Global algebraic Poincaré–Bendixson annulus for the Rayleigh equation

. We consider the Rayleigh equation ¨ x + λ ( ˙ x 2 /3 − 1 ) ˙ x + x = 0 depending on the real parameter λ and construct a Poincaré–Bendixson annulus A λ in the phase plane containing the unique limit cycle Γ λ of the Rayleigh equation for all λ > 0. The novelty of this annulus consists in the fact that its boundaries are algebraic curves depending on λ . The polynomial defining the interior boundary represents a special Dulac–Cherkas function for the Rayleigh equation which immediately implies that the Rayleigh equation has at most one limit cycle. The outer boundary is the diffeomorphic image of the corresponding boundary for the van der Pol equation. Additionally we present some equations which are linearly topologically equivalent to the Rayleigh equation and provide also for these equations global algebraic Poincaré–Bendixson annuli.


Introduction
The British physicist and Nobel prize winner J. W. Strutt, better known as Lord Rayleigh, published fundamental results to a broad spectrum of physical phenomena.In his monograph "Theory of Sounds" [18] he used the linear differential equation with constant coefficients for the description of acoustic oscillations of a clarinet.The nonlinear modification of this equation Corresponding author.Email: grin@grsu.bywhere λ is a real parameter, is known under the name Rayleigh equation [3,17].Its corresponding system dx dt = −y, which is invariant under the transformation t → −t, y → −y, λ → −λ has been studied by several authors [1,2,8,9,13,15,[21][22][23].
The existence of a limit cycle (isolated closed orbit) of a planar autonomous system is established usually by the construction of an annulus A in the phase plane with the following properties: (i).A contains no equilibrium of the system under consideration.(ii).The boundary of A consists of two simple closed curves (in what follows called ovals) such that any trajectory of the considered system meeting the boundary of A will enter A either for increasing or for decreasing t.An annulus with the properties (i) and (ii) is called a Poincaré-Bendixson annulus since the application of the Poincaré-Bendixson theorem [5,16] to that annulus provides the existence of at least one limit cycle in A. The crucial problem in that approach is the construction of the ovals forming the boundary of A. In numerous publications (see e.g.[5,6,14,16,19,20]) these ovals consist of piecewise smooth curves constructed in a sophisticated way.In this paper we are concerned with the construction of such ovals which are differentiable curves having only a finite number of points where the trajectories of the underlying system touch the ovals.We call such ovals as crossing ovals.Recently, two papers have been published [7,10] in which a procedure for the construction of algebraic crossing ovals for planar polynomial systems is described.For both papers it is characteristic that they need the approximation of at least one orbit by a polynomial in t.In what follows, we present an approach to construct algebraic crossing ovales for the Rayleigh system (1.2) and some of its topologically equivalent systems, which is completely different from that one presented in the cited papers [7,10].
The structure of our paper is as follows: in Section 2 we describe a method for the construction of an algebraic crossing oval for a class of polynomial systems.For this reason we introduce the concept of Dulac-Cherkas functions including one method for their construction.Section 3 is devoted to the construction of a crossing oval by means of a diffeomorphically equivalent system.In Section 4 we derive some linearly diffeomorphically equivalent systems to the Rayleigh system (1.2) and present the corresponding Poincaré-Bendixson annuli.

Construction of an interior boundary for a Poincaré-Bendixson annulus of the Rayleigh system (1.2)
Our approach to construct an interior boundary for a Poincaré-Bendixson annulus for system (1.2) is based on the use of a Dulac-Cherkas function.For this reason we introduce in the next subsection the definition of a Dulac-Cherkas function and compose some of its properties.

Definition and properties of Dulac-Cherkas functions
We consider the planar differential system under the assumption (A) Let G be an open subset of R 2 , let Λ be some open interval, let P, Q ∈ C 1 0 (x,y) λ (G × Λ, R).We denote by X the vector field defined by (2.1).First we recall the definition of a Dulac function.
Definition 2.1.Suppose the assumption (A) to be valid.A function B belonging to the class does not change sign in G and vanishes only on a set N λ ⊂ G of measure zero for λ ∈ Λ.
The class of Dulac functions has been generalized by L. A. Cherkas in 1997 (see [4]).The corresponding generalized Dulac function, which is called Dulac-Cherkas function nowadays, is defined as follows.
Definition 2.2.Suppose the assumption (A) to be valid.A function 2) can be relaxed by assuming that Φ may vanish in G on a set N λ of measure zero, and that no oval of this set is a limit cycle.
For the sequel we introduce the subset W λ of G defined by From the Definition 2.2 we get immediately Lemma 2.5.Suppose the assumption (A) to be valid.Let Ψ be a Dulac-Cherkas function of system (2.1) in G for λ ∈ Λ.Then any oval of W λ having only a finite number of points where (grad Ψ, X) vanishes is a crossing oval for system (2.1) and can be used as a boundary for a Poincaré-Bendixson annulus.
The following theorem is a special case of a more general result established in [11].
Theorem 2.6.Suppose the assumption (A) to be valid.Let G be a simply connected region, let Ψ be a Dulac-Cherkas function of (2.1) in G for λ ∈ Λ such that W λ contains exactly one oval O λ in G.
Then in the case κ < 0 system (2.1) has for λ ∈ Λ at most one limit cycle in G, and if it exists, it surrounds W λ and is hyperbolic.

This theorem implies
Corollary 2.7.Under the assumptions of Theorem 2.6 the oval O λ can be used as interior boundary for a Poincaré-Bendixson annulus of system (2.1) provided it is a crossing oval.
The problem how to construct a Dulac-Cherkas function for the Rayleigh system (1.2) will be treated in the next subsection.We note that the presented procedure can be applied to a more general class of planar polynomial systems.

Construction of Dulac-Cherkas functions for system (1.2)
We consider system (1.2) in R 2 for λ > 0. The corresponding vector field X reads X(x, y, λ) := (−y, x + λy − λy 3 /3). (2.4) We look for a Dulac-Cherkas function in the form where we assume that for all λ > 0 the function Ψ 2 is not identically zero.Using (2.4) and (2.5) we obtain for the function Φ defined in (2.2) the representation where the functions Φ k are defined by the relations ) where the symbol ′ indicates the differentiation with respect to y.One approach to guarantee that Φ is a definite function in R 2 for λ > 0 is to require Φ k to be identically zero for 1 ≤ k ≤ 3 and that Φ 0 is definite.Applying this approach we get from (2.10) the linear differential equation such that it holds Taking into account (2.11) and (2.12) we obtain from (2.9) whose solution reads Taking into account (2.14), (2.13) and (2.12) we get from (2.8) (2.20) Now we have to determine c 0 and c 2 such that Φ 0 (y, λ, −1) is a definite function and that the corresponding Dulac-Cherkas function Ψ has the property that its zero-level set W λ contains an oval surrounding the origin.Setting c 0 = − 8 3 c 2 , where by (2.12) c 0 ̸ = 0 holds, we have which has for λ > 0 the same sign for all y and vanishes only at y = ± √ 2. Thus it holds Lemma 2.8.The polynomial is a Dulac-Cherkas function for system (1.2) in R 2 for λ > 0.

Construction of an interior boundary for a Poincaré-Bendixson annulus of system (1.2)
The set W λ of the Dulac-Cherkas function Ψ in (2.22) is defined by First we note that W 0 is the circle x 2 + y 2 = 8/3.From (2.23) we get further that for all λ > 0 the set W λ is symmetric with respect to the origin and that the intersection of W λ with the straight lines y = ± √ 3 is empty for any λ > 0. For the following we denote by S 2 √ 3 in R 2 the strip symmetric to the x-axis and with thickness 2 √ 3. We obtain from (2.23) the result  In order to prove that the oval I λ is a crossing oval, we note that we have by (2.2) According to (2.21) there exist four points on I λ , where the vector field X touches the oval I λ .Therefore, I λ is a crossing oval and we get from Corollary 2.7 Theorem 2.10.The oval I λ represents for λ > 0 an interior boundary for a Poincaré-Bendixson annulus of system (1.2).

Construction of an outer boundary for a Poincaré-Bendixson annulus of the Rayleigh system (1.2)
For the construction of an outer boundary of a Poincaré-Bendixson annulus for the Rayleigh system (1.2) a similar but more sophisticated procedure could be applied as it has been used for the van der Pol system in our paper [12].In what follows we describe another approach based on the concept of diffeomorphically equivalent systems.In the following subsection we present the definition of topological equivalence of phase portraits and some important consequence.

Definition of topological equivalence and some important consequences
Our basic assumption reads as follows Consider the topological structure of the trajectories of the system in G 1 and the topological structure of the trajectories of the system Definition 3.1.Suppose assumption ( Ã) to be valid.Let Λ 1 be a subinterval of Λ.The systems (3.1) and (3.2) are called topologically equivalent for λ ∈ Λ 1 if for λ ∈ Λ 1 there is a homeomorphism h λ mapping G 1 onto G 2 and which maps the trajectories of system (3.1)onto the trajectories of system (3.2) and there is a strictly increasing homeomorphism g λ mapping R onto itself such that τ = g λ (t).If h λ is a diffeomorphism then the systems are called diffeomorphically equivalent.
The following result is a consequence of the well known fact that the composition of a local diffeomorphism with a diffeomorphism is still a local diffeomorphism.Theorem 3.2.Suppose that the assumption ( Ã) is valid and that the systems (3.1) and (3.2) are diffeomorphically equivalent for λ ∈ Λ 1 .Let O λ be a crossing oval for system (3.1) for λ ∈ Λ 1 .Then the image of O λ under the diffeomorphism d λ is a crossing oval for system (3.2) for λ ∈ Λ 1 .
In order to be able to apply Theorem 3.2 for the construction of an outer boundary for a Poincaré-Bendixson annulus for the Rayleigh system (1.2) we use the following lemma.

Lemma 3.3. The van der Pol system
is for λ > 0 diffeomorphically equivalent to the Rayleigh system (1.2).
Proof.Applying the diffeomorphism d λ mapping R 2 onto itself defined by we get from (3.3) which coincides with the Rayleigh system (1.2).

Construction of an outer boundary for a Poincaré-Bendixson annulus of the Rayleigh system (1.2)
In the paper [12] we have proved the following result Theorem 3.4.For λ > 0, the oval ) is a crossing oval forming an outer boundary of a global algebraic Poincaré-Bendixson annulus for the van der Pol system (3.3).
According to Lemma 3.3, the van der Pol system (3.3) is for λ > 0 diffeomorphically equivalent to the Rayleigh system (1.2),where the corresponding diffeomorphism d λ is defined in (3.4).By Theorem 3.2, the image of the crossing oval V λ for the van der Pol system (3.3)under the diffeomorphism d λ is for λ > 0 a crossing oval O λ of the Rayleigh system (1.2).From (3.4) and (3.6)

Global algebraic Poincaré-Bendixson annuli for systems diffeomorphically equivalent to the Rayleigh system
If we apply for λ > 0 the linear diffeomorphism to the Rayleigh system (1.2) we obtain the system which is diffeomorphically equivalent to system (1.2) for λ > 0. and form a global algebraic Poincaré-Bendixson annulus Āλ containing the unique limit cycle Γλ of system (4.2).
If λ tends to zero we get from (4.3) and (4.4) that both ovals shrink to the origin which reflects the property of system (4.2) that the limit cycle Γλ bifurcates from the origin when λ passes zero (Andronov-Hopf bifurcation).This distinguishes system (4.2) from the Rayleigh system where the limit cycle Γ λ bifurcates from the circle x 2 + y 2 = 2 when λ passes zero.which is a singularly perturbed system in case of small ε.Thus, the unique limit cycle Γε represents a relaxation oscillation for small ε.If we apply the linear diffeomorphism (4.5) to the ovals I λ and O λ we obtain a global algebraic Poincaré-Bendixson annulus Âε for system (4.6).

Lemma 2 . 9 . 2 √ 3 ,
The set W λ defined in (2.23) consists in R 2 for λ > 0 of three different branches: the oval I λ surrounding the origin and located in the strip S the unbounded branch W 1 λ located in the first quadrant in the region y > √ 3 and the symmetric branch W 3 λ in the third quadrant in the region y < − √ 3 .

Figure 2 .
Figure 2.1 shows the branches of W λ for λ = 1.3.In order to prove that the oval I λ is a crossing oval, we note that we have by(2.2)

Figure 2 . 1 :
Figure 2.1: Three branches of the set W λ including the oval I λ for λ = 1.3.

(3. 7 )Theorem 3 . 5 .
It can be verified that the derivative of O λ along system (1.2) is negative on O λ except at four points.Thus we have the result The algebraic oval O λ defined in (3.7) is for λ > 0 an algebraic crossing oval of the Rayleigh system (1.2) forming the outer boundary of a Poincaré-Bendixson annulus.Together with the algebraic oval I λ it determines a global algebraic Poincaré-Bendixson annulus A λ containing the unique limit cycle Γ λ of the Rayleigh system (1.2).