The Limit Cycles of the Higgins–Selkov Systems

In this paper, we investigate the problem of limit cycles for general Higgins–Selkov systems with degree n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n+1$$\end{document}. In particular, we first prove the uniqueness of limit cycles for a general Liénard system, which allows for discontinuity. Then, by changing the Higgins–Selkov systems into Liénard systems, theorems and some techniques for Liénard systems can be applied. After, we prove the nonexistence of limit cycles if the bifurcation parameter is outside an open interval. Finally, we complete the analysis of limit cycles for the Higgins–Selkov systems showing its uniqueness.


Introduction and Main Results
In the qualitative theory of planar polynomial differential systems, it is well known how difficult is to study the famous Hilbert's 16th problem, see Ilyashenko (2002) (2003) and Zhang et al. (1992). Up to now there are seldom works having solved the problem of exact number of limit cycles for polynomial differential systems.
The most important physiological function of carbohydrates is to provide energy for organisms' life activities. Glucose catabolism is the main way for organisms to obtain energy. There are three main pathways for the oxidative decomposition of glucose in organisms. Among them, the anaerobic oxidation of glucose is called glycolysis. We consider the following polynomial differential system of arbitrary degreė x = 1 − x y n , y = ay(−1 + x y n−1 ) (1) which was proposed first by Higgins (1964) and modified further by Selkov (1968) for studying the biological nonlinear glycolytic oscillations, and was called the Higgins-Selkov system. Here n is a positive integer and a is a real parameter. Artés et al. (2018) characterized the global dynamics described in the Poincaré disc for system (1) as n = 2 and a ∈ R\(1, 3). Moreover, there are two conjectures stated in Artés et al. (2018) on the the number of limit cycles of systems (1) when a ∈ (1, 3). After, Chen and Tang (2019) proved these conjectures, which complete the global phase portraits of system (1) when n = 2.
Recently, Brechmann and Rendall (2018) researched the uniqueness of limit cycles for system (1) and additionally proved that no limit cycles exist when a ∈ (0, 1/(n − 1)). Llibre and Mousavi (2021) classified the phase portraits of system (1) for n = 3, 4, 5, 6 in the Poincaré disc for all the values of the parameter a and determined in function of the parameter a the regions of the phase space with biological meaning.
The aim of this paper is to give a clearer study and answer for the existence and the exact number of limit cycles of system (1). We have the following main results.
Remark that the bifurcation diagrams of the limit cycles for system (1) are similar to those in Fig. 1 for n = 3, 5 and in Fig. 2 for n = 4, 6 of Llibre and Mousavi (2021), respectively.
An outline of this paper is as follows: A theorem on the uniqueness of limit cycles for general Liénard systems is presented in Sect. 2, which we need in our study of the limit cycles of the Higgins-Selkov system. In Sect. 3, we obtain the existence and the exact number of limit cycles of the Higgins-Selkov system and then prove our main theorem.

Preliminaries
In order to study the number of limit cycles for system (1), we need the following preliminary results. We first recall the uniqueness theorem of Zhang (1958) or in Zhang (1986) on the number of limit cycles of the following generalized Liénard systemsẋ (2) LetĜ Theorem 2 Consider the generalized Liénard system (2) for x ∈ (−∞, +∞), when φ(y),F(x) andĝ(x) satisfy the following conditions: Then system (2) has at most one limit cycle. Moreover the limit cycle is stable when it exists.
In fact, we can find many differential systems of the form (2), but many of them do not satisfy the conditions of Theorem 2. Thus we propose the following three questions: (a) WhenĜ(−∞) =Ĝ(+∞) = +∞ and the other conditions of Theorem 2 hold, does the conclusion of Theorem 2 still hold? (b) When eitherf (x) orĝ(x) has a discontinuity point x 0 of the second kind (i.e., lim x→x 0 +ĝ (x) or lim x→x 0 −ĝ (x) does not exist) and the other conditions of Theorem 2 hold, does the conclusion of Theorem 2 still hold? (c) Whenĝ(x) has a discontinuity point at x = 0 of the first kind (i.e., lim x→0+ĝ (x) = lim x→0−ĝ (x)) and the other conditions of Theorem 2 hold, does the conclusion of Theorem 2 still hold?
For these reasons, we give the following theorem without the aforementioned conditions. Theorem 3 Consider system (2) in the interval (α, β), where α and β eventually can be −∞ and +∞, respectively. Assume that φ(y),F(x) andĝ(x) satisfy the following conditions: Lipschitz in any finite interval and g 0 (0) = 0.
Then system (2) has at most one limit cycle in (α, β). Moreover the limit cycle is stable when it exists.
Proof Since the vector field of system (2) is Lipschitz for c = 0, its solutions exist and are unique except at x = 0. Since the vector field of system (2) is discontinuous at the line := {(x, y) : x = 0} for c > 0, we need to study the dynamics on and we will adapt the Filippov method, see di Bernardo et al. (2008) and Kuznetsov et al. (2003). Let where ·, · denotes the inner product. As defined in di Bernardo et al. (2008) and Kuznetsov et al. (2003), the crossing set is The sliding set s is the complement to c , which is given by Therefore except at the origin, all orbits crossing any point are unique. In other words, all periodic orbits are crossing. Assume that γ is a periodic orbit of system (2). Then we have that γ is hyperbolic if γ div(−φ(y) −F(x),ĝ(x))dt = 0, see, for instance, Theorem 1.23 of Dumortier et al. (2006). Moreover, γ is stable (resp. unstable) if γ div(−φ(y) −F(x),ĝ(x))dt < 0 (resp. > 0).
In order to prove the uniqueness of limit cycles of system (2), assume that system (2) has at least two limit cycles, where γ 1 , γ 2 are the innermost limit cycles and γ 1 lies in the bounded region surrounded by γ 2 .
Moreover, we claim that the equationf (x) = 0 has at most one positive root and one negative root, where a connect set of roots is viewed as one root. Otherwise, assume thatf (x) has two positive zeros x 1 and x 2 such that 0 < x 1 < x 2 . Then, there exists a real , which contradicts the nondecreasing off (x)/ĝ(x). Thus, the claim is proved.
We claim that any periodic orbit must surround the point (x 4 , 0). So no periodic orbits exist if x 4 does not exist. Let It is to note thatĝ(x)F(x) < 0 for all x ∈ (α, x 4 ). Assume that system (2) exhibits a periodic orbit γ , which lies in the strip x ∈ (α, x 4 ). Then, we can find that Thus, the claim is proved. Now we will prove that Consider the two limit cycles Fig. 1. Notice that the limit cycle γ i intersects the graphic of the function y = φ −1 (−F(x)) at the points C i and I i for i = 1, 2, respectively. Since we only need to prove which is equivalent to (5), where for any constant b ∈ R. It is clear that However, the origin is a source and the periodic orbit γ 1 is internally stable because Assume that the line x = x J 1 intersects with the graphic of the function y = φ −1 (−F(x)) at the points B i and D i for i = 1, 2, respectively. Notice that Let y = y 1 (x) and y = y 2 (x) be the orbit segments A 1 B 1 and A 2 B 2 , respectively. Since y 1 < y 2 and the function φ(x) is increasing, we have φ(y 1 ) < φ(y 2 ). Then, we have It is similar to prove that where P 2 , Q 2 ∈ γ 2 and x P 2 = x Q 2 = x I 1 .
Let x = x 1 (y) and x = x 2 (y) be the orbit segments D 1 C 1 B 1 and D 2 C 2 B 2 , respectively. Then, we have where M 2 , N 2 ∈ γ 2 , y M 2 = y D 1 and y N 2 = y B 1 . Since f 1 (x) ≥ 0 for all x < x I 1 , we have Therefore, (6) holds from (9)-(12). Notice that the origin is a source and the periodic orbit γ 1 is internally stable. Thus, γ 1f (x)dt ≥ 0. It follows from (5) (2006)], it is impossible for the existence of two consecutive stable limit cycles. Therefore, system (2) has at most two limit cycles. Moreover, γ 1 is semi-stable and γ 2 is stable if they exist.
In order to induce a contradiction for the case that γ 1 is semi-stable, we construct an auxiliary vector field for small | |. We can check that the vector field (−φ(y) −F(x),ĝ(x)) is rotated with respect to the parameter ; see Zhang et al. (1992, Chapter 4.3) or Perko (1975). Consider the following systemẋ System (13) is exactly system (2) if = 0. Moreover, we can check that system (13) still satisfies all assumptions of Theorem 3. In other words, system (13) has at most two limit cycles. Further, we can find that γ 1 will split into at least two limit cycles for < 0 by Zhang et al. (1992, Theorem 3.4 of Chapter 4). Then, system (13) can have three limit cycles, a contradiction with the previous result. Therefore, we have proven that system (2) has at most one periodic orbit in the case Case (II) Second, we consider the case that there must be x 3 < 0 iff (x 3 ) > 0. Since the proof is similar to the Case (I), we omit it.
Case (III) We consider the case that x 3 can be negative or positive iff (x 3 ) > 0. We claim that the equationF(x) = 0 has either one nonzero root or two nonzero roots. Notice that the equationF(x) = 0 cannot have two positive roots or two negative roots. Otherwise if there exist two different points x 41 , x 42 ∈ (0, β) or ∈ (α, 0) such that which contradicts the nondecreasing of the functionf (x)/ĝ(x). On the other hand, if the equationF(x) = 0 has not nonzero roots, we get dE/dt ≥ 0 for x ∈ (α, β) from (4), implying that no periodic orbits exist. If the equationF(x) = 0 has a unique nonzero root, we consider that the nonzero root is x + ∈ (0, β) for simplicity. If system (2) has a periodic orbit, it must surround (x + , 0). Otherwise, this is a contradiction with the fact that dE/dt ≥ 0. When equationF(x) = 0 has one positive root x + ∈ (0, β) and a negative root x − ∈ (α, 0), if system (2) has a periodic orbit, it must surround at least one of the points (x + , 0) and (x − , 0). Otherwise again we have a contradiction with the fact that dE/dt ≥ 0. Without loss of generality, we can assume that any limit cycle surrounds (x + , 0).
Assume that system (2) has three limit cycles γ 1 , γ 2 , γ 3 as the ones shown in Fig. 2, where γ 1 is the innermost one, γ 3 surrounds γ 1 and γ 2 , the points the periodic orbit γ i intersects the graphic of the function y = φ −1 (−F(x)) at the points C i and J i for i = 1, 2, 3, respectively. Notice that In a similar way to the proof of Case (I), we shall obtain that system (2) has at most one periodic orbit in the strip x ∈ (x J 4 , β). Moreover, the periodic orbit is stable if it exists. Now we shall prove that Notice that the function y = φ(x) has the same properties as in Case (I) when x > 0, as it is shown in Figs. 1 and 2. Thus, we can obtain that To prove the first inequality of (14), it suffices to prove inequality (6). Using the auxiliary function f 1 (x) in (7) for the Case (I) again, we can prove that From (15) and (16), we get Moreover, we can calculate that doing a similar calculation as in Case (I) for x > 0. Thus, from (15) and (18), we get the second inequality of (14), i.e., It follows from (17) and (19) that (14) holds. Since the origin is a source, we have from inequality (14). However, it is impossible to have two consecutive stable limit cycles. Therefore, system (2) cannot have three periodic limit cycles and there are at most two limit cycles.
We divide the rest of the proof in two subcases. First, we consider the subcase that γ 1 only surrounds one of the points (x + , 0) and (x − , 0). In Case (I), we have proved that for this kind of periodic orbits, as γ 1 , at most one can exist, and it is stable. Thus, its consecutive periodic orbit γ 2 is internally unstable and then γ 2f (x)dt ≤ 0. Moreover inequality (17) holds and γ 2 is stable, indicating a contradiction. Therefore, the periodic orbit γ 2 does not exist and system (2) has exact one periodic orbit γ 1 if such periodic orbit exists. Now we consider the subcase that system (2) has no such kind of periodic orbits like γ 1 . We assume that system (2) has two periodic orbits γ 2 and γ 3 , which surround both points (x + , 0) and (x − , 0). Since the origin is a source, we have by inequality (19). Therefore, γ 2 is semi-stable and γ 3 is stable. Using the auxiliary vector field (13) again, we can get that system (13) still satisfies all conditions of this theorem and has at most two limit cycles. However, by the rotated properties of system (13), the semi-stable γ 2 will split into at least two limit cycles for = 0 by Zhang et al. (1992, Theorem 3.4 of Chapter 4). Then, system (13) can have three limit cycles, a contradiction. Thus, we have proven that system (2) has at most one periodic orbit in the case (III) and the proof is completed.
Notice that the vector field is Lipschitz if c = 0 in Theorem 3. Thus, the results of Theorem 3 also hold when system (2) is Lipschitz or further smooth. The modified Liénard system (21) of the Higgins-Selkov (1) is Lipschitz except at the line x = 1, which is a discontinuity point of the second kind for the functions in the system. So we need to apply Theorem 3 for showing the uniqueness of the limit cycles.

Proof of Theorem 1
Notice that system (1) cannot have periodic orbits when a ≤ 0, because the unique equilibrium (1, 1) is a saddle as a < 0 orẏ ≡ 0 as a = 0. Thus, in the following we only consider the case a > 0. Moreover, the periodic orbits of system (1) must lie in the region since the x-axis is invariant andẋ| x=0 = 1.
In order to simplify the computations and the analysis we do the following change of coordinates changing system (1) intoẋ Obviously the periodic orbit of system (20) only exists in the region x 1 > 0, becausė y 1 = 1 > 0 andẋ 1 = 0 on the line x 1 = 0. Moreover, system (20) can be changed into the following Liénard systemẋ where doing the transformation (x 1 , y 1 , t 1 ) → (x + 1, ay + (a + 1), t/(x + 1) n ). Here we only need to consider x > −1 for the problem of limit cycles of system (21), because system (21) is equivalent to system (1) as a > 0 and x > −1. Note that x = −1 is a discontinuous line for system (21). From Brechmann and Rendall (2018) system (21) has no periodic orbits when a ∈ (0, 1/(n − 1)). In the following, we prove that system (21) may have periodic orbits only if a > 1/(n − 1). Here we cannot use the methods of Chen and Tang (2019), because it is difficult to decide when the equations have solutions or not for an arbitrary integer n. We need to use a new method and technique.
Proposition 5 For a > 0, the amplitude of the stable or unstable limit cycle of system (21) surrounding the origin varies monotonically with respect to the parameter a.
Proof Notice that we can change system (21) into the following equivalent forṁ by the transformation of coordinates (x, y, t) → x, y/ √ a, √ at , where From the calculation in Llibre and Mousavi (2021), we have the value of the following determinant for a 2 < a 1 , a 2 , a 1 ∈ (0, +∞) and x > −1.
Thus, the vector field of system (22) is a generalized rotated vector field [see Zhang et al. (1992, Chapter 4.3) or Perko (1975)] with respect to the parameter a if x > −1. Moreover, from Zhang et al. (1992, Theorem 3.5, Chapter 4), the amplitude of the stable or unstable limit cycle of system (22) surrounding the origin varies monotonically with respect to the positive parameter a.
Proof By Artés et al. (2018) and Chen and Tang (2019), system (21) has no periodic orbits for a ≤ 1 when n = 2. In the rest of this proof, we only consider n ≥ 3. We only need to consider system (21) and its limit cycles in the region x > −1. Assume that for n ≥ 3 and −1 < It follows from (23) that Furthermore, it follows from (25) and (26) that Moreover, we calculate from (27) and (28) that Substituting (29) into (24), we have Let Then, we have that where Now we consider the case a = 1/(n − 1). From (31) and (32), we get Then, we claim that dH dx (x, 1/(n − 1)) < 0.
In other words, Eq. (30) has no solutions for x 2 > 0, and then, Eq. (23) have no solutions {x 1 , x 2 } such that −1 < x 1 < 0 < x 2 if n ≥ 3 and a = 1/(n − 1). Thus, from continuity we have F( Moreover, we have that F(0) = 0 and xg(x) > 0. Therefore, by Proposition 2.1 of Chen and Chen (2015), system (21) has no periodic orbits for a = 1/(n − 1). Now consider the case a < 1/(n − 1). When a ≤ 0, either the unique equilibrium (1, 1) of system (1) is a saddle or the system has an invariant line through equilibrium (1, 1), which implies nonexistence of periodic orbits. The vector field of equivalent system (22) of system (21) is a generalized rotated vector field with respect to a for x > −1 and a > 0 by the proof of Proposition 5. Moreover, the amplitude of the stable or unstable limit cycle surrounding the origin of (21) varies monotonically with respect to a. Assume that system (21) exhibits limit cycles for a = a 0 ∈ (0, 1/(n − 1)), where γ is the innermost limit cycle. Since the origin of (21) is stable, then γ is internally unstable. Note that the amplitude of an unstable limit cycle decreases as a increases by Zhang et al. (1992, Theorem 3.5, Chapter 4). When a increases from a = a 0 to a = 1/(n − 1), the origin keeps stability. Therefore, γ does not disappear for a = 1/(n − 1). This is a contradiction to our above analysis as a = 1/(n − 1), and the proof is completed.
When a = a n := 2 n − 1 2 n − 2 , we give the following lemma for the region where periodic orbits exist. Obviously, a n > 1 for n ≥ 3.
Proposition 7 When a = a n for n ≥ 3, periodic orbits of system (21) only exist in the strip x ∈ (−1, 1.6).
Proof Note that any periodic orbit of system (1) must lie in the first quadrant and the y-axis of system (1) is changed into the line y = x − 1 of system (21). Therefore, the periodic orbits of system (21) cannot intersect the line y = x − 1. Assume that is a periodic orbit of system (21) and intersects with the curve y = F(x) at the point (x * , F(x * )) in the right half-plane. Then, x ≤ x * as (x, y) ∈ . If x * ≤ 1, we have that lies in the strip x ∈ (−1, 1] and this lemma is proven. In the following, we consider the case x * > 1. Let y =ỹ(x) < F(x) denote the orbit segment of as 0 ≤ x ≤ x * . For x ≥ 1 and a = a n , we have thatỹ(x) > x − 1 and which impliesỹ where ϕ 1 (x) = x − 1 (x+1) n and ϕ 2 (x) = x − 1 (x+1) n−1 . Notice that for x ≥ 1 we get the maximum value of the positive function ϕ 1 (x) − ϕ 2 (x) at x = 1, implying that the function ϕ 1 (x)/ϕ 2 (x) has its maximum value a n also at x = 1 and the inequality (33) is obtained.
Proposition 8 System (21) has no periodic orbits when a ≥ a n for n ≥ 3.
When a ∈ (1/(n − 1), a n ), we will study the existence and uniqueness of limit cycle in the following proposition.
Proposition 9 For n ≥ 3, there exists a constant a * ∈ (1/(n − 1), a n ) such that system (21) has a unique limit cycle when a ∈ (1/(n − 1), a * ) and no periodic orbits when a ∈ (a * , +∞). Moreover, the limit cycle is stable and hyperbolic, and its amplitude increases with a.
Proof By Zhang et al. (1992, Theorem 3.5, Chapter 4) and Proposition 5, the amplitude of the stable limit cycle surrounding the origin of (21) is monotonous with respect to a as x > −1 and a > 0. From Llibre and Mousavi (2021), the Hopf bifurcation occurs and a stable limit cycle appears when a varies from 1/(n − 1) to 1/(n − 1) + , where > 0 is small. The amplitude of the stable limit cycle is sufficiently small for small enough > 0. Thus, the amplitude of the stable limit cycle increases as a increases.
On the other hand, system (21) has no periodic orbits when a ∈ (−∞, 1/(n − 1)] ∪ [a n , +∞) by Propositions 6 and 8, and has a unique finite equilibrium at the origin. Therefore, there exists a * ∈ (1/(n − 1), a n ) such that the amplitude of the stable limit cycle approaches infinity when a = a * − for sufficiently small > 0 by the continuity of the vector field and the monotonous properties of amplitude of the stable limit cycle in parameter a.
From Propositions 5-9, we can obtain Theorem 1. At last, some numerical examples are provided as follows to verify our theoretical results for n = 3.
Consider parameters in the supercritical Hopf bifurcation curve of system (1) for n = 3, i.e., a = 1/(n − 1) = 0.5. Then, the unique equilibrium (1, 1) is a stable weak focus, as shown in Fig. 4a. When a = 0.55, equilibrium (1, 1) is an unstable hyperbolic focus and the Hopf bifurcation generates a stable hyperbolic limit cycle, as shown in Fig. 4b. When a = 0.6, equilibrium (1, 1) is still an unstable hyperbolic focus and the stable hyperbolic limit cycle disappears at infinity, as shown in Fig. 4c.