5) satisfying inequality (2) for a given value of k in the case of a closed bounded domain Ω is equivalent to the inequality max min C j 1 (x,y) Ω Φ(x, y, C) > 0. (6) 12

The linear dependence of the function Φ on the additional variables C j, j = 1,...,n, permits one to find the maximin (6) from the corresponding linear programming problem: to find max L, L R, with the restrictions on a grid chosen in the domain Ω. n C j Φ j (x,y) L, C j 1, (x,y) Ω (7) j=1 To solve the problem (7) we can apply e.g. simplex methods and construct the function Ψ in the form of polynomials or in the form of splines. 13

Application of DC function to some classes of system (1) If Ω = Ω x = {(x,y) : x [x 1,x N0 ],y R} we construct the function Ψ in the form Ψ(x,y) = n Ψ j (x)y j, Ψ j C 1 (R), (8) j=0 for systems with l 1. dx dt = y, dy l dt = h j (x)y j, j=0 Φ(x,y) h j C 0 (R), m Φ i (x)y i, (9) i=0 where Φ i (x) is a function of the known coefficient functions h 0 (x),...,h l (x), of the unknown coefficient functions Ψ 0 (x),..., Ψ n (x), of their first derivatives Ψ 0(x),..., Ψ n(x), and of k. The highest power m of y in (29) is m = max{n + 1,n + l 1}. To determine the functions Ψ j (x),j = 0,...,n, and the real number k we reduce Φ(x,y) to the following cases 1. Φ(x,y) = Φ 0 (x), Φ i (x) = 0 for i = 1,...,m. (10) 2. Φ(x,y) = Φ 0 (x) + Φ 1 (x)y + Φ 2 (x)y 2, 3. Φ(x,y) = Φ 0 (x) + Φ 2 (x)y 2 + Φ 4 (x)y 4, for l = 1 and l = 2 the relations (10) represent a system of n + 1 linear differential equations to determine the n + 1 functions Ψ j,j = 0,...,n. 14

In case l = 1 we have Liènard system and the system (10) can be solved successively by simple quadratures, starting with Ψ n. ẋ = y, ẏ = h 0 (x) + h 1 (x)y, (11) In case l = 2 the system (10) can also be integrated by solving inhomogeneous linear differential equations, starting with Ψ n. We note that the functions Ψ j depend on the parameter k, but we get no restriction on k in the process of solving this system. To fulfill the condition (2) we have to choose k and the integration constants appropriately. x l [x 1,x N0 ], i = 1,N 0. L max, n C i Φi (x l ) L 0, C 1, (12) i=0 15

dx dt = y F(x), dy dt = x, (13) F(x) = Tñ+1 (x) = 1 2ñ 1(2xT ñ(x) Tñ 1 (x)), T 0 (x) = 1, T 1 (x) = x, ñ 1, (14) Example We put k = 1 and ñ = 11 in (14) for system (13). If N 0 = 89, 1 x 1. In this case the highest degree of Φ(x) is 100, min = 9.6468 10 7. There exist 5 limit cycles. 16

In case l = 3 (Kukles system) the first equation of the system (10) is an algebraic equation which determines the constant k uniquely as k = n. The remaining equations represent a system of n + 1 linear differential equations. Its general solution depends on n + 1 3 integration constants which can be used to try to fulfill the relations (2). If this is not possible we have to look for corresponding conditions on the functions h i. The paper [Cherkas L.A., Grin A., 2010] contains two algorithms to construct Φ(x,y) > 0: for odd y p all Φ p (x) = 0, for even y p all Φ p (x) 0 and Φ 0 (x) > 0. 17

For Kukles system with P 0 (x) C 1 we look for ẋ = yp 0 (x), ẏ = h 0 (x) + h 1 (x)y + h 2 (x)y 2 + h 3 y 3, (15) n Ψ = Ψ i (x)y n i,n N, (16) i=1 To fulfil Φ(x,y) > 0 we require Φ w (x) < ε for odd y w, ε sufficiently small and Φ v (x) 0 for even y v and Φ 0 (x) > 0. In this case we take all Ψ i (x) in the form Ψ i (x) = m i j=0 C ijx j, C ij R, m i N and solve the linear programming problem L max, m m C j Φ vj (x l ) L > 0, C j Φ wj (x l ) ε < 0, j=0 j=0 the vector C j consists of coefficients C ij from all Ψ i (x) and has dimension m = m 1 +... + m n + n. Example For system (15) with P 0 (x) = 1 + x 2 /6, h 0 (x) = x(1 + x 2 ), h 1 (x) = 1 x 2, h 2 (x) = (x 1)/100, h 3 = 1 we constructed Ψ in the form (16) by using k = 1, n = 3, m 1 = 6, m 2 = 7, m 3 = 8, ε = 0.0000001, N 0 = 100, [ 1.5; 1.5]. For the solution (C,L ) the equation Ψ = 0 defines unique oval and polynomial Φ(x,y) > 0 on the whole plane. It allows to prove the uniqueness of limit cycle globally. 18

In case l 4 system (31) consist of n + 1 linear differential equations and l 2 algebraic equations to determine k and the functions Ψ 0,..., Ψ n. Thus, this system has generically no solution. But using reduction to Φ(x,y) = Φ 0 (x) we can construct classes of mentioned systems with DC function in the form Ψ = Ψ 0 (x) + Ψ 1 (x)y + Ψ 2 (x)y 2 that means existence at most one limit cycle if Φ 0 (x) > 0 for all real values of x. [Cherkas L.A., Grin A., Schneider K.R. 2011] 19

We have tested the derived approaches to classes of system (1) where P(x,y) and Q(x,y) are periodic functions in the variable x, including the case of the Abel equation and the generalized pendulum equation dy dx = q 0(x) + q 1 (x)y +... + q w (x)y w, (17) dy dx = q 0(x) + q 1 (x)y +... + q w (x)y w. (18) y 20

An extension of mentioned approaches to systems with cylindrical phase space We consider systems of two autonomous differential equations on the cylinder Z := S 1 R under the assumption dϕ dt = P(ϕ,y), dy dt = Q(ϕ,y), (19) (A 1 ). P and Q belong to the class C 1 (Ω,R), and are 2π-periodic in the first variable. We denote by Ω the finite region on Z bounded by two closed curves γ 1 and γ 2 which surround the cylinder Z and do not intersect. Two kinds of limit cycles of (19) in Ω: A limit cycle of the first kind is contractible to a point in Ω: it requires the existence of an equilibrium of (19) in Ω. A limit cycle of the second kind surrounds the cylinder Z: it does not need the existence of any equilibrium in Ω. 21

Let f be the vector field defined on Ω by system (19), let D be a subregion of Ω. Definition 0.15 A function B C 1 (D,R) with the properties (i). B(ϕ,y) = B(ϕ + 2π,y) (ϕ,y) D, (ii). div(bf) := (BP) + (BQ) = (gradb,f) + Bdivf 0 ( 0) in D, ϕ y where div(bf) vanishes only on a subset of D of measure zero, is called a Dulac function of system (19) in D. Theorem 0.16 Let B be a Dulac function of (19) in the region D Ω. If the boundary D of D is connected and contractible to a point, then (19) has no limit cycle in D. In case that D consists of two closed curves in Ω which do not intersect and which are not contractible to a point then (19) has no limit cycle of the first kind in D and at most one limit cycle of the second kind of (19) in D. 22

Definition 0.17 A function Ψ C 1 (D,R) with the properties (i). Ψ(ϕ,y) = Ψ(ϕ + 2π,y) (ϕ,y) D, (ii). The set W := {(ϕ,y) D : Ψ(ϕ,y) = 0} has measure zero. (iii). There is a real number k 0 such that Φ(ϕ,y) := (gradψ,f) + kψdivf 0 ( 0) in D, (20) where the set V := {(ϕ,y) D : Φ(ϕ,y) = 0} has the properties (a). V has measure zero, (b). If Γ is a limit cycle of (19), then it holds Γ V Γ. (iv). (gradψ,f) W 0 (21) is called a Dulac-Cherkas function of (19) in D. Remark 0.18 If the inequalities in (20) hold strictly, then also condition (21) is valid. 23

We assume that the bounded region D Ω is formed by the curves 1 and 2 surrounding the cylinder Z. Theorem 0.19 Let Ψ be a DC function of (19) in D. Then it holds: (i). If the set W is empty, then system (19) has at most on limit cycle of the second kind in D. (ii). If the set W contains at least two branches connecting the curves 1 and 2, then system (19) has no limit cycle of the second kind in D. (iii) If the curve W consists in D of s closed branches (ovals) W 1,W 2,...,W s surrounding the cylinder Z and if D contains no equilibrium of (19), then system (19) has at least s 1 but not more than s + 1 limit cycle of the second kind in D. Theorem 0.20 Let Ψ be a DC function of (19) in D. Then the curve W can contain s 2 ovals surrounding the cylinder only if k is negative. From (20) we get such that we have Hence, it holds divf = Φ dψ dt kψ Γ divfds = 1 k Γ Φ Ψ ds. sign divfds = signk. Γ Remark 0.21 In case that k is negative we have divfds < 0. Thus, we can conclude that Γ is a simple limit cycle, that means Γ Γ is hyperbolic. 24

The construction of a DC function in a closed finite region Ω Z can be reduced to a linear programming problem L max, n c j φ j (ϕ p,y p ) L 0, c 1, (22) j=1 on a grid of nodes (ϕ p,y p ),p = 1,...,N 0, in the region Ω. As base functions on the cylinder Z we use the functions y i cos(lϕ) and y i sin(lϕ) where i and l are positive integer, such that As an example we consider the system n+1 [ m ( Ψ(ϕ,y) := y i 1 ail cos((l 1)ϕ) + b il sin((l 1)ϕ) )]. (23) dϕ dt = y2 + 1, i=1 dy dt l=1 = (y 1.5 0.3sinϕ)(y + 1.5 + 0.3sinϕ)(y 0.3sinϕ). (24) We estimate the number of limit cycles of the second kind for system (24) in the region Ω : π ϕ π, 2π y 2π, using the ansatz (23) with m = 4,n = 4 and k = 0.5 in the corresponding function Φ. We cover Ω with a uniform grid containing 900 nodes. 25

Analytical construction of a DC function for a class of systems We consider the vector field X l (ϕ,y) on the cylinder Z defined by the system dϕ dt = y, dy l dt = h j (ϕ)y j, l 1 (25) j=0 under the assumption that the functions h 0,...,h l are continuous and 2π-periodic, and that For a DC function Ψ(ϕ,y) of (25) in Z we make the ansatz h l (ϕ) 0. (26) Ψ(ϕ,y) = n Ψ j (ϕ)y j, n 1 (27) j=0 assuming that the functions Ψ 0,..., Ψ n are continuously differentiable and 2π-periodic. Ψ n (ϕ) 0, (28) We describe an algorithm to determine the functions Ψ j (ϕ) in (27) and the constant k such that the corresponding function Φ(ϕ,y) does not depend on y. For the sequel we represent Φ(ϕ,y) in the form Φ(ϕ,y) m Φ i (ϕ)y i, (29) i=0 the highest power m of y equals m = max{n + 1,n + l 1}. (30) 26

Our goal is to determine Ψ j (ϕ),j = 0,...,n, and the number k in such a way that Φ i (ϕ) 0 for i = 1,...,m. (31) Then it holds Φ(ϕ,y) Φ 0 (ϕ) Ψ 1 (ϕ)h 0 (ϕ) + kψ 0 (ϕ)h 1 (ϕ). (32) If we additionally require Φ 0 (ϕ) 0 ( 0) for 0 ϕ 2π (33) and if Φ 0 (ϕ) vanishes only at finitely many points of ϕ, then Ψ is a DC function of (25) in Z. For l 3 system (31) consists of l 2 algebraic equations and n + 1 linear differential equations 0 (n + lk)h l (ϕ)ψ n (ϕ), 0 (k(l 1) + n)h l 1 (ϕ)ψ n (ϕ) + (n 1 + lk)h l (ϕ)ψ n 1 (ϕ), 0 (n 1 + k(l 1))h l 1 (ϕ)ψ n 1 (ϕ) + (n + k(l 2))h l 2 (ϕ)ψ n (ϕ) + (n 2 + lk)h l (ϕ)ψ n 2,... 0 Ψ n(ϕ) + nh 2 (ϕ)ψ n (ϕ) + (n 1)h 3 (ϕ)ψ n 1 (ϕ) +... + (n l)h l+2 (ϕ)ψ n l (ϕ) + 2kh 2 (ϕ)ψ n (ϕ) + 3kh 3 (ϕ)ψ n 1 (ϕ) +... + klh l (ϕ)ψ n l+2 (ϕ),... 0 Ψ 1(ϕ) + (1 + 2k)h 2 (ϕ)ψ 1 (ϕ) + 3kh 3 (ϕ)ψ 0 (ϕ) + (2 + k)h 1 (ϕ)ψ 2 (ϕ) + 3h 0 (ϕ)ψ 3 (ϕ), 0 Ψ 0(ϕ) + 2kh 2 (ϕ)ψ 0 (ϕ) + (k + 1)h 1 (ϕ)ψ 1 (ϕ) + 2h 0 (ϕ)ψ 2 (ϕ). (34) 27

Thus, for l 4 from (30) we get m = n + l 1 and system (34) has generically no solution. The algebraic equation Φ m (ϕ) 0 determines the constant k uniquely as k = n l. We consider the system (25) in the case l = 5 dϕ dt = y, dy dt = h 0(ϕ) + h 1 (ϕ)y + h 2 (ϕ)y 2 + h 3 (ϕ)y 3 + h 4 (ϕ)y 4 + h 5 (ϕ)y 5. Our aim is to determine the functions h i, 0 i 5, by means of a DC function Ψ such that (35) has a given number of limit cycles surrounding the cylinder. We use the ansatz Ψ(ϕ,y) = Ψ 0 (ϕ) + Ψ 1 (ϕ)y + Ψ 2 (ϕ)y 2, (36) (35) Ψ 2 (ϕ) 0 for 0 ϕ 2π. (37) Φ(ϕ,y) Φ 0 (ϕ) = Ψ 1 (ϕ)h 0 (ϕ) + kψ 0 (ϕ)h 1 (ϕ). (38) k = 2 5, (39) h 2 (ϕ) = h 3 (ϕ) = h 4 (ϕ) = 5Ψ 1(ϕ)h 5 (ϕ), (40) 2Ψ 2 (ϕ) 5 ( 4Ψ 2 (ϕ) 2Ψ 0 (ϕ) + 3Ψ2 1(ϕ) Ψ 2 (ϕ) ) h 5 (ϕ), (41) 5 [( 9 ) 6Ψ 2 (ϕ) 2 Ψ 0(ϕ) + 3Ψ2 1(ϕ) Ψ1 (ϕ)h 5 (ϕ) ] Ψ 8Ψ 2 (ϕ) Ψ 2 (ϕ) 2(ϕ), (42) 28

5 [ h 1 (ϕ) = 8Ψ 2 (ϕ) + Ψ 1(ϕ) + Ψ 1(ϕ)Ψ 2(ϕ) 6Ψ 2 (ϕ) ( 3Ψ 2 0(ϕ) + 3Ψ 0(ϕ)Ψ 2 1(ϕ) 2Ψ 2 (ϕ) Ψ4 1(ϕ) 16Ψ 2 2(ϕ) ) h5 (ϕ) ], Ψ 2 (ϕ) (43) 1 [ h 0 (ϕ) = Ψ 2Ψ 2 (ϕ) 0(ϕ) + 3Ψ 1(ϕ)Ψ ( 1(ϕ) Ψ 2(ϕ) Ψ 2 1 (ϕ) 8Ψ 2 (ϕ) Ψ 2 (ϕ) 16Ψ 2 (ϕ) + 2Ψ ) 0(ϕ) 3 ( 15 + 8 Ψ2 0(ϕ) 5Ψ 0(ϕ)Ψ 2 ) 1(ϕ) + 3Ψ4 1(ϕ) Ψ1 (ϕ)h 5 (ϕ) ]. 16Ψ 2 (ϕ) 128Ψ 2 2(ϕ) Ψ 2 2(ϕ) The 2π-periodic continuous function h 5 can be chosen arbitrarily. (44) Φ 0 (ϕ) = 3h ( 5(ϕ) Ψ 3 4Ψ 0(ϕ) + 3Ψ2 0(ϕ)Ψ 2 1(ϕ) 2 2(ϕ) 4Ψ 2 (ϕ) 1 ( + Ψ 1 (ϕ)ψ 2Ψ 2 (ϕ) 0(ϕ) + Ψ 1(ϕ)Ψ 0 (ϕ) 2 ). Ψ 2(ϕ)Ψ 3 1(ϕ) 16Ψ 2 2(ϕ) 3Ψ 2(ϕ)Ψ 1 (ϕ)ψ 0 (ϕ) 4Ψ 2 (ϕ) 3Ψ 0(ϕ)Ψ 4 1(ϕ) 16Ψ 2 2(ϕ) + 3Ψ 1(ϕ)Ψ 2 1(ϕ) 8Ψ 2 (ϕ) ) + Ψ6 1(ϕ) 64Ψ 3 2(ϕ) (45) If we choose Ψ 2 (ϕ) = 1 2, Ψ 1(ϕ) = 1, Ψ 0 (ϕ) = cosϕ 10, (46) then the equation Ψ(ϕ,y) = 0 describes two ovals W 1 := {(ϕ,y) Z : y = 1 + 21 2 cos ϕ}, W 2 := {(ϕ,y) Z : y = 1 21 2 cos ϕ}. 29

We set h 5 (ϕ) 1 (47) and get h 0 (ϕ) 6203 8 h 1 (ϕ) 5395 8 305 cos ϕ 2 h 2 (ϕ) 295 2 285 cos ϕ 2 + 15 cos2 ϕ 2 + sin ϕ, (48) + 15 cos2 ϕ, (49) 2 + 15 cos ϕ, (50) h 3 (ϕ) 85 2 + 5 cosϕ, (51) Φ 0 (ϕ) 27783 8 3969 cosϕ 4 h 4 (ϕ) 5. (52) + 189 cos2 ϕ 2 + sinϕ 3 cos 3 ϕ Theorem 0.22 System (35) with h i defined by (47) - (52) has a unique limit cycle of second kind on the cylinder Z which is hyperbolic and unstable and located between W 1 and W 2. Now we use the ansatz Ψ(ϕ,y) = Ψ 0 (ϕ) + Ψ 1 (ϕ)y + Ψ 2 (ϕ)y 2 + Ψ 3 (ϕ)y 3 + Ψ 4 (ϕ)y 4, Ψ 4 (ϕ) 0. (53) Φ(ϕ,y) Φ 2 (ϕ)y 2 + Φ 1 (ϕ)y + Φ 0 (ϕ). (54) 30

If we introduce the function Ψ : Z R by and setting h 5 = 10 we obtain Ψ(ϕ,y) := y4 2 + ( cosϕ 10 2) y 2 + 1 10, (55) dϕ dt = y, dy dt = sin ϕ 20 + ( 55 2 + 5 cos ϕ 2 cos2 ϕ 16 (56) ) ( 5 cos ϕ) y + 50 y 3 10y 5 2 ( 87 cosϕ Φ(ϕ,y) 54 + 9 cos2 ϕ 10 20 ( + sin ϕ cos ϕ sin ϕ + 5 100 3 cos3 ϕ ) y 2 400 ) y + 11 5 cos ϕ + cos2 ϕ 5 200. The set W consists of four branches surrounding the cylinder W 1 := {(ϕ,y) Z : y = 2 cos ϕ 10 + cos2 ϕ 40 cos ϕ + 380 }, 10 W 2 := {(ϕ,y) Z : y = 2 cos ϕ 10 cos2 ϕ 40 cos ϕ + 380 }, 10 W 3 := {(ϕ,y) Z : y = 2 cosϕ 10 cos2 ϕ 40 cos ϕ + 380 }, 10 W 4 := {(ϕ,y) Z : y = 2 cos ϕ 10 + cos2 ϕ 40 cos ϕ + 380 }. 10 (57) Theorem 0.23 On the cylinder Z system (56) has four limit cycles of second kind which are hyperbolic. 31

Conclusions 1. l 1 dx dt = p j (x)y j, j=0 dy l dt = h j (x)y j, h j C 0 (R), p j C 1 (R), j=0 2. dx dt = y εf(x), dy dt = g(x), (58) dx dt = y, dy dt = x + ε(h 0(x) + h 1 (x)y + h 2 (x)y 2 + h 3 (x)y 3 ), (59) 32

Thank you! 33