Weierstraß-Institut. für Angewandte Analysis und Stochastik. Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN

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1 Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN Construction of generalized pendulum equations with prescribed maximum number of limit cycles of the second kind Klaus R. Schneider 1 Alexander Grin submitted: June Weierstraß-Institut Mohrenstraße Berlin Germany Klaus.Schneider@wias-berlin.de Yanka Kupala State University of Grodno Ozheshko Street 3003 Grodno Belarus grin@grsu.by No. 7 Berlin Mathematics Subject Classification. 34C0 34C3. Key words and phrases. generalized pendulum equation limit cycles of the second kind limit cycle of multiplicity three bifurcation behavior Dulac-Cherkas function.

2 Edited by Weierstraß-Institut für Angewandte Analysis und Stochastik WIAS) Leibniz-Institut im Forschungsverbund Berlin e. V. Mohrenstraße Berlin Germany Fax: World Wide Web:

3 1 ABSTRACT. Consider a class of planar autonomous differential systems with cylindric phase space which represent generalized pendulum equations. We describe a method to construct such systems with prescribed maximum number of limit cycles which are not contractible to a point limit cycles of the second kind). The underlying idea consists in employing Dulac-Cherkas functions. We also show how this approach can be used to control the bifurcation of multiple limit cycles. 1. INTRODUCTION We consider on a cylinder Z := {ϕ y) : ϕ [0 π] y R} the generalized pendulum system 1.1) dϕ dt = y dy dt = l h j ϕ µ)y j l 3 j=0 depending on the real parameter µ I where I is some interval. We assume that the functions h j : R I R 0 j l are continuous and additionally continuously differentiable and π-periodic in the first variable. We denote by f l the corresponding vector field Moreover we suppose f l = y l h j ϕ µ)y ). j j=0 1.) h l ϕ µ) 0 for ϕ µ) [0 π] I. The most difficult problem in the qualitative investigation of system 1.1) is the localization and estimation of the number of limit cycles which represent isolated closed orbits of 1.1) with finite primitive period. It is well known that we have to distinguish two kinds of limit cycles of 1.1) on Z. A limit cycle Γ on Z is called a limit cycle of the first kind if Γ is contractible to a point on Z Γ is called a limit cycle of the second kind if Γ surrounds the cylinder Z that is it is not contractible to a point [1 ]. For the investigation of limit cycles of the first kind the well-known methods for planar autonomous systems can be applied. Especially the existence of a limit cycle of the first kind of 1.1) on Z requires the existence of an equilibrium of 1.1) on Z. In contrast to that fact the existence of a limit cycle of the second kind on Z does not need the existence of any equilibrium. If we study the behavior of 1.1) in dependence on the parameter µ then it may happen that there are some critical values of the parameter µ related to a qualitative change of the phase portrait of 1.1). These values are called bifurcation points. For example they may be related to the occurrence of new equilibria or limit cycles. An important but difficult problem is to determine the number of limit cycles in dependence on the parameter µ. A well-known tool to prove the absence of limit cycles is the Bendixson criterion which has been extended by H. Dulac see e.g. [10]). A further development is due to L. Cherkas who introduced a class of functions - nowadays also called Dulac-Cherkas functions - which can be used to estimate the number of limit cycles in planar systems and to determine their localization and stability [3 7].

4 In the papers [4 6] the method of Dulac-Cherkas functions has been extended to the study of limit cycles of the second kind. In this paper our goal is to show how Dulac-Cherkas functions can be used to construct systems 1.1) having a prescribed number of limit cycles of the second kind and to control their bifurcations. The paper is organized as follows. In section we recall some basic facts about Dulac-Cherkas functions. Section 3 contains the description of our general idea. By using functions Ψϕ y µ) which are quadratic polynomials in y as Dulac-Cherkas functions in Section 4 we construct for the case l = 3 systems 1.1) which have at most one or three limit cycles. To derive systems 1.1) with more limit cycles in Section we deal with the construction of Dulac-Cherkas functions Ψϕ y µ) which are polynomials in y of degree four in the case l =.. PRELIMINARIES We consider on the cylinder Z the system dϕ dy.1) = P ϕ y µ) dt under the assumption dt = Qϕ y µ) f := P Q) A 1 ). P Q : Z I R are continuous continuously differentiable in the first two variables and π-periodic in the first variable. Let D be a subregion of Z. Definition.1. Suppose hypothesis A 1 ) to be valid. A function B : Z I R with the same smoothness as P Q and having the properties i). Bϕ y µ) = Bϕ + π y µ) ϕ y µ) D I ii). divbf) := BP ) + BQ) = gradb f) + Bdivf 0 0) ϕ y in D for µ I where divbf) vanishes only on a subset of D of measure zero is called a Dulac function of system.1) in D for µ I. The following result can be found in []. Theorem.. Suppose hypothesis A 1 ) to be valid. Let B be a Dulac function of.1) in the bounded region D Z for µ I. If the boundary D of D is connected and contractible to a point then.1) has no limit cycle of the first kind in D. In case that D consists of two closed curves 1 and on Z which do not meet and which are not contractible to a point then.1) has no limit cycle of the first kind in D and at most one limit cycle of the second kind in D. Now we give a generalization of a Dulac function which can be found in [6]. It is basically due to L. Cherkas [3] hence we call it Dulac-Cherkas function. Definition.3. Suppose hypothesis A 1 ) to be valid. A function Ψ : Z I R having the same smoothness as P Q and with the properties

5 3 i). Ψϕ y µ) = Ψϕ + π y µ) ϕ y µ) D I. ii). For µ I the set Wµ) := {ϕ y) D : Ψϕ y µ) = 0} has measure zero. iii). There is a real number κ 0 such that for µ I.) Φϕ y µ κ) := gradψ f) + κψdivf 0 0) in D iv)..3) where the set V κ µ) := {ϕ y) D : Φϕ y µ κ) = 0} has the properties a). V κ µ) has measure zero. b). If Γµ) is a limit cycle of.1) then it holds Γµ) V κ µ) Γµ). gradψ f) Wµ) 0 is called a Dulac-Cherkas function of system.1) in D for µ I. Remark.4. If the inequalities in.) hold strictly then also condition.3) is valid. Remark.. If Φ does not depend on y that is Φϕ y µ κ) = Φ 0 ϕ µ κ) and if Φ 0 vanishes only in finitely many points ϕ i µ κ) then the conditions on the set V κ µ) are fulfilled. Remark.6. From.3) it follows that the curves belonging to the set Wµ) are crossed transversally by the trajectories of system.1). In what follows we focus on limit cycles of the second kind. Therefore we assume for what follows A ). The boundary of the considered region D consists of two closed curves u upper closed curve) and l lower closed curve) surrounding the cylinder Z and satisfying u u =. The following results have been proved in [6]. They show how the topological structure of Wµ) influences the topological structure of the limit cycles of the second kind of system.1). Theorem.7. Suppose the hypotheses A 1 ) and A ) to be valid. Let Ψ be a Dulac-Cherkas function of.1) in D for µ I. If the set Wµ) is empty then system.1) has at most one limit cycle of the second kind in D. Now we assume A 3 ).The set Wµ) consists in D of s isolated closed curves ovals) W 1 µ) W µ)... W s µ) surrounding the cylinder Z and which do not touch the boundaries u and l.

6 4 Without loss of generality we may assume the following ordering of these curves: W i µ) is located on Z above W i+1 µ). With this ordering we associate the following notation: The region on Z between W i µ) and W i+1 µ) is denoted by A i µ) the region between u and W 1 µ) is denoted by A 0 µ) the region between W s µ) and l is denoted by A s µ). Fig. 1 illustrates the case s =. Fig.1. Regions A i in the case s =. Theorem.8. Assume the hypotheses A 1 ) A 3 ) to be valid and that D contains no equilibrium of.1). Then system.1) has at least s 1 but at most s + 1 limit cycles of the second kind in D more precisely the region A i µ) i = 1... s 1 contains a unique limit cycle Γ i µ) of the second kind each of the regions A 0 µ) and A s µ) may contain a unique limit cycle of the second kind. Furthermore the limit cycle Γ i µ) in A i µ) is hyperbolic and asymptotically stable unstable) if κφϕ y µ κ)ψϕ y µ) < 0 > 0) in A i µ). 3. GENERAL IDEA As we mentioned in the introduction our goal is to show how Dulac-Cherkas functions can be used to construct systems 1.1) with a given number of limit cycles of the second kind and to control their bifurcations. For this purpose we suppose that the given Dulac-Cherkas function has the structure n 3.1) Ψϕ y µ) = Ψ j ϕ µ)y j n 1 j=0 where the functions Ψ j : R I R 0 j n are continuous and twice continuously differentiable and π-periodic in the first variable and where 3.) Ψ n ϕ µ) 0 for ϕ µ) [0 π] I.

7 According to this assumption the corresponding function Φϕ y µ) determined by.) has the structure 3.3) Φϕ y µ κ) = m Φ i ϕ µ κ)y i m = n + l 1. i=0 To get an explicit representation of the coefficient functions Φ i we rewrite the natural number l in 1.1) in the form l = 3 + s. Then using 1.1) and 3.1) we get 3.4) Φ gradψ f l ) + κψdivf l n+1 Ψk 1 ) n++s = A 0 + κb 0 + ϕ + A k + κb k y k + A k + κb k )y k k=1 k=n+ where 3.) A k = i+j=k 0 j n 1 0 i 3+s=l j + 1)Ψ j+1 h i B k = i+j=k 0 j n 1 0 i 3+s=l i + 1)h i+1 Ψ j. Under our assumptions n 1 l 3 we can represent the coefficients Φ m Φ 1 Φ 0 in the explicit form 3.6) Φ m = n + lκ)h l ϕ µ)ψ n ϕ µ) Φ 1 = Ψ 0ϕ µ) + kh ϕ µ)ψ 0 ϕ µ) + κ + 1)h 1 ϕ µ)ψ 1 ϕ µ) + h 0 ϕ µ)ψ ϕ µ) Φ 0 = Ψ 1 ϕ µ)h 0 ϕ µ) + κψ 0 ϕ µ)h 1 ϕ µ) where the prime denotes the differentiation with respect to ϕ. For Ψ to be a Dulac-Cherkas function of 1.1) in D for µ I it is sufficient that one of the inequalities 3.7) is fulfilled for ϕ y µ) D I. Φϕ y µ κ) 0 Φϕ y µ κ) 0 Our key idea is to choose the constant κ and the functions h j 0 j l in such a way that some of the coefficient functions Φ i vanish identically such that Φ takes a structure which permits us in a simple way to derive conditions guaranteeing that Φ is positive or negative definite. Since one of our goals is to show that Dulac-Cherkas functions can be used to control the existence of limit cycles of the second kind we have to take into account that by Theorem.8 the existence of at least two limit cycles of the second kind for system 1.1) requires that the set Wµ) contains at least one oval. From that reason we consider in the sequel functions Ψ in 3.1) for n = and n = 4 with the property that the set Wµ) has two or four ovals respectively.

8 6 In what follows we demonstrate our key idea in the cases l = n = l = 3 n = and l = n = 4. In the case l = n = the functions Φ k 0 k 6 read according to 3.4) as follows 3.8) Φ 6 = + κ)h Ψ Φ = + 4κ)h 4 Ψ κ)h Ψ 1 Φ 4 = + 3κ)h 3 Ψ κ)h 4 Ψ 1 + κh Ψ 0 Φ 3 = + κ)h Ψ κ)h 3 Ψ 1 + 4κh 4 Ψ 0 + Ψ Φ = + 1κ)h 1 Ψ κ)h Ψ 1 + 3κh 3 Ψ 0 + Ψ 1 Φ 1 = + 0κ)h 0 Ψ κ)h 1 Ψ 1 + κh Ψ 0 + Ψ 0 Φ 0 = 1 + 0κ)h 0 Ψ 1 + 1κh 1 Ψ 0. Our goal is to vanish Φ 6... Φ 1 identically by a suitable choice of κ and of the functions h i. Taking into account 3.) we get from 3.8) κ = h 4 = 1 + κ)h Ψ 1 + 4κ)Ψ 3.9) h 3 = 1 + 4κ)h 4Ψ 1 + κh Ψ 0 + 3κ)Ψ h = 1 + 3κ)h 3Ψ 1 + 4κh 4 Ψ 0 + Ψ + κ)ψ h 1 = 1 + κ)h Ψ 1 + 3κh 3 Ψ 0 + Ψ 1 + 1κ)Ψ h 0 = 1 + 1κ)h 1Ψ 1 + κh Ψ 0 + Ψ 0 + 0κ)Ψ. As a consequence we obtain 3.10) Φ ϕ y µ ) = Φ 0 ϕ µ ) = Ψ 1 ϕ µ)h 0 ϕ µ) Ψ 0ϕ µ)h 1 ϕ µ) such that the inequalities 3.7) read now 3.11) Φ 0 ϕ µ ) 0 Φ 0 ϕ µ ) 0 that is they do not depend on y. The case l = 3 n = can be treated analogously using the relations above. We obtain

9 7 3.1) Φ 4 = + 3κ)h 3 Ψ Φ 3 = + κ)h Ψ κ)h 3 Ψ 1 + Ψ Φ = + 1κ)h 1 Ψ κ)h Ψ 1 + 3κh 3 Ψ 0 + Ψ 1 Φ 1 = + 0κ)h 0 Ψ κ)h 1 Ψ 1 + κh Ψ 0 + Ψ 0 Φ 0 = 1 + 0κ)h 0 Ψ 1 + 1κh 1 Ψ 0. Our goal is to vanish Φ 4... Φ 1 identically by a suitable choice of κ and the functions h i. Taking into account 3.) we get from 3.1) κ = ) h = 1 + 3κ)h 3Ψ 1 + Ψ + κ)ψ h 1 = 1 + κ)h Ψ 1 + 3κh 3 Ψ 0 + Ψ 1 + 1κ)Ψ h 0 = 1 + 1κ)h 1Ψ 1 + κh Ψ 0 + Ψ 0 + 0κ)Ψ. The inequalities 3.7) have the form as in 3.11) where Φ 0 is defined by Φ 0 ϕ µ ) = Ψ 1 ϕ µ)h 0 ϕ µ) 3 3 Ψ 0ϕ µ)h 1 ϕ µ). Next we consider the case l = n = 4. Here the functions Φ k 0 k 8 read according to 3.4) as follows Φ 8 =4 + κ)h Ψ 4 Φ 7 =4 + 4κ)h 4 Ψ κ)h Ψ 3 Φ 6 =4 + 3κ)h 3 Ψ κ)h 4 Ψ κ)h Ψ Φ =4 + κ)h Ψ κ)h 3 Ψ κ)h 4 Ψ κ)h Ψ 1 + Ψ ) Φ 4 =4 + 1κ)h 1 Ψ κ)h Ψ κ)h 3 Ψ κ)h 4 Ψ 1 + κh Ψ 0 + Ψ 3 Φ 3 =4 + 0κ)h 0 Ψ κ)h 1 Ψ κ)h Ψ κ)h 3 Ψ 1 + 4κh 4 Ψ 0 + Ψ Φ =3 + 0κ)h 0 Ψ κ)h 1 Ψ κ)h Ψ 1 + 3κh 3 Ψ 0 + Ψ 1 Φ 1 = + 0κ)h 0 Ψ κ)h 1 Ψ 1 + κh Ψ 0 + Ψ 0 Φ 0 =1 + 0κ)h 0 Ψ 1 + 1κh 1 Ψ 0.

10 8 Taking into account 1.) and 3.) we get from 3.14) that Φ 8 vanishes identically if we choose 3.1) κ = 4. From 3.14) and 3.) we get further that we can use the functions h 0... h 4 to vanish the functions Φ 3... Φ 7 accordingly. h 4 = 3 + κ)h Ψ κ)Ψ 4 h 3 = + κ)h Ψ κ)h 4 Ψ κ)Ψ ) h = 1 + κ)h Ψ κ)h 4 Ψ κ)h 3 Ψ 3 + Ψ κ)ψ 4 h 1 = κh Ψ κ)h 4 Ψ κ)h 3 Ψ κ)h Ψ 3 + Ψ κ)Ψ 4 h 0 = 4κh 4Ψ κ)h 3 Ψ κ)h Ψ κ)h 1 Ψ 3 + Ψ 4Ψ 4. Thus the inequalities in 3.7) read 3.17) Φ ϕ y µ 4 ) Φ ϕ µ 4 )y + Φ 1 ϕ µ 4 )y + Φ 0 ϕ µ 4 ) 0 0) where we can use the corresponding discriminant in order to derive conditions that Φϕ y µ κ) is definite. Another possibility to simplify the form of the inequalities in 3.7) is at first to determine κ h 4 h 3 h and h 0 as above and reduce Φϕ y µ) to the following form 3.18) Φ ϕ y µ 4 ) = Φ 4 + Φ ϕ µ 4 ) y 4 ϕ µ 4 )y + Φ 1 ϕ µ 4 )y + Φ 0 ϕ µ 4 ) 0 0). Then under the additional assumption we can determine h 1 by Ψ 1 ϕ µ) 0 for ϕ µ) [0 π] I h 1 = h 0Ψ 8 h Ψ 0 + Ψ )Ψ 1 = 10h 0Ψ 8h Ψ 0 + Ψ 0 Ψ 1 from the identity Φ 1 ϕ µ 4 ) 0. Thus the inequalities in 3.7) can be written as follows 3.19) Φ 4 ϕ µ 4 )y 4 + Φ ϕ µ 4 )y + Φ 0 ϕ µ 4 ) 0 0)

11 where the sign of y plays no role and we can use different methods to guarantee that Φ is definite. In all these cases we will choose the functions Ψ j ϕ µ) in such a way that the corresponding set Wµ) consists of a prescribed number of ovals and that the function Φϕ y µ κ) satisfies one of the inequalities 3.11) or 3.17) or 3.19). Then under the assumptions of Theorem.8 the constructed system 1.1) has not more than a prescribed number of limit cycles of the second kind in D CONSTRUCTION OF SYSTEMS 1.1) IN THE CASE l = 3 HAVING NOT MORE THAN THREE LIMIT CYCLES OF THE SECOND KIND Now we apply the described method to construct systems 4.1) dϕ dt = y dy dt = h 0ϕ µ) + h 1 ϕ µ)y + h ϕ µ)y + h 3 ϕ µ)y 3 having not more than three limit cycles of the second kind. For this purpose we use for Ψ the ansatz 4.) Ψϕ y µ) = Ψ 0 ϕ µ) + Ψ 1 ϕ µ)y + Ψ ϕ µ)y where the functions Ψ i satisfy the assumptions formulated at the beginning of section 3. According to our treatment in section 3 we have the case l = 3 n =. Thus the set Wµ) := {ϕ y) D : Ψϕ y µ) = 0} consists of at most two ovals surrounding the cylinder and the function Φ introduced in 3.3) represents a polynomial of degree m = 4 where the coefficient functions Φ i are defined in 3.1). If the constant κ and the functions h i 0 i satisfy the relations 3.13) then the functions Φ 4 Φ 3 Φ and Φ 1 vanish identically and the inequalities 3.7) have the form 3.11). In our case we have 4.3) Φ 0 ϕ µ ) = Ψ 1 ϕ µ)h 0 ϕ µ) 3 3 Ψ 0ϕ µ)h 1 ϕ µ) where the functions h 0 h 1 and h are defined by 4.4) h = 3 ) h3 Ψ 1 Ψ Ψ 4.) h 1 = 1 [ 3Ψ1 ) h3 Ψ 1 Ψ + 6h3 Ψ 0 3Ψ 4Ψ Ψ 1] 4.6) h 0 = 1 3h3 Ψ 0 Ψ 1 h 3Ψ 3 1 Ψ Ψ Ψ 8Ψ 0 + Ψ 1Ψ 1 + Ψ 1Ψ Ψ 0Ψ ). 4Ψ 8Ψ Ψ From 4.3) 4.) and 4.6) we get

12 10 4.7) Φ 0 = 1 Ψ h3 Ψ 0 Ψ 1 Ψ 3Ψ 0Ψ 1 Ψ Ψ h 3Ψ 4 1 8Ψ + Ψ 1Ψ 1 4Ψ Ψ 0Ψ 1 + Ψ 1Ψ 0 h 3 Ψ 0 ). + Ψ3 1Ψ 8Ψ Using these relations we have the following result. Theorem 4.1. Let the functions Ψ 0 Ψ 1 Ψ : R R R be continuous and twice continuously differentiable and π-periodic in the first variable let the functions h h 1 and h 0 be defined by 4.4) 4.) and 4.6) respectively let h 3 : R R R be any continuous function continuously differentiable and π-periodic in the first variable and such that the relation 1.) is satisfied. If there exists an interval I and a region D on Z whose boundary consists of two closed curves which surround the cylinder and do not meet such that for µ I i) the function Φ 0 ϕ µ ) defined in 4.7) has the same sign in D and vanishes only in 3 finitely many points ϕ i µ) ii) gradψ f) Wµ) 0 iii) system 4.1) has no equilibrium in D then system 4.1) has in D at most three limit cycles of the second kind for µ I. Proof. Under the assumptions of Theorem 4.1 and taking into account the Remarks.4 and. the function Ψϕ y µ) defined by 4.) is for µ I a Dulac-Cherkas function on the whole cylinder and the corresponding set Wµ) contains at most two ovals surrounding the cylinder. Applying Theorem.8 the proof is complete. In the following we choose the functions Ψ 0 Ψ 1 Ψ in a special way in order to construct a system 4.1) having for µ = µ a limit cycle of the second kind with multiplicity three exhibiting different bifurcation behavior for increasing and decreasing parameter µ. The underlying idea is to construct a Dulac-Cherkas function Ψϕ y µ) with the property that the topological structure of the corresponding set Wµ) changes when the parameter µ crosses some critical value µ. We will exploit this property to construct a system 4.1) having µ as a bifurcation point related to a change of the number of limit cycles of the second kind. We put 4.8) 4.9) 4.10) Hence we have 4.11) and the set Ψ ϕ µ) 1 Ψ 1 ϕ µ) := 1 + µ cos ϕ) Ψ 0 ϕ µ) := 1 + µ cos ϕ) µ = 1 4 Ψ 1 µ. Ψϕ y µ) = y µ cos ϕ) µ Wµ) := {ϕ y) Z : Ψϕ y µ) = 0}

13 11 consists for µ > 0 of the two ovals 4.1) W 1 µ) := {ϕ y) Z : y µ cos ϕ µ = 0} 4.13) W µ) := {ϕ y) Z : y µ cos ϕ + µ = 0 for µ = 0 from the oval 4.14) W 0 := {ϕ y) Z : y = 1} and Wµ) is empty for µ < 0. Thus we have a change in the topological structure of the set Wµ) when µ crosses zero. Now we determine a class of systems 4.1) for which the function Ψϕ y µ) in 4.11) is a Dulac-Cherkas function. By 4.8) ) and using the relations Ψ 0 Ψ 0 = Ψ 1 Ψ 1 Ψ 1 = µ sin ϕ we obtain from 4.3) - 4.7) for the functions h i i = 0 1 and for Φ 0 the relations 4.1) h ϕ µ) = 3h 3 ϕ µ)1 + µ cos ϕ) 4.16) h 1 ϕ µ) = 3 [ h3 ϕ µ)1 + µ cos ϕ) + µsin ϕ h 3 ϕ µ) ] 4.17) h 0 ϕ µ) = µ cos ϕ)[ h 3 ϕ µ)1 + µ cos ϕ) + µsin ϕ 3h 3 ϕ µ) ] Φ 0 ϕ µ ) µ sin ϕ h 3 ϕ µ) ). 3 Thus under the assumption that h 3 : R R R is continuous continuously differentiable and π-periodic in the first variable and obeys the condition 4.18) h 3 ϕ µ) > 1 ϕ µ)

14 1 one of the inequalities. holds strictly and we can conclude that the function Ψ in 4.11) is a Dulac-Cherkas function for the system 4.19) dϕ dt =y dy dt =1 1 + µ cos ϕ)[ h 3 ϕ µ)1 + µ cos ϕ) + µsin ϕ 3h 3 ϕ µ)) ] + 3 [ h3 ϕ µ)1 + µ cos ϕ) + µsin ϕ h 3 ϕ µ) ] y + 3h 3 ϕ µ)1 + µ cos ϕ)y + h 3 ϕ µ)y 3 on Z for all µ R \ {0}. For µ = 0 system 4.19) takes the form dϕ dt = y dy 4.0) dt = h 3ϕ µ)y + 1) 3. Under the condition 4.18) the second equation has the unique equilibrium y = 1 which is an equilibrium of multiplicity three. Hence we can conclude that system 4.19) has for µ = 0 a unique closed trajectory which is a limit cycle Γ0) := {ϕ y) Z : y = 1} of the second kind of multiplicity three. Its stability depends on the sign of h 3 ϕ µ). Our goal is to study the bifurcation behavior of Γ0) when µ crosses the value 0 by means of Theorem.8. A basic assumption of that theorem is the non-existence of an equilibrium. It is obvious that any equilibrium ϕ 0 µ) 0) of system 4.19) satisfies the equation 4.1) 1 + µ cos ϕ) [ h 3 ϕ µ)1 + µ cos ϕ) + µsin ϕ 3h 3 ϕ µ) ] = 0. The following lemmata are obvious. Lemma 4.. The first factor in 4.1) has no root for µ < 1 and there is a sufficiently small positive number µ 1 < 1 such that the second factor has no real root for µ < µ 1. Thus system 4.1) has no real real root for µ < µ 1. Lemma 4.3. To given µ there is a sufficiently large positive number C 0 µ) such that all closed curves y = C and y = C on Z with C C 0 µ) are crossed by the trajectories of 4.19) transversally and in the direction of increasing y. If we denote by Aµ) the region on Z bounded by the curves y = C 0 µ) and y = C 0 µ) then we get from Lemma 4.3 by applying the Poincaré-Bendixson Theorem the following corollary. Corollary 4.4. System 4.19) has for µ < µ 1 in Aµ) at least one limit cycle. By apply Theorem.8 Lemma 4. and Lemma 4.3 we get the following result. Theorem 4.. From the limit cycle Γ0) of multiplicity three of the system 4.19) there bifurcates a unique simple limit cycle for decreasing µ there bifurcate three simple limit cycles for increasing µ.

15 13. CONSTRUCTION OF SYSTEMS 1.1) IN THE CASE l = HAVING NOT MORE THAN FOUR We consider system.1) in the case l = LIMIT CYCLES OF THE SECOND KIND dϕ dt = y.1) dy dt = h 0ϕ) + h 1 ϕ)y + h ϕ)y + h 3 ϕ)y 3 + h 4 ϕ)y 4 + h ϕ)y. In the following subsection we construct a Dulac-Cherkas function such that system.1) has a unique limit cycle of the second kind in the last subsection we construct a Dulac-Cherkas function such that system.1) has exactly four limit cycles of the second kind..1. Construction of a system 1.1) in the case l = n = having a unique limit cycle of the second kind. We use the ansatz.) where Ψϕ y) = Ψ 0 ϕ) + Ψ 1 ϕ)y + Ψ ϕ)y.3) Ψ ϕ) = 1 Ψ 1ϕ) = 1 Ψ 0 ϕ) = cos ϕ 10. In that case the curve W consists of the two ovals W 1 := {ϕ y) Z : y = cos ϕ} W := {ϕ y) Z : y = 1 1 cos ϕ}. The region bounded by W 1 and W is denoted by A 1 the regions on Z located above W 1 and below W are denoted by A 0 and A respectively. Obviously we have.4) Ψϕ y) < 0 in A 1. If we determine the constant κ and the functions h i according to 3.9) where we additionally set h ϕ) 1 then we obtain.).6) h 0 ϕ) κ = 30 cos ϕ + 1 cos ϕ + sin ϕ h 1 ϕ) 39 8 h ϕ) 9 8 cos ϕ + 1 cos ϕ + 1 cos ϕ h 3 ϕ) 8 + cos ϕ h 4 ϕ).

16 14 The corresponding system.1) reads explicitly.7) dϕ dt =y dy dt = cos ϕ + 1 cos ϕ + sin ϕ 8 cos ϕ + 1 cos ϕ ) y 9 ) + 1 cos ϕ y cos ϕ ) y 3 + y 4 + y. From 3.10) we get Φ ϕ y ) = Φ 0 ϕ) The relation cos ϕ cos ϕ + sin ϕ 3 cos 3 ϕ..8) Φ 0 ϕ) > 0 ϕ can be easily verified. Thus we can conclude that the function Ψϕ y) = cos ϕ 10 + y + 0.y is a Dulac-Cherkas function for system.7) on Z. From.6) we get that the function h 0 is positive for all ϕ. That implies that system.7) has no equilibrium on Z. Thus we get from Theorem.8 that system.7) has at least one limit cycle of the second kind. Furthermore we obtain from.8).).4).9) κφ 0 ϕ)ψϕ y) > 0 in A 1. Hence system.7) has a unique limit cycle of the second kind in A 1. We denote by N the curve defined by.10) N :={ϕ y) Z : dy dt = cos ϕ 30 cos ϕ + 1 cos ϕ )y + 1 cos ϕ + sin ϕ cos ϕ)y cos ϕ)y3 + y 4 + y = 0}. It is obvious that any limit cycle of system.1) must either cut the curve N or be a subset of that curve. Using the method of Sturmian chains it can be shown that the curve N consists of a unique branch located in A 1. Thus using this fact relation.9) and Theorem.8 we have the following result. Theorem.1. System.7) has a unique limit cycle of the second kind. This limit cycle is hyperbolic orbitally unstable and located in the region A 1.

17 .. Construction of a system in the case l = n = 4 having exactly four limit cycles. 1 We use for Ψϕ y) the ansatz.11) with Ψϕ y) = 4 Ψ j ϕ)y j j=0.1) Ψ 0 ϕ) 1 10 Ψ 1ϕ) 0 Ψ ϕ) = cos ϕ 10 such that we have.13) Ψ 3 ϕ) 0 Ψ 4 ϕ) 1 Ψϕ y) = cos ϕ 10 ) y + 1 y4. According to 3.3) Φϕ y) can be represented in the form 8.14) Φϕ y) = Φ j ϕ)y j j=0 where the functions Φ j ϕ) are defined by 3.14). If we set h ϕ) 10 and determine κ h 0... h 4 by 3.1) and 3.16) we obtain.1) κ = 4 h 4ϕ) 0 h 3 ϕ) = 0 cos ϕ h ϕ) 0 h 1 ϕ) = + cos ϕ 1 16 cos ϕ h 0 ϕ) = 1 sin ϕ. 0 Using these relations the corresponding system.1) reads.16) dϕ dt = y dy dt = sin ϕ cos ϕ cos ϕ 16 ) cos ϕ) y + 0 y 3 10y and from 3.14) we get.17) Φϕ y) = 11 cos ϕ + cos ϕ cos ϕ cos ϕ 10 0 sin ϕ 3 cos3 ϕ 400 cos ϕ sin ϕ ) y. ) y Using the discriminant it can be shown by simple calculations that Φϕ y) is strictly positive for all ϕ and y. Thus according to Remark.4 Ψϕ y) is a Dulac-Cherkas function of system

18 16.16) on Z. The set W consists of four closed curves W i surrounding the cylinder Z: W 1 := {ϕ y) Z : y = W := {ϕ y) Z : y = W 3 := {ϕ y) Z : y = W 4 := {ϕ y) Z : y = cos ϕ 10 + cos ϕ 10 cos ϕ 10 cos ϕ 10 + cos ϕ 40 cos ϕ } 10 cos ϕ 40 cos ϕ } 10 cos ϕ 40 cos ϕ } 10 cos ϕ 40 cos ϕ }. 10 As usually we denote the region bounded by W i and W i+1 by A i. It follows from.16) that there is a sufficiently large number C such that dy dy < 0 on the closed curve y = C and > 0 dt dt on the closed curve y = C. The region bounded by C and W 1 will be denoted by A 0 and region bounded by W 4 and C will be denoted by A 4. Since there is no equilibrium in the regions A 0 A 1 A 3 A 4 by using Theorem.8 we get the following result. Theorem.. System.16) has in the regions A 0 and A 4 a unique hyperbolic limit cycle of the second kind which is asymptotically orbitally stable and in the regions A 1 and A 3 a unique hyperbolic limit cycle of the second kind which is orbitally unstable. Theorem.3. The region A contains no limit cycle and no homoclinic curve of the second kind of system.16). Proof. The region A contains the closed curve y = 0 on which there are located two equilibria of system.16): the saddle 0 0) and and the stable node π 0). The eigenvalues corresponding to the saddle have different signs they read as λ λ Thus the corresponding saddle value λ = λ 1 + λ is negative. In the same way as we have proved Theorem 4.1 in [] we can prove that any limit cycle of the second kind of system.16) cannot meet the closed curve y = 0. We denote the region on Z bounded by the closed curves W and y = 0 by A + and by A the region bounded by the closed curves y = 0 and W 3. Now we prove that there is no limit cycle of the second kind of system.16) in A +. First we note that Ψϕ y) > 0 in A. This follows immediately from.13). Thus Ψϕ y) 1/κ is a Dulac function in A see Lemma.7 in [6]) and by Theorem. in [6] there is at most one closed orbit of the second kind of system.16) in A + either a limit cycle or a homoclinic orbit of the second kind). Now we assume that there is a limit cycle Γ of the second kind in A +. Since Φϕ y) is strictly positive in A and κ < 0 by.1) we get from Theorem.8 that Γ is asymptotically orbitally stable. Next we denote by S + the separatrix of the saddle 0 0) located in the region A + and having 0 0) as ω-limit set. Now we look for the

19 α-limit set of the separatrix S +. For this purpose we consider the curve N defined by N :={ϕ y) Z : dy dt = sin ϕ cos ϕ cos ϕ) y.18) cos ϕ ) y 3 10y = 0}. It can be shown that the maximum of the branch N of the curve N located in A satisfies y < That means the branch N passes the points π 0) 0 0) and has the same sign as the curve y = 1/0 sin ϕ. Since we can prove that the straight line y = 0.008ϕ π in the interval π ϕ π/ is a curve without contact we can conclude that the α-limit set of the separatrix S + is not located in A. Thus there must be an invariant set in A+ which is not orbitally stable and which is the α-limit set of the separatrix S +. Since the equilibrium π 0) is a stable focus only a homoclinic orbit H of the second kind to the saddle 0 0) or an unstable limit cycle Γ u of the second kind are possible α-limit sets of the separatrix S +. But each of these possibilities leads to the existence of a second closed orbit in A + which contradicts to the fact that at most one closed orbit can exists in A +. Thus our assumption of the existence of a stable limit cycle of the second kind leads to a contradiction. The same proof works also in the region A. From the Theorems. and.3 we get Theorem.4. System.16) has exactly four limit cycles of the second kind. They are hyperbolic and located in the regions A i i = The limit cycles in the region A 0 and A 4 are asymptotically orbitally stable the other ones are unstable. 17 REFERENCES [1] A.A. ANDRONOV A.A. VITT S.E. KHAJKIN The theory of oscillations Nauka Moscow [] N.N. BAUTIN E.A. LEONTOVICH Methods and rules for the qualitative study of dynamical systems on the plane Nauka Moscow [3] L.A. CHERKAS Dulac function for polynomial autonomous systems on a plane Differential Equations ) [4] L.A. CHERKAS A.A. GRIN Dulac functions for dynamical systems on the cylinder in Russian) Bulletin of Grodno State University 007) 3-8. [] L. A. CHERKAS A. A. GRIN Limit cycle function of the second kind for autonomous systems on the cylinder Differential Equations ) [6] L.A. CHERKAS A.A. GRIN K.R. SCHNEIDER A new approach to study limit cycles on a cylinder Dynamics of continuous discrete and impulsive systems Series A: Mathematical Analysis ) [7] A.A. GRIN K.R. SCHNEIDER On some classes of limit cycles of planar dynamical systems Dynamics of Continuous Discrete and Impulsive Systems ser. A ) [8] A.A. GRIN K.R. SCHNEIDER On the construction of a class of generalized Kukles systems having at most one limit cycle Journal of Mathematical Analysis and Applications ) [9] A.A. GRIN K.R. SCHNEIDER Study of the bifurcation of a multiple limit cycle of the second kind by means of a Dulac-Cherkas function: a case study WIAS Preprint 6 016). [10] L. PERKO Differential Equations and Dynamical Systems New York Springer Third Edition 001.