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NON-EXISTENCE AND UNIQUENESS OF LIMIT CYCLES FOR PLANAR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH HOMOGENEOUS NONLINEARITIES

In this paper we study the limit cycles of the planar polynomial di erential systems ẋ = ax− y + Pn(x, y), ẏ = x+ ay +Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n ≥ 2, and a ∈ R. Consider the functions φ(θ) = Pn(cos θ, sin θ) cos θ +Qn(cos θ, sin θ) sin θ, ψ(θ) = Qn(cos θ, sin θ) cos θ − Pn(cos θ, sin θ) sin θ, ω1(θ) = aψ(θ)− φ(θ), ω2(θ) = (n− 1) ( 2aψ(θ)− φ(θ) ) + ψ′(θ). First we prove that these di erential systems have at most 1 limit cycle if there exists a linear combination of ω1 and ω2 with de nite sign. This result improves previous knwon results. Furthermore, if ω1(ν1aψ − ν2φ) ≤ 0 for some ν1, ν2 ≥ 0, we provide necessary and su cient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these di erential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin.

= ax − y + Pn(x, y), y = x + ay + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n ≥ 2, and a ∈ R. Consider the functions φ(θ) = Pn(cos θ, sin θ) cos θ + Qn(cos θ, sin θ) sin θ, ψ(θ) = Qn(cos θ, sin θ) cos θ − Pn(cos θ, sin θ) sin θ, First we prove that these dierential systems have at most 1 limit cycle if there exists a linear combination of ω1 and ω2 with denite sign. This result improves previous knwon results. Furthermore, if ω1(ν1aψ − ν2φ) ≤ 0 for some ν1, ν2 ≥ 0, we provide necessary and sucient conditions for the non-existence, and the existence and uniqueness of the limit cycles of these dierential systems. When one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin.

Introduction and statements of main results
In this work we study the limit cycles of the planar polynomial dierential systems with homogeneous nonlinearities of the form dx dt =ẋ = ax − y + P n (x, y), where P n and Q n are homogeneous polynomials of degree n ≥ 2. The study of the limit cycles of these dierential systems is a particular case of the second part of Hilbert's 16th problem, which is one of the main problems in the qualitative theory of the planar polynomial dierential systems.
Such particularity provides an opportunity to analyze the Hilbert's 16th problem restricted to the limit cycles surrounding the origin of the dierential systems (1) in a relatively simple way.
There are several works studying the limit cycles which bifurcate from the ones of the dierential systems (1) under small perturbations, see for instance [5,24,33,34,38] and the references therein. But here we are interested in the results for the non-bifurcation case, now we summarize the more representatives.
Moreover when such a limit cycle exists, it is hyperbolic and surrounds the origin.
We remark that Theorem 1 generalizes the known results (I) and (III), and provides more information about the result (III) under additional assumptions. Furthermore we will show in Proposition 13 that, the condition µ 1 ω 1 + µ 2 ω 2 ≥ 0 (≤ 0) in Theorem 1, actually leads to the tangency or transversality of the vector eld dened by the dierential system (1) on an algebraic curve. This provides a geometric meaning to this condition. Conversely, if this tangency or transversality is valid, then in some case we can obtain the best upper bound for the number of limit cycles of the system, see the following corollary.
Corollary 2. Suppose ψ(θ) ̸ = 0 and there exists µ ∈ R\{0} such that ψ(θ)r n−1 + µ = 0 is an invariant curve or a transversal section for the vector eld of system (2). Then the dierential system (1) has at most one limit cycle. Moreover when this limit cycle exists it is hyperbolic and surrounds the origin.
We remark that Corollary 2 says that the tangency or transversality of the vector eld of system (1) on the algebraic curves (Q n x − P n y) + µ(x 2 + y 2 ) = 0 with µ ∈ R\{0} and (x, y) ̸ = (0, 0), provides useful information to know the uniqueness of the limit cycles of the dierential system (1). This result is quite similar to the classical result, which says that a quadratic polynomial dierential system has at most 1 limit cycle if it has an invariant straight line, see [17,18,40].
We also have a partial improvement for the known result (II), see the next theorem.
Moreover when such a limit cycle exists, it is hyperbolic and surrounds the origin.
In what follows by choosing dierent values for the parameters µ 1 , µ 2 , ν 1 and ν 2 in Theorems 1 and 3, we recover some of the classical results on the limit cycles of the dierential systems (1), and also some new ones.
Corollary 4. The polynomial dierential system (1) has at most one limit cycle if one of the following conditions holds. Moreover when this limit cycle exists, it is hyperbolic and surrounds the origin. We provide three applications of our results. The rst one shows the uniqueness of limit cycles and compares our results with the results (I)-(IV). The second one studies the interval of the parameter a, in which the dierential system (1) has no no limit cycles, or exactly one limit cycle. The last application concerns with the non-existence of limit cycles. These applications are done in section 3.
There are two powerful tools in the proof of our main results. One of them is the Abel dierential equation where A(t), B(t) and C(t) are C ∞ ([0, κ]) functions with κ > 0, and x ∈ R, because using the transformation introduced by Cherkas [12], which really goes back to Liouville [31], the dierential equation (3) becomes the Abel dierential equation There are many works on the Abel dierential equation because of its importance in the qualitative theory of the dierential equations, see [14, 610, 15, 19, 2123, 27 30, 37, 39] and the references therein. Nevertheless, up to now, the major part of results on the limit cycles of the Abel dierential equations require that some coecient of the Abel dierential equation does not change sign. We want to point out that the proofs of our theorems use a result of [25] on the Abel dierential equations where some coecients can change their signs.
On the other hand we will see that the results on the limit cycles of the Abel dierential equations not always can be used, especially when the function ψ(θ) has zeros.

5
In such case we have to investigate directly the dierential equation (3), which seems more dicult. Usually in order to control the maximum number of limit cycles for such equations, an ecient way is to nd some suitable auxiliary functions, see for example Theorem 5 of [4], or Lemma 2.1 of [25], or Proposition 2.2 of [28].
The second main tool that we shall use in this paper, is a formula on the divergence from [26]. According to this formula, if the auxiliary function for the Abel dierential equation (6) is known when ψ(θ) ̸ = 0, then we can obtain an auxiliary function for the dierential equation (3) when the function ψ(θ) has zeros. In this way the proofs using the dierential equation (3) can be done in a similar way to the proofs using the Abel dierential equation (6).
The organization of this paper is as follows. In section 2 we provide some preliminary results. The proofs of Theorems 1 and 3, and of Corollaries 2 and 4 are given in section 3.

Preliminaries
For proving our theorems and corollaries we need some preliminary results, we state them one by one in the following.
2.1. Some basic denitions and results. Consider the dierential equation where L ∈ C ∞ ([0, κ] × R) and κ is a positive constant. Denote by x(t, x 0 ) the solution of (7) such that x(0, The solution x(t, x 0 ) is a periodic solution of the dierential equation (7), if it is dened is a periodic solution of equation (7). A periodic orbit is a limit cycle if it is isolated in the set of all periodic orbits of the dierential equation (7).
The function H(x 0 ) = x(κ, x 0 ) is the return map of the dierential equation (7). It is well-known that where the prime represents the rst-order derivative (see Lloyd [37] for instance). When , the solution x(t, x 0 ) provides a limit cycle, which is called a hyperbolic limit cycle (resp. a limit cycle with multiplicity n). For more details see for instance [20].
Clearly a limit cycle of equation (7) is hyperbolic if and only if it has multiplicity 1.
Consider the function G : U → R dened by is a periodic solution of the dierential equation (7) in U , then (ii) If G U ≥ 0 (resp. ≤ 0) and there exists a non-empty open set E ⊆ [0, κ] such that G U ∩(E×R) ̸ = 0, then the dierential equation (7) has at most 1 limit cycle in U , which is hyperbolic and unstable (resp. stable).
The proof of Lemma 5 is easy, and it follows in a similar way to the proof of Lemma 2.1 of [25].
Now we state a second result which follows essentially from Lemma 5. We denote by W (·, ·) the Wronskian determinant of two functions. Theorem 6. Suppose there exist two smooth functions λ 1 (t) > λ 2 (t) such that for i = 1, 2 does not change signs, and Then the Abel dierential equation (4) has at most 2 non-zero limit cycles, counted with multiplicities.
It is easy to prove that I S U ≥ 0 (≤ 0), where U is an arbitrary connected component such that I S U ∩(E×R) ̸ = 0. Note that f | U ̸ = 0. By Lemma 5 U contains at most 1 limit cycle of the dierential equation (7). Therefore the number of non-zero limit cycles of (7) is no more than 6 (2 in the curves x = λ 1 (t) and x = λ 2 (t), and 4 coming at most one for each connected components of the set . By virtue of further analysis (including bifurcation method and comparison principle), this upper bound can be reduced to 2 and it is sharp. Thus we obtain Theorem 6, for more details see [25].
There is no doubt that the Abel dierential equation is a powerful tool for studying the limit cycles of the dierential system (1). However not always we can use it. For this reason we also need to apply Lemma 5 directly to the dierential equation (3), but the diculty is still to nd a suitable auxiliary function for applying the Lemma 5 to equation (3). To this end we introduce the following result given in [26].
and let F : V 2 → R be a C 1 function. Assume that Q and P are two vector elds on V 1 and V 2 , respectively, with P • T = DT Q. Then . Hence a straightforward calculation shows that The conclusion of the lemma follows.
For ending this subsection we state two more lemmas. The rst provides information on the stability at innity for the polynomial dierential system (1) when ψ(θ) ̸ = 0 (it is known that the innity is a periodic orbit of the system in this case), the second provides the non-existence of limit cycles under convenient assumptions.
then the periodic orbit at the innity of the polynomial dierential system (1) is stable (resp. unstable).
Lemma 8 is in fact a particular case of Proposition 4 of [24].
then the dierential equation (16) has no limit cycles in region [ , then the dierential equation (16) has no periodic orbits in [ Proof. The lemma can be easily proved by comparing the solutions of the dierential equation (16) with the solutions of the dierential equation dx/dt = p(t)l(x). For more details we refer the readers to the paper [27].
2.2. Limit cycles of a particular Abel dierential equation. In this subsection we consider the Abel dierential equation (4) with One of the particularities of this Abel dierential equation is that a First we apply Lemma 5 and Theorem 6 to obtain some estimates for the number of limit cycles of the Abel dierential equation (4) with S(t, x) given in (17). In order to do that we consider the function Thus a direct calculation shows that Now we dene and we have the next proposition.
Proposition 10. Suppose that S is dened as in (17). Then the Abel dierential equation (4) has at most 1 limit cycle (counted with multiplicity) in V , if one of the following conditions holds: We will divide the proof into three cases.
Observe that equation (21) can be rewritten aṡ When E 1 = ∅ it follows from statement (i) of Lemma 9 that equation (21) has no limit cycles in V , i.e. the Abel equation (4) has no limit cycles in V .
When E 1 ̸ = ∅ we claim that at most one connected component of V (resp. V ) contains the limit cycles of (21) (resp. (4)). In fact this is trivial if Thus, by statement (ii) of Lemma 9, either {(t, y)|y > 1} or {(t, y)|y < 0} contains no limit cycles of (21), which leads to our assertion.
On the other hand, take α = −1, β = −2 and c ≡ 0 in (19). In each connected Therefore when E 1 ̸ = ∅ Lemma 5 implies that each connected component of V has at most 1 limit cycle of the Abel equation (4) counted with multiplicity. Consequently, the number of limit cycles of (4) in V is no more than 1.
It follows from Lemma 9 that there are no (resp. at most one) connected components of V containing limit cycles of the Abel equation (4) when E 1 = ∅ (resp. E 1 ̸ = ∅). Now take α = −2, β = −1 and c = − ln |a 1 | in (19). In each connected components of Thus from Lemma 5 and the above argument the Abel equation (4) has at most 1 limit cycle (counted with multiplicity) in V when E 1 ̸ = ∅.
From assumption λ 1 > λ 2 and λ 1 , λ 2 ̸ = 0. In addition we have that where W (·, ·) represents the Wronskian determinant for two functions. So 4λ 1 λ 2 According to Theorem 6 the Abel equation (4) has at most 2 non-zero limit cycles, counted with multiplicities. Note that x = b 1 /a 1 is one of them and not located in V . Hence the conclusion holds.
(ii) Let E 2 = {t|a 2 ̸ = 0}. When E 2 = ∅ we know that (4) is a Riccati equation and therefore it has at most 1 non-zero limit cycle. In what follows we consider the case that E 2 ̸ = ∅.
Take α = β = −1 and c ≡ 0 in (19). Then in each connected components of V we have by assumption that According to Lemma 5 the Abel equation (4) has at most 1 limit cycle in each connected components of V , counted with multiplicity.
When a 1 b 1 > 0 or a 1 has zeros the conclusion clearly holds because V is connected in these cases.
For the case a 1 b 1 < 0 we again use (21) and rewrite the equation aṡ Following a way similar to the argument in case (i.a), we can verify using Lemma 9 that (21) has no limit cycles in {(t, y)|y > 1} or in {(t, y)|y < 0}. That is all the limit cycles of the Abel equation (4) can only appear in one connected component of V , and therefore the number is at most 1.
(iii) If a 1 a 2 ≡ 0 then (4) is reduced to a Riccati equation and therefore the number of non-zero limit cycle is no more than 1. If a 1 a 2 ̸ ≡ 0 and a 1 ̸ = 0, then a 2 ≥ 0 (≤ 0). The case goes back to statement (ii). The remaining case is a 1 a 2 ̸ ≡ 0 with a 1 having zeros. We know that V is connected. Take α = −2, β = 0 and c ≡ 0 in (19). In V we have by assumption that Lemma 5 implies that the Abel equation (4) has at most 1 limit cycle V , counted with multiplicity. Now we consider the non-existence of limit cycles for the Abel equation (4). Suppose that S is dened as in (17). Rewrite the equation as d dt Then for a non-zero periodic orbit x = x(t) of the Abel equation (4) we have In particular note that b 1 (a 1 x − b 1 )/x ̸ = 0 in V . Thus it is easy to obtain the following criterion.
Proposition 11. Suppose that S is dened as in (17). Then the Abel equation (4) has is a periodic solution of the Abel equation (4) in V . Then according to (23) and the assumption we have χ(t, x(t)) ≡ 0. Note that the function χ is linear in the variable x and the region V is open. We get that a 2 ≡ 0 and therefore b 2 ≡ 0. Thus from (22) all the orbits of (4) in V are periodic. Then the proposition is proved.
Proof. (i.a) By assumption we have aψ − φ = ω 1 ≥ 0 (≤ 0). According to (27) and statement (ii) of Proposition 10, equation (6) has at most 1 limit cycle in V 2 , counted with multiplicity. That is system (1) has at most 1 limit cycle surrounding the origin, and this limit cycle is hyperbolic if it exists.
(i.b) Without loss of generality suppose that µ 2 = 1. Take η = µ 1 /(n − 1) + 1. We get from (27) and by the assumptions that a 1 = ψ ̸ = 0 and Therefore the number of limit cycles of (6) in V 2 is at most 1 (counted with multiplicity) from statement (i) of Proposition 10. The conclusion immediately follows.
In order to prove statement (ii) rst we observe that Second we dene the function χ(θ, ρ) = (aψ − φ)ρ − a. (ii.a) By assumption either aψ − φ = ω 1 ≡ 0 or φ ≡ 0. Thus either χ ≡ −a or χ = a(ψρ − 1). In any case we get that χ does not change sign in each connected component of V 2 . Consequently equation (6) has no limit cycles in V 2 by Proposition 11. System (1) has no limit cycles surrounding the origin.
(ii.b.2) When ψ < 0 we have that (i) leads to aψ ̸ = 0. It is known by (29) that ω 1 = aψ − φ ≥ 0 (≤ 0). Hence statement (i.a) shows that system (1) has at most 1 limit cycle surrounding the origin, and such limit cycle is hyperbolic when it exists. Moreover, when ω 1 ̸ = 0 the equation 1 + ψr n−1 = a + φr n−1 = 0 has no solutions and therefore the origin is the unique singularity of system (1). Recall that φ ̸ ≡ 0, and aφ/ψ ≥ 0 from (29). According to Lemma 8 and a direct calculation the stabilities at the origin and at innity of the system are the same. So such a limit cycle exactly exists. (29) implies that either (aψ−φ)ψ ≥ 0 and a < 0, or (aψ−φ)ψ ≤ 0 and a > 0. Together with (27) and statement (iii) of Proposition 10, equation (6) has at most 1 limit cycle in V 2 , counted with multiplicity, i.e. system (1) has at most 1 limit cycle surrounding the origin, and this limit cycle is hyperbolic if it exists.
For the case that ν 1 < ν 2 we again consider the sign of the function χ on V 2 . Recall that χ is linear in the variable ρ and χ(θ, 0) = −a. If we denote by E + = {θ|ψ > 0}, Consequently sgn(χ)| V 2 ≡ −sgn(a). Applying Proposition 11 we get that equation (6) has no limit cycles in V 2 , i.e. no limit cycles of system (1) surround the origin. First we note that the condition µ 2 1 + µ 2 2 ̸ = 0 and µ 1 ω 1 + µ 2 ω 2 ≥ 0 (≤ 0) for system (1), actually leads to some tangency or transversality of the system on some curves. This observation is originated from the geometric condition of Theorem 6 and its application during the proof of statement (i) of Proposition 10. More precisely we have the following proposition.
In order to prove the second part we observe that γ is a curve and (D v γ)| γ=0 ≥ 0 (≤ 0) from assumption. Thus an orbit of system (1) which intersects γ = 0, either is contained in a connected component of γ = 0, or crosses γ = 0 and then stays in one connected component of γ ̸ = 0. Note that an orbit crossing γ = 0 can not be periodic, so the conclusion follows.

Remark 14. Proposition 13 provides a geometrical meaning to conditions (I) and (III).
More precisely the condition (I), i.e. the denite sign of µ 1 ω 1 + µ 2 ω 2 with µ 1 = 1 and µ 2 = 0 implies that: The orbits of system (1) which intersect the curve ψr n−1 + 1 = 0, all go across the curve from the same side to the other side. This property was also mentioned by the authors of [16]. Nevertheless, for the condition (III) it seems that such similar property was never mentioned in the previous works. Actually condition (III) (the denite sign of ω 2 ) means that: The orbits of system (1) which intersect the curve 2ψr n−1 + 1 = 0, all go across the curve from the same side to the other side. We would like to emphasize that, both of the two geometric properties are equivalent to the original conditions when ψ < 0.  .
We begin to show the idea of nding a suitable auxiliary function for the dierential equation (3).
So an essential reason in order that statement (i.b) of Proposition 12 holds is Here F is dened as in (30). On the other hand we can verify that T is a dieomorphism where η 1 = κ 1 − κ 2 and η 2 = −κ 2 , then (31) and Lemma 7 imply ∂R ∂r Thus when ψ ̸ = 0 the number of limit cycles of system (1) can also be estimated by applying Lemma 5 to equation (3) and using F .
When the assumption ψ ̸ = 0 in statement (i.b) of Proposition 12 is changed by ψ = 0 at some points, the inequality (31) is unknown because the functions λ i 's of Theorem 6 cannot be found. Nevertheless the divergence formula of Lemma 7 formally provides an auxiliary function F given in (32), which allows to verify the inequality (33) directly, and then to obtain the conclusion using Lemma 5. More precisely we have the next proposition.
Proof. Due to the previous arguments we only need to consider the periodic orbits of the dierential equation (3) which are located in the region V 1 .
When ψ ≡ 0 equation (3) is reduced to a Bernoulli equation dr/dθ = ar + φr n . Our claim can be directly checked from the expression of the general solution of this equation.
When ψ ̸ ≡ 0 take η 1 = µ 1 /(n − 1) + 2µ 2 and η 2 = µ 1 /(n − 1) + µ 2 . Then by Proposition 13 each periodic orbit either does not intersect the curve η 1 ψr n−1 + η 2 = 0, or it is a connected component of this curve. Furthermore, since ψ has zeros, the curve has no compact connected component. Hence such periodic orbits are all contained in one connected component of the region Now let F be the function determined by (32). For (θ, r) ∈ U we get ∂R ∂r Hence, if the set E = {θ|µ 1 ω 1 + µ 2 ω 2 ̸ = 0} is not empty, then by Lemma 5 the number of limit cycles of the dierential equation (3) in U is at most 1. Moreover, when such a limit cycle exists, it is hyperbolic.
2.4. The distribution of limit cycles of system (1). Until now we have studied the limit cycles surrounding the origin of system (1). Now we consider the existence of limit cycles which do not surround the origin. Proposition 16. Any arbitrary periodic orbit of system (1) must surround the origin if one of the following conditions holds.
It follows from the assumptions that η 2 1 + η 2 2 ̸ = 0. Thus g is a smooth function dened in the region U = Observe that η 1 ψr n−1 + η 2 = 0 is either a simple closed curve surrounding the origin, or a non-closed curve, or does not exist. By Proposition 13 each periodic orbit which does not surround the origin is located in a connected component of U . Hence if µ 1 ω 1 +µ 2 ω 2 ̸ ≡ 0 (i.e. µ 1 ω 1 + µ 2 ω 2 has at most nitely many zeros), then our conclusion holds by Bendixson-Dulac Theorem, see for instance Theorem 7.12 of [20]. If µ 1 ω 1 + µ 2 ω 2 ≡ 0, then div (gv) ≡ 0. There exists a rst integral which only is not dened in the curve η 1 ψr n−1 + η 2 = 0, in other words it is dened in U . Thus the region U contains no limit cycles which do not surround the origin of the system, and the proposition follows in this case.
For the case that ω 1 ̸ ≡ 0 we can check that statements (i), (ii) and (29) in the proof of Proposition 12 still hold by assumptions. Thus ω 1 ψ ≥ 0 (≤ 0). Now denote by γ = ψr n−1 + 1. In each connected component of the curve γ = 0, we have ψ < 0 and therefore ω 1 | γ=0 does not change sign and has nitely many zeros. By Proposition 13 we get that Hence any periodic orbit of system (2) does not intersect the curve γ = 0. Note that a singularity of the system is contained in γ = 0 if it is not the origin. Consequently all periodic orbits of system (2) must surround the origin (this observation is also stated in [16]). Therefore statement (ii) holds.

Proof of the main results
There are two goals in this section. The rst is to provide the proofs of Theorems 1 and 3, and of Corollaries 2 and 4. The second is to give some examples for illustrating the application of our results.
Proof of Theorem 1. The distribution of the limit cycles of system (1) is obtained directly by statement (i) of Proposition 16. Together with statement (i) of Proposition 12 and Proposition 15, the system has at most 1 limit cycle, and such a limit cycle is hyperbolic when it exists. Now we focus on the exact number of the limit cycles of system (1) when ψ > 0.
First, by (2) the origin is the unique singularity of the system. Moreover, taking into account the sign of a and Lemma 8, the stabilities at the origin and at innity of the system are opposite (resp. the same) if a ∫ 2π 0 (φ/ψ)dθ > 0 (resp. < 0). In these cases it is easy to verify the exact number of the limit cycles of the system. Hence statement (a) and statement (b) hold except the case a ∫ 2π 0 (φ/ψ)dθ = 0. Second, from the denition of the functions ω 1 , ω 2 , φ and ψ we have that For the case that a ∫ 2π 0 (φ/ψ)dθ = 0, the proof is divided into two cases.
Finally we provide three examples, the rst one shows the existence and non-existence of limit cycles, and the second (resp. third) one shows the reachability of the upper bound of limit cycles when ψ < 0 (resp. ψ has zeros).
Similar to the previous example, the system has at most 1 limit cycle. Also one can check that x 2 + y 2 = 1 is a limit cycle of the system. This veries the reachability of the upper bound when ψ < 0.
Thus −4kω 1 + ω 2 = 2k + cos(2θ) > 0 and the number of limit cycle of the system is at most 1. Observe that this system has a limit cycle x 2 + y 2 = 1. The reachability of the upper bound is true when ψ has zeros.
Proof of Theorem 3. This theorem follows directly from statement (ii) of Proposition 12 and statement (ii) of Proposition 16.
Now we show a concrete example with the existence and uniqueness of limit cycles.
Using Theorems 1 and 3 we shall prove Corollaries 2 and 4.
Proof of Corollary 4. The conclusion can be easily veried using Theorems 1 and 3, and a direct computation.
Obviously all of these three equalities have indenite signs, which violate the conditions of the results (I)-(IV). That is condition (iii) of Corollary 4 is indeed a new result.
In the second example we shall study the interval of the parameter a in which the system has no limit cycles, or exactly one limit cycle.
Finally we give the third example for showing the non-existence of limit cycles in a dierential system.
By statement (iv) of Theorem 3 there is no limit cycle of system (44).