By considering the upper bound of the amplitude of limit cycles on

Teljes szövegt

(1)Electronic Journal of Qualitative Theory of Differential Equations 2017, No. 34, 1–14; doi: 10.14232/ejqtde.2017.1.34. http://www.math.u-szeged.hu/ejqtde/. An upper bound for the amplitude of limit cycles of Liénard-type differential systems Fangfang Jiang1 , Zhicheng Ji2 and Yan Wang B 2 1 School 2 School. of Science, Jiangnan University, Wuxi, 214122, China of IoT Engineering, Jiangnan University, Wuxi, 214122, China. Received 21 October 2016, appeared 9 May 2017 Communicated by Gabriele Villari Abstract. In this paper, we investigate the position problem of limit cycles for a class of Liénard-type differential systems. By considering the upper bound of the amplitude of limit cycles on {( x, y) ∈ R2 : x < 0} and {( x, y) ∈ R2 : x > 0} respectively, we provide a criterion concerning an explicit upper bound for the amplitude of the unique limit cycle of the Liénard-type system on the plane. Here the amplitude of a limit cycle on {( x, y) ∈ R2 : x < 0} (resp. {( x, y) ∈ R2 : x > 0}) is defined as the minimum (resp. maximum) value of the x-coordinate on such a limit cycle. Finally, we give two examples including an application to predator–prey system model to illustrate the obtained theoretical result, and Matlab simulations are presented to show the agreement between our theoretical result with the simulation analysis. Keywords: Liénard-type system, limit cycle, amplitude, upper bound. 2010 Mathematics Subject Classification: 34A26, 34A34, 34C07.. 1. Introduction. Liénard differential equation d2 x dx + f ( x ) + g( x ) = 0 (1.1) 2 dt dt is one of the more studied differential systems, which is originated from physics and proposed by French physicist Liénard in 1928. By the Liénard transformation, the second order differential equation (1.1) can be transformed into the following equivalent two dimensional Liénard system dx dy = y − F ( x ), = − g ( x ), (1.2) dt dt where F ( x ) =. Rx 0. f (s)ds. The more general differential system is of the form dx = h ( y ) − F ( x ), dt. B Corresponding. dy = − g ( x ), dt. author. Email: wangyan88@jiangnan.edu.cn. (1.3).

(2) 2. F. F. Jiang, Z. C. Ji and Y. Wang. which is called as Liénard-type system or generalized Liénard system. The qualitative theory of (1.2) and (1.3) is an important and challenging problem which has attracted great interests of many researchers. So far there have been rich achievements concerning the existence and uniqueness of limit cycles, the number of limit cycles, and the bifurcation of limit cycles, see for example [2, 4, 6, 9, 12, 13, 15–17, 24–29], and references therein. The Liénard system (1.2) and Liénard-type system (1.3) are two important classes of nonlinear systems of ordinary differential equations, because these two dynamical systems often appear in several branches of science such as biology, chemistry, mechanics and engineering. Indeed (1.2) and (1.3) are suitable mathematical models for many practical applications in the real world, and the relevant subjects have been the focus of many recent studies. On the application side, many planar models in mechanics, engineering, biology, chemical reaction and ecological models can be transformed into (1.2) or (1.3), and the existence and uniqueness of limit cycles in the original system can be proved via the existence and uniqueness of limit cycles for the transformational system, see for example [5, 7, 10, 14, 21–23] and references therein. Hence it deserves considerable attention to investigate the nonlinear Liénard system (1.2) and Liénardtype system (1.3). On the other hand, one of the classical problems in the qualitative theory of planar ordinary differential systems is to characterize the number and relative position of limit cycles. This problem restricted to planar polynomial differential systems is the second part of the well known Hilbert’s 16th problem, for more details we refer to [11] for example. When the functions F ( x ) and g( x ) in (1.2) are real coefficient polynomials in the variable x and have degree n and m respectively, then (1.2) is called as a generalized Liénard polynomial differential system. In such case, the investigation for the number of limit cycles becomes a discussion to the second part of the Hilbert’s 16th problem. Up to now, there are a lot of works concerning the number of limit cycles of the generalized Liénard polynomial differential systems [1, 3, 8, 18–20]. However, to the best of our knowledge there are few papers involving the position problem of limit cycles. Especially, there is hardly any result on the relative position of limit cycles to the nonlinear Liénard-type system (1.3). So we focus on the amplitude of limit cycles of (1.3). Inspired by [24], in which the authors studied the location problem of a limit cycle for a class of Liénard systems (1.2) with symmetry. In this paper we investigate the same problem to the general Liénard-type system (1.3), and we obtain an explicit upper bound for the amplitude of the unique limit cycle of (1.3) under the sufficient assumptions proving the existence and uniqueness of limit cycles of (1.3). Hence, in some sense this paper is a generalization. As we know, there are many rich achievements concerning the existence and uniqueness of limit cycles of (1.3). In the existing papers, to guarantee the existence and uniqueness of solutions of the system, it is often assumed that g( x ) is continuous, F ( x ) is continuously differentiable on (b, a) with −∞ ≤ b < 0 < a ≤ +∞, and h(y) is continuously differentiable for y ∈ R. Beyond that it is always assumed that. (i ) xg( x ) > 0 for x 6= 0, and denote by G ( x ) =. Rx 0. g(s)ds satisfying G (±∞) = +∞;. (ii ) there exist x1 and x0 satisfying b < x1 < 0 < x0 < a, such that F ( x1 ) = F (0) = F ( x0 ) = 0 and xF ( x ) > 0 for x ∈ ( x0 , a) ∪ (b, x1 ), xF ( x ) < 0 for x ∈ ( x1 , x0 ); (iii ) F 0 ( x ) > 0 for x ∈ ( x0 , a) ∪ (b, x1 ); (iv) h0 (y) > 0 for y ∈ R, h(0) = 0 and h(±∞) = ±∞..

(3) The position problem of limit cycles. 3. We note that the existence of limit cycles can be proved by various methods based on the well known Poincaré–Bendixson annular region theorem by constructing a trapping zone where a limit cycle is located. While for the uniqueness of the limit cycle, it can be established by various techniques based on the geometric properties of the isocline curve h(y) = F ( x ). From the above assumptions, it is easy to obtain the existence and uniqueness of a limit cycle of (1.3). Then in this paper, we are concerned with the position problem of the unique limit cycle of (1.3). By considering the amplitudes of the unique limit cycle on half planes {( x, y) ∈ R2 : x > 0} and {( x, y) ∈ R2 : x < 0} respectively, we obtain an explicit upper bound for the amplitude of the unique limit cycle of (1.3) on R2 . The paper is organized as follows. In the next section, we present some relevant preliminaries on the existence and uniqueness of limit cycles of the Liénard-type system. In Section 3, we give several relevant lemmas which can be used to derive the upper bound of amplitudes of the unique limit cycle on {( x, y) ∈ R2 : x > 0} and {( x, y) ∈ R2 : x < 0} respectively. Then we provide a criterion concerning the position of the unique limit cycle on R2 . In Section 4, we give two examples including an application to predator–prey system model and the corresponding Matlab simulations to illustrate the obtained result. Conclusion is stated in Section 5.. 2. Preliminaries. Consider the following Liénard-type system  dx   = h ( y ) − F ( x ), . dt. (2.1).    dy = − g( x ).. dt. Suppose that the following assumptions hold:. ( H1) there exist x0 > 0 and x1 < 0 such that F ( x1 ) = F (0) = F ( x0 ) = 0, and F ( x )( x − x0 ) > 0 for x ∈ (0, +∞) \ { x0 }, F ( x )( x − x1 ) > 0 for x ∈ (−∞, 0) \ { x1 } and F (±∞) = ±∞; ( H2) F ∈ C1 (R, R) satisfying F 0 ( x ) > 0 for x ∈ ( x0 , +∞) ∪ (−∞, x1 ); ( H3) g ∈ C (R, R) satisfying xg( x ) > 0 for x 6= 0, and denote by G ( x ) , G (±∞) = +∞;. Rx 0. g(s)ds satisfying. ( H4) h ∈ C1 (R, R) satisfying h0 (y) > 0, yh(y) > 0 for y 6= 0, and h(±∞) = ±∞. By ( H1)–( H4), it is easy to obtain that the system (2.1) has a unique limit cycle surrounding the unique equilibrium point O(0, 0). Since x 0 = h(y) for x = 0, it follows that all orbits of (2.1) other than the origin run around the origin O in a clockwise fashion on R2 . Moreover, according to the theorem of the existence and uniqueness of solutions then for any initial point P( x0 , y0 ) ∈ R2 \{O} the system (2.1) has a unique solution ϕ( P, t) satisfying ϕ( P, 0) = P for t ∈ (− T1 , T2 ), where (− T1 , T2 ) is the maximum existence interval of the solution. The corresponding orbit is denoted by L P = { ϕ( P, t) : − T1 < t < T2 }. Similarly, we let L+ P = { ϕ( P, t) : 0 ≤ t < T2 } denote the positive half orbit starting from the point P, and let L− P = { ϕ( P, t) : − T1 < t ≤ 0} denote the negative half orbit. For convenience, we also define the.

(4) F. F. Jiang, Z. C. Ji and Y. Wang. 4. following regions: Σ0 = {( x, y) : x = 0, −∞ < y < +∞},. Σ F = {( x, y) ∈ R2 : h(y) = F ( x )},. Σ0+ = {( x, y) : x = 0, y > 0},. Σ0− = {( x, y) : x = 0, y < 0},. Σ+ = {( x, y) ∈ R2 : x > 0},. Σ− = {( x, y) ∈ R2 : x < 0},. 2 Σ+ F = {( x, y ) ∈ R : x > 0, F ( x ) > h ( y )},. 2 Σ− F = {( x, y ) ∈ R : x > 0, F ( x ) < h ( y )},. e + = {( x, y) ∈ R2 : x < 0, F ( x ) > h(y)}, Σ F. e − = {( x, y) ∈ R2 : x < 0, F ( x ) < h(y)}. Σ F. By the geometric properties of solutions of (2.1), it follows that for any orbit starting from − e+ P(0, y) ∈ Σ0+ (resp. Σ0− ) then the positive orbit L+ P enters Σ F (resp. Σ F ) when t > 0 small, runs around the origin O on Σ+ (resp. Σ− ) in a clockwise fashion, and crosses Σ F \ {O} transversally for the first time and only once in a finite time. After L+ P makes a half turn around the origin on Σ+ (resp. Σ− ), it again intersects with Σ0− (resp. Σ0+ ) for the first time in a finite time. By the − e+ direction of vector field of the system, it follows that L+ P continues to enter Σ F (resp. Σ F ) and + runs clockwise on Σ− (resp. Σ+ ), intersecting successively with Σ F \ {O} and Σ0 (resp. Σ0− ) in a finite time. After making a circle on R2 , if L+ P returns to the point P (0, y ) for the first time, then the orbit corresponding to the initial point P is the unique limit cycle of (2.1). In the next section, we investigate the position of the unique limit cycle of (2.1), i.e. we obtain the amplitude of the unique limit cycle, where the amplitude of a limit cycle on R2 is defined as the maximum absolute value of the x-coordinate on such a limit cycle.. 3. Main results. In this section, we investigate the upper bound for the amplitude of the unique limit cycle of (2.1). Let λ( x (t), y(t)) = H (y(t)) + G ( x (t)), (3.1) Ry. where H (y) = 0 h(s)ds and G ( x ) = solutions of (2.1) then. Rx 0. g(s)ds satisfying H (0) = 0 and G (0) = 0. Along the. dλ( x (t), y(t)) = −h(y(t)) g( x (t)) + g( x (t))[h(y(t)) − F ( x (t))] = − g( x (t)) F ( x (t)). dt. (3.2). Since F ( x )( x − x0 ) > 0 for x ∈ (0, +∞) \ { x0 } and F ( x )( x − x1 ) > 0 for x ∈ (−∞, 0) \ { x1 }, dλ( x (t),y(t)) it follows from ( H3) that > 0 for x ∈ ( x1 , 0) ∪ (0, x0 ). This implies that the unique dt limit cycle of (2.1) is not completely contained in the strip region {( x, y) ∈ R2 : x1 < x < x0 }, i.e. the limit cycle must encircle ( x1 , 0) and ( x0 , 0) as interior points. Hence for the purpose of the amplitude of the unique limit cycle of (2.1), we only consider the orbits crossing Σ F + transversally for x > x0 and x < x1 . Consider a positive orbit L+ A starting from A (0, y A ) ∈ Σ0 + − (see Figure 3.1). It follows that L A enters Σ F when t > 0 small, intersects with Σ F for x > x0 at B( x B , y B ), and after L+ A makes a half turn around the origin on Σ+ then it eventually intersects − e+ with Σ0 at C (0, yC ). Later L+ A starting from the point C continues to enter Σ F , and intersect + successively with Σ F and Σ0 at D ( x D , y D ) with x D < x1 and E(0, y E ). By the properties of planar autonomous systems, we let ∆+ ( x B ) = H (y A ) − H (yC ) denote ˘ in Figure 3.1), the positive half trajectory arc of (2.1) on Σ+ crossing Σ F for x B > x0 (see ABC and let ∆− ( x D ) = H (yC ) − H (y E ) denote the positive half trajectory arc on Σ− crossing Σ F ˘ in Figure 3.1). The main result concerning an upper bound for the for x D < x1 (see CDE amplitude of the unique limit cycle of (2.1) as follows..

(5) The position problem of limit cycles. 5. Theorem 3.1. Let ( H1)–( H4) hold, then there exists an explicit upper bound x ∗ = max{ x B , | x D |} for R the amplitude of the unique limit cycle of (2.1), where x B and x D are determined by 0x F ( x ) g( x )dx = 0. In other words, the unique limit cycle locates in the strip region {( x, y) ∈ R2 : | x | < x ∗ }. We first give the following several relevant lemmas, which can be used to derive the amplitude of the unique limit cycle of (2.1). y h(y)=F(x) A E. B. x. O. D. x=x 0 x=x. C 1. Figure 3.1: An orbit of (2.1) crossing Σ F transversally when x > x0 and x < x1 .. Lemma 3.2. For any x B ∈ ( x0 , +∞), then the function ∆+ ( x B ) is strictly monotone increasing satisfying ∆+ ( x B ) → +∞ as x B → +∞; while for any x D ∈ (−∞, x1 ) it follows that the function ∆− ( x D ) is strictly monotone decreasing satisfying ∆− ( x D ) → +∞ as x D → −∞. ˇ Proof. Consider any two orbits of (2.1) Aˇ 1 B1 C1 D1 E1 and A2 B2 C2 D2 E2 , which cross Σ F transversally when x > x0 and x < x1 (see Figure 3.2). Let these two orbits start respectively from A1 (0, y A1 ), A2 (0, y A2 ) ∈ Σ0+ with y A2 > y A1 > 0. When x > x0 the orbits intersect with Σ F at B1 ( x B1 , y B1 ) and B2 ( x B2 , y B2 ) satisfying x B2 > x B1 > x0 , make a half turn in a clockwise fashion on Σ+ then they again intersect with Σ0− at C1 (0, yC1 ) and C2 (0, yC2 ). By the properties of planar autonomous systems, it follows that yC2 < yC1 < 0. Later the orbits continue to enter Σ− , and intersect with Σ F at D1 ( x D1 , y D1 ) and D2 ( x D2 , y D2 ) satisfying x D2 < x D1 < x1 . After making a circle then the orbits eventually intersect with Σ0+ at E1 (0, y E1 ) and E2 (0, y E2 ) satisˇ fying y E2 > y E1 > 0. In Figure 3.2, F1 , G1 and F2 , G2 are where Aˇ 1 B1 C1 D1 E1 and A2 B2 C2 D2 E2 intersect with the straight line lx0 , {( x, y) : x = x0 , −∞ < y < +∞} respectively, and F1 and M2 , G1 and N2 have the same value of y-coordinate. Similarly, H1 and I1 are the inter-. section points of Aˇ 1 B1 C1 D1 E1 with l x1 , {( x, y ) : x = x1 , − ∞ < y < + ∞ }, H2 and I2 are the intersection points of Aˇ 2 B2 C2 D2 E2 with l x1 , and H1 and J2 , I1 and K2 have the same values of y-coordinate. ¸ ¸ ˘ We first analyze the right half trajectory arcs A 1 B1 C1 and A2 B2 C2 on Σ+ . For A1 F1 and 0 ˘ A 2 F2 , it follows from h ( y ) > 0 that for the same vale of x-coordinate then the positive value ˘ ˘ of h(y) − F ( x ) along A 2 F2 is greater than the one along A1 F1 . So one has that. λ( A2 ) − λ( F2 ) =. Z. F ( x ) g( x ) F ( x ) g( x ) dx > dx = λ( A1 ) − λ( F1 ). h ( y ( x )) − F ( x ) h ( y ( x )) − F ( x ) ¯ ¯ A1 F1 A2 F2 Z. (3.3).

(6) F. F. Jiang, Z. C. Ji and Y. Wang. 6. ˘ ˘ It is similar to G 1 C1 and G2 C2 then. λ( G2 ) − λ(C2 ) > λ( G1 ) − λ(C1 ).. (3.4). 0 ˘ ˙ Along F 1 G1 and M2 N2 , since F ( x ) > 0 for x > x0 , it follows that for the same value of y˘ ˙ coordinate then the value of F ( x ) along M 2 N2 is greater than the one along F1 G1 . Furthermore,. λ( M2 ) − λ( N2 ) =. Z. − F ( x )dy > ˘ M2 N2. Z ¯ F1 G1. − F ( x )dy = λ( F1 ) − λ( G1 ) > 0.. (3.5). Moreover, since F ( x )( x − x0 ) > 0 for x ∈ (0, +∞) \ { x0 }, it follows that λ( F2 ) − λ( M2 ) = λ( N2 ) − λ( G2 ) =. Z. − F ( x )dy = F¯ 2 M2. Z. − F ( x )dy = ¯ N 2 G2. Z yF 2 yM Z y N2 2. F ( x (y))dy > 0, (3.6) F ( x (y))dy > 0.. yG2. Adding from (3.3) to (3.6) then λ( A2 ) − λ(C2 ) > λ( A1 ) − λ(C1 ).. (3.7). By (3.1) it follows that H (y A2 ) − H (yC2 ) > H (y A1 ) − H (yC1 ), i.e. ∆+ ( x B2 ) > ∆+ ( x B1 ). Which together with x B2 > x B1 > x0 then the function ∆+ ( x B ) is strictly monotone increasing for any x B > x0 . On the other hand, from h(+∞) = +∞ it follows that λ( A2 ) − λ( F2 ) =. Z. F ( x ) g( x ) dx → 0, ¯ A2 F2 h ( y ( x )) − F ( x ). ˘ as the value of y-coordinate along A 2 F2 tends to + ∞ uniformly, and λ ( G2 ) − λ (C2 ) → 0 due ˘ to h(−∞) = −∞. While for F2 G2 it follows from F (+∞) = +∞ that. 0 < λ( F2 ) − λ( G2 ) =. Z. − F ( x )dy = ¯ F2 G2. Z yF 2 yG2. F ( x (y))dy → +∞,. as the value of y-coordinate of F2 , y F2 → +∞ and the value of y-coordinate of G2 , yG2 → −∞. Hence the first part of Lemma 3.2 holds. ¸ ¸ For the left half trajectory arcs C 1 D1 E1 and C2 D2 E2 on Σ− . With the similar analysis then. λ(C2 ) − λ( H2 ) > λ(C1 ) − λ( H1 ), λ( J2 ) − λ(K2 ) > λ( H1 ) − λ( I1 ),. λ( I2 ) − λ( E2 ) > λ( I1 ) − λ( E1 ), λ( H2 ) − λ( J2 ) > 0,. λ(K2 ) − λ( I2 ) > 0,. and so λ(C2 ) − λ( E2 ) > λ(C1 ) − λ( E1 ).. (3.8). That is ∆− ( x D2 ) > ∆− ( x D1 ) and then the conclusion holds. The proof is complete. Remark 3.3. If we let ∆− (| x D |) = H (yC ) − H (y E ) with | x D | ∈ (| x1 |, +∞) denote the positive ˘ on Σ , it follows from (3.8) that the function ∆ (| x |) is strictly half trajectory arc CDE − − D monotone increasing tending +∞ as | x D | → +∞..

(7) The position problem of limit cycles. 7. y A2 A1. I2 K2. h(y)=F(x) F2. E2 E1. B2 M2. F1. B1. I1. x. O J2 D2. D1. G1 H1. H2. x=x1. N2 G2. C1 C2. x=x0. Figure 3.2: Two orbits of (2.1) which cross Σ F transversally when x > x0 and x < x1 .. Similarly, we denote by ∆( x ) = ∆+ ( x B ) + ∆− ( x D ) = H (y A ) − H (y E ) for the orbit ˝ ABCDE (see Figure 3.1) of (2.1). Note that if y A = y E i.e. ∆( x ) = 0, the corresponding orbit is a periodic orbit. Furthermore, the amplitude of the periodic orbit is exactly max{ x B , | x D |} with x D < x1 < x0 < x B . If 0 < y E < y A i.e. ∆( x ) > 0, we claim that there exists at least one periodic orbit contained in the compact region Ω, and max{ x B , | x D |} is a upper bound of the amplitude of the periodic orbit, where Ω denotes the closed region encircled by ˝ ABCDE ∪ EA ˝ with ABCDE being a directed arc from A to E successively and EA being a directed line segment from E to A. It follows from (3.2) that the origin O(0, 0) is a source point, and then Ω is a positive invariant region. Hence according to the planar Poincaré–Bendixson annular region theorem then (2.1) has at least one periodic orbit in Ω. If 0 < y A < y E i.e. ∆( x ) < 0, it follows that Ω is encircled by a periodic orbit. So max{ x B , | x D |} is a low bound for the amplitude of such a periodic orbit. Our next task is to find max{ x B , | x D |} as small as possible such that ∆( x ) ≥ 0. We only analyze in detail the relevant properties to the right half trajectory arcs of (2.1) on Σ+ , while for the left half trajectory arcs on Σ− is symmetric. For convenience, we let L B = ˘ starting from B( x , y ) ∈ Σ { ϕ( B, t) : α < t < β} denote the right half trajectory arc ABC B B F ˜ (see Figure 3.1), and denote by L+ = { ϕ ( B, t ) : 0 ≤ t < β } the positive trajectory arc BC B − ˜ contained in Σ+ and L = { ϕ ( B, t ) : α < t ≤ 0} is denoted by the negative trajectory arc AB F B − + − contained in Σ F . Then L B = L B ∪ L B . Here ϕ( B, t) is the unique solution of (2.1) satisfying ϕ( B, 0) = B, and (α, β) denotes the existence interval of the solution on Σ+ . Moreover, we + also present these two trajectory arcs L− B and L B by the graphs of the functions y = y ( x ) and y = y( x ) for x ∈ [0, x B ] respectively. Hence the following properties are true: (1) y( x ) is strictly monotone decreasing for x ∈ (0, x B ), satisfying h(y( x B )) = F ( x B ). (2) y( x ) < h−1 ( F ( x )) < y( x ) for x ∈ (0, x B ), satisfying y(0) = yC < 0 < y(0) = y A . (3) y( x ) is strictly monotone increasing for x ∈ (0, x B ), satisfying h(y( x B )) = F ( x B ). The following several lemmas are used to determine the upper bound of the amplitude for the unique limit cycle of (2.1) on Σ+ . Let v( x ) = h(y( x )) − F ( x ) and v( x ) = F ( x ) − h(y( x )), then v( x B ) = v( x B ) = 0 and v( x ) > 0, v( x ) > 0 for x ∈ [0, x B )..

(8) F. F. Jiang, Z. C. Ji and Y. Wang. 8. Lemma 3.4. Consider y = y( x ) for x ∈ [0, x B ], when x B > x0 it follows that H (y A ) >. 1 v ( x0 ). Z xB 0. F ( x ) g( x )dx + G ( x B ) + H (y B ).. (3.9). Proof. By differentiating λ( x, y) = H (y) + G ( x ) with respect to x and taking y = y( x ) then d g( x ) F ( x ) [ H (y( x )) + G ( x )] = − . dx v( x ). (3.10). Take integral from x = 0 to x = x B in (3.10), we obtain that H (y B ) = −. Z xB g( x ) F ( x ). v( x ). 0. dx − G ( x B ) + H (y A ).. (3.11). Since F ( x )( x − x0 ) > 0 for x ∈ (0, x B ) \ { x0 }, it follows from the property (1) that. Furthermore −. g( x ) F ( x ) v( x ). v( x ) = h(y( x )) − F ( x ) > v( x0 ),. x ∈ [0, x0 ),. v( x ) = h(y( x )) − F ( x ) < v( x0 ),. x ∈ ( x0 , x B ].. < − g(vx()xF0()x) for x ∈ (0, x B ] \ { x0 }, and then (3.9) holds due to (3.11).. Lemma 3.5. The function v( x ) is monotone decreasing for x ∈ [0, x B ]. Proof. We first claim that there exists at least one x1 ∈ (0, x0 ) such that F 0 ( x1 ) = 0. Suppose on the contrary that F 0 ( x ) 6= 0 for any x ∈ (0, x0 ). By F (0) = 0 and F ( x )( x − x0 ) > 0 for x ∈ (0, +∞) \ { x0 }, it follows that F 0 (0) < 0. Together with the continuous dependence of the derived function F 0 ( x ) on (0, x0 ) then F 0 ( x ) < 0 for x ∈ U + (0), where U + (0) denotes some small right-neighborhood of 0. Hence we suppose that F 0 ( x ) < 0 for every x ∈ (0, x0 ). Let U − ( x0 ) denote some small left-neighborhood of x0 , it follows from F ( x0 ) = 0 that F ( x ) > 0 for every x ∈ U − ( x0 ). This contradicts with F ( x ) < 0 for x ∈ (0, x0 ). Hence the claim holds. Without loss of generality, we suppose that there exist xi , i = 1, 2, . . . , m with m being an odd integer satisfying 0 < x1 < x2 < · · · < xm < x0 such that F 0 ( xi ) = 0. By F 0 ( x ) < 0 for x ∈ U + (0), F 0 ( x ) > 0 for x ∈ U − ( x0 ) ∪ [ x0 , x B ] and F ∈ C1 ((0, +∞), R), it follows that F0 (x) < 0 0. F (x) > 0. for x ∈ (0, x1 ) ∪ ( x2 , x3 ) ∪ · · · ∪ ( xm−1 , xm ), for x ∈ ( x1 , x2 ) ∪ ( x3 , x4 ) ∪ · · · ∪ ( xm , x B ].. For convenience, let us introduce the notations I = [0, x1 ] ∪ [ x2 , x3 ] ∪ · · · ∪ [ xm−1 , xm ] and I = ( x1 , x2 ) ∪ ( x3 , x4 ) ∪ · · · ∪ ( xm , x B ]. We next show that dy( x ) dv( x ) = F 0 ( x ) − h0 (y) ≤0 dx dx dv( x ) dx. (3.12). dy( x ) dx. > 0 for x ∈ (0, x B ]. Since again h0 (y) > 0, it follows < 0 for any x ∈ I. While for x ∈ I, we from three steps to analyze.. By the property (3), it follows that that. for x ∈ [0, x B ].. dy( x ). (i) If F 0 ( x ) ≤ h0 (y) dx for any x ∈ ( x1 , x2 ), it follows that dvdx(x) ≤ 0 for x ∈ ( x1 , x2 ). Otherwise there exist a finite number xi ∈ ( x1 , x2 ) with x1 being the first point such that F 0 ( xi ) > h0 (y) F 0 ( x ) > h0 (y) dv( x ) dx. dy( x ) dx. dy( xi ) dx .. By the continuous dependence of the derived function F 0 ( x ),. for x ∈ ( x1 , x2 ). Furthermore. ≤ 0 for any x ∈ ( x1 , x2 ).. dy( x2 ) dx. < 0, this is a contradiction. Hence.

(9) The position problem of limit cycles. 9. 1 (ii) For x ∈ ( x2i−1 , x2i ), i = 2, 3, . . . , m− 2 , it is similar to the case (i ) and so omitted. dy( x ) dx. dv( x ) ≤ 0 for x ∈ ( xm , x B ]. dx xi ) dy(e Otherwise there exist some points xei ∈ ( xm , x B ] such that F 0 ( xei ) > h0 (y) dx . By the dy( x ) continuous dependence of the derived function F 0 ( x ), then F 0 ( x ) > h0 (y) dx for x ∈ U − ( x B ) with U − ( x B ) being a small left-neighborhood of x B . Furthermore F 0 ( x ) → +∞ dv( x ) as x → x − B , it is a contradiction. Hence dx ≤ 0 for any x ∈ ( xm , x B ].. (iii) If F 0 ( x ) ≤ h0 (y). for any x ∈ ( xm , x B ], it follows that. In conclusion (3.12) holds. The proof is complete. Lemma 3.6. Consider y = y( x ) for x ∈ [0, x B ], when x B > x0 it follows that 1 H (yC ) ≤ − v ( x0 ). Z xB 0. F ( x ) g( x )dx + G ( x B ) + H (y B ).. (3.13). Proof. With the similar way to Lemma 3.4, it follows that d g( x ) F ( x ) [ H (y( x )) + G ( x )] = , dx v( x ) and take integral from x = 0 to x = x B in the above equality then H (yC ) = G ( x B ) + H (y B ) −. Z xB g( x ) F ( x ) 0. v( x ). dx.. (3.14). From Lemma 3.5 and F ( x )( x − x0 ) > 0 for x ∈ (0, x B ) \ { x0 }, it follows that. − g( x ) F ( x ) − g( x ) F ( x ) ≤ , v( x ) v ( x0 ). x ∈ (0, x B ].. Hence together with (3.14) then (3.13) holds. The proof is complete. Correspondingly, for the left half trajectory arcs of (2.1) on Σ− then we have also several ˘ relevant lemmas. Let L D = { ϕ( D, t) : µ < t < ν} denote the left half trajectory arc CDE + starting from D ( x D , y D ) ∈ Σ F (see Figure 3.1), and denote by L D = { ϕ( D, t) : 0 ≤ t < ν} the ˜ contained in Σ e − , and L− = { ϕ( D, t) : µ < t ≤ 0} is denoted by the positive trajectory arc DE F D e + . Moreover, we also present these two half orbits negative trajectory arc ¯ CD contained in Σ F + L− and L by the graphs of the functions y = y( x ) and y = y( x ) for x ∈ [ x D , 0] respectively. D D Hence the following properties are true: (1) y( x ) is strictly monotone increasing for x ∈ ( x D , 0), satisfying h(y( x D )) = F ( x D ). (2) y( x ) < h−1 ( F ( x )) < y( x ) for x ∈ ( x D , 0), satisfying y(0) = yC < 0 < y(0) = y E . (3) y( x ) is strictly monotone decreasing for x ∈ ( x D , 0), satisfying h(y( x D )) = F ( x D ). Similarly, let v( x ) = h(y( x )) − F ( x ) and v( x ) = F ( x ) − h(y( x )). Then v( x D ) = v( x D ) = 0 and v( x ) > 0, v( x ) > 0 for any x ∈ ( x D , 0]. Lemma 3.7. Consider y = y( x ) for x ∈ [ x D , 0], when x D < x1 it follows that H (yC ) >. 1 v ( x1 ). Z 0 xD. F ( x ) g( x )dx + G ( x D ) + H (y D )..

(10) F. F. Jiang, Z. C. Ji and Y. Wang. 10. Lemma 3.8. The function v( x ) is monotone increasing for x ∈ [ x D , 0]. Lemma 3.9. Consider y = y( x ) for x ∈ [ x D , 0], when x D < x1 it follows that 1 H (y E ) ≤ − v ( x1 ). Z 0 xD. F ( x ) g( x )dx + G ( x D ) + H (y D ).. Based on the above several lemmas, now we are ready to prove our main result (i.e. Theorem 3.1) concerning the position of the unique limit cycle of (2.1). Proof of Theorem 3.1. Consider the right half trajectory arcs of (2.1) on Σ+ . We first show that R x∗. + ∗ satisfying x ∗ > x such that there exists a unique x+ 0 + 0 g ( x ) FR( x ) dx = 0. By F ( x )( x − x0 ) > 0 for x ∈ (0, +∞) \ { x0 }, it follows that 0x g( x ) F ( x )dx < 0 for 0 < x ≤ R d x0 . Since again dx ( 0x g( x ) F ( x )dx ) > 0 for x > x0 , it follows from F (+∞) = +∞ that there R exist sufficiently large x with x > x0 such that 0x g( x ) F ( x )dx > 0. Hence by the property of R strictly monotone increasing of the continuous function 0x g( x ) F ( x )dx, there exists a unique. R x∗. x + ∗ ∗ > x such that x+ 0 0 g ( x ) F ( x ) dx = 0 and 0 g ( x ) F ( x ) dx > 0 for x > x+ . On the other hand, ∗ it follows from Lemma 3.2 that there exists a unique x B (> x0 ) such that ∆+ ( x ∗B ) = 0 and ∆+ ( x B ) > 0 for x B > x ∗B . Moreover, by Lemmas 3.4 and 3.6 one has that. R. ∆+ ( x B ) = H (y A ) − H (yC ) >. Ç. 1 1 + v ( x0 ) v ( x0 ). åZ x B. F ( x ) g( x )dx.. (3.15). 0. ∗ Since the function 0x g( x ) F ( x )dx is strictly monotone increasing for x > x0 , we let x B = x+ R xB ∗ ) > 0, i.e. x ∗ is an upper bound for the such that 0 F ( x ) g( x )dx = 0. Furthermore ∆+ ( x+ + amplitude of the unique limit cycle of (2.1) on Σ+ . ∗ satisfying Similar to the left half trajectory arcs of (2.1) on Σ− . Then there exists a unique x− R 0 ∗ < x such that x− 1 x ∗ g ( x ) F ( x ) dx = 0, and from Lemmas 3.7 and 3.9 it follows that. R. −. ∆− ( x D ) = H (yC ) − H (y E ) >. Ç. 1 1 + v ( x1 ) v ( x1 ). åZ 0. F ( x ) g( x )dx.. (3.16). xD. ∗ such that ∆ ( x ∗ ) > 0. By combining with (3.15) and (3.16) then the unique So we let x D = x− − − limit cycle of (2.1) locates in the strip region {( x, y) ∈ R2 : | x | < x ∗ } with x ∗ = max{ x B , | x D |}. The proof is complete.. Remark 3.10. In [24], the authors studied the location problem of a limit cycle for a class of Liénard systems with symmetry. However, in this paper we investigate the same problem for a more general Liénard-type differential system (2.1), and it is worth noting that we do not require that the orbits of the system is symmetric in the origin. Moreover, we obtain an explicit upper bound for the amplitude of the unique limit cycle of (2.1) merely under the sufficient conditions proving the existence and uniqueness of limit cycles of (2.1). Hence in some sense this paper improves and generalizes the result in [24].. 4. Application. Example 4.1. Consider the following celebrated predator–prey system model with nonmonotonic functional response Å ã  dx x xy   , = rx 1 − −  dt K c + x2 Å ã  µx   dy = y −D , 2. dt. c+x. (4.1).

(11) The position problem of limit cycles. 11. where r, c, µ, K and D are positive parameters. In [21], Ruan and Xiao have studied that (4.1) has√a unique limit cycle surrounding the unique positive equilibrium point ( x1 , y1 ) with x1 =. µ−. µ2 −4cD2 , 2D. y1 = r (1 −. x1 K )( c. + x12 ) if µ+. 16 2 µ2 > cD , 3. ». 2µ − µ2 − 4cD2 >K> 2D. ». µ2 − 4cD2 . 2D. Now we verify the position of the unique limit cycle of (4.1). By making a transformation y − y 1 = y 1 ( eY − 1 ) ,. x − x1 = − X,. xdt = (c + x2 )dT,. then (4.1) is transformed into the Liénard-type system form  dX   = h (Y ) − F ( X ) , . dT. (4.2).    dY = − g( X ),. dT. and g( X ) = DX (Xx1−−xX1 + x2 ) . √ Let c√= 1/2, D = 1, µ√= 2 and choose √ K = 3/2, r = 3/2, it follows that x1 = 1 − 2/2, x2 = 1 + 2/2, x3 = 2 − 2/2 and y1 = 2/2. We still let x, y, t be the variables of (4.2) and √ √ √ √ x (√ x + 2) 2 y 3 2 3 2 . By some then h(y) = 2 (e − 1), F ( x ) = x [ x − 2 (1 − 2) x + 2 − 2 ] and g( x ) = 1− 2/2− x simple but tedious computations it follows that. with h(Y ) = y1 (eY − 1), F ( X ) =. Z A− x ñ. Z x. F ( x ) g( x )dx = 0. rX 2 2 K ( X + ( K − 3x1 ) X + c + 3x1 − 2Kx1 ). t4 − (5A + B)t3 + (10A2 + 4AB + C )t2 − (10A3 + 6A2 B + 3AC + E)t. A. A5 + A4 B + A3 C + A2 E + (5A + 4A B + 3A C + 2AE) − dt, t 4. 3. ô. 2. √ √ √ √ with A = 1 − 22 , B = 52 2 − 32 , C = 5 − 3 2 and E = 2 2 − 3. By Matlab and Theorem 3.1, it follows that the unique limit cycle of (4.2) locates in the strip region {( x, y) ∈ R2 : | x | < 0.89}. Indeed Matlab simulation shows the result shown in Figure 4.1.. 2 1.5 1. y. 0.5 0 −0.5 −1 −1.5 −2 −1.5. −1. −0.5. 0. 0.5. 1. x. Figure 4.1: The unique limit cycle of (4.2) surrounding the origin..

(12) F. F. Jiang, Z. C. Ji and Y. Wang. 12. Example 4.2. Consider the following Liénard-type system  dx   = y3 − ( x 3 − x ), . dt. (4.3).    dy = − x.. dt. Since F ( x ) = x3 − x and g( x ) = x are odd functions, it easily verify that the system (4.3) has a unique symmetric limit cycle surrounding the origin (see Figure 4.2). From Z x. F ( x ) g( x )dx = x. 3. Å. 0. 1 2 1 x − 5 3. ã. = 0,. √. 15 3. it follows that x =. is the unique positive root. Hence by Theorem 3.1, the unique limit. cycle of (4.3) locates in the strip region {( x, y) ∈ R2 : | x | <. √. 15 3 }. (see Figure 4.2).. 2 1.5 1. y. 0.5 0 −0.5 −1 −1.5 −2 −2. −1.5. −1. −0.5. 0 x. 0.5. 1. 1.5. 2. Figure 4.2: The unique symmetric limit cycle of (4.3). Remark 4.3. From the above two Figures we note that the result obtained from Theorem 3.1 is not the best upper bound. The interested readers can further seek some new methods to study a good upper bound for the amplitude of a limit cycle in Liénard-type system (2.1).. 5. Conclusion. In this paper, we have investigated the position problem of the unique limit cycle of Liénardtype system (2.1). By considering the amplitudes of the unique limit cycle on Σ− and Σ+ respectively, we have obtained an explicit upper bound for the amplitude of the unique limit cycle on the plane. Compared with the assumptions in [24], we have obtained the result merely under the sufficient conditions of the existence and uniqueness of limit cycles of (2.1). Hence in some sense, we have generalized and improved the result in [24]. Finally, we have given two examples including an application to predator–prey system model and the corresponding Matlab simulations to illustrate the effectiveness of our theoretical result. However, it should be noted that from these two figures then the result (i.e. Theorem 3.1) is not the best. Hence the interested readers can seek some new methods to further study the upper bound of the amplitude of limit cycles of (2.1)..

(13) The position problem of limit cycles. 13. Acknowledgements This work is supported by the China Postdoctoral Science Foundation (2017M611685), the Fundamental Research Funds for the Central Universities (JUSRP11723), the National Natural Science Foundation of China (61572238), and the Provincial Outstanding Youth Foundation of JiangSu Province (BK20160001).. References [1] J. Alavez-Ramirez, G. Blé, J. López-López, J. llibre, On the maximum number of limit cycles of a class of generalized Liénard differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22(2012), No. 3, 1250063, 14 pp. MR2916691; url [2] G. Chang, T. Zhang, M. Han, On the number of limit cycles of a class of polynomial systems of Liénard type, J. Math. Anal. Appl. 408(2013), 775–780. MR3085071; url [3] B. García, J. Llibre, J. S. Pérez del Río, Limit cycles of generalized Liénard polynomial differential systems via averaging theory, Chaos Soliton Fract. 62/63(2014), 1–9. MR3200747; url [4] Q. M. He, The existence and uniqueness of limit cycles for the equations ẋ = φ(y) − F ( x ), ẏ = − g( x ), Chinese Ann. Math. Ser. A 7(1986), 222–237. MR858591 [5] S.-B. Hsu, T.-W. Hwang, Uniqueness of limit cycles for a predator–prey system of Holling and Leslie type, Canad. Appl. Math. Quart. 6(1998), 91–117. MR1661821 [6] K. Ch. Huang, On the existence of limit cycles of the system dx/dt = h(y) − F ( x ), dy/dt = − g( x ), Acta Math. Sinica 23(1980), 483–490. MR592275 [7] T. Hwang, H. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions, J. Phys. A 38(2005), 8211–8223. MR2168299; url [8] F. Jiang, J. Shi, J. Sun, On the number of limit cycles for discontinuous generalized Liénard polynomial differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 25(2015), No. 10, 1550131, 10 pp. MR3402863; url [9] F. Jiang, J. Sun, On the uniqueness of limit cycles in discontinuous Liénard-type systems, Electron. J. Qual. Theory Differ. Equ. 2014, No. 71, 1–12. MR3304197; url [10] Y. Kuang, H. Freedman, Uniqueness of limit cycles in Gause-type models of predator– prey systems, Math. Biosci. 88(1988), 67–84. MR930003; url [11] J. Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13(2003), No. 1, 47–106. MR1965270; url [12] C. Li, J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations 252(2012), 3142–3162. MR2871796; url [13] Y. Liu, M. Han, V. Romanovski, Some bifurcation analysis in a family of nonsmooth Liénard systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 25(2015), No. 4, 1550055, 16 pp. MR3342115; url.

(14) 14. F. F. Jiang, Z. C. Ji and Y. Wang. [14] P. Liu, J. Shi, Y. Wang, X. Feng, Bifurcation analysis of reaction–diffusion Schnakenberg model, J. Math. Chem. 51(2013), 2001–2019. MR3103426; url [15] J. Llibre, A. C. Mereu, M. A. Teixeira, Limit cycles of the generalized polynomial Liénard differential equations, Math. Proc. Cambridge Philos. Soc. 148(2010), 363–383. MR2600146; url [16] J. Llibre, M. Ordóñez, E. Ponce, On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry, Nonlinear Anal. Real World Appl. 14(2013), 2002–2012. MR3043136; url [17] J. Llibre, E. Ponce, F. Torres, On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities, Nonlinearity 21(2008), 2121–2142. MR2430665; url [18] J. Llibre, C. Valls, On the number of limit cycles of a class of polynomial differential systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468(2012), 2347–2360. MR2949385; url [19] J. Llibre, C. Valls, On the number of limit cycles for a generalization of Liénard polynomial differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23(2013), No. 3, 1350048, 16 pp. MR3047963; url [20] J. Llibre, C. Valls, Limit cycles for a generalization of polynomial Liénard differential systems, Chaos Solitons Fractals 46(2013), 65–74. MR3011852; url [21] S. Ruan, D. Xiao, Global analysis in a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61(2001), 1445–1472. MR1813689; url [22] J. Wang, J. Shi, J. Wei, Predator–prey system with strong Allee effect in prey, J. Math. Biol. 62(2011), 291–331. MR2771175; url [23] D. Xiao, Z. Zhang, On the uniqueness and nonexistence of limit cycles for predator–prey systems, Nonlinearity 16(2003), 1185–1201. MR1975802; url [24] L. Yang , X. Zeng, An upper bound for the amplitude of limit cycles in Liénard systems with symmetry, J. Differential Equations 258(2015), 2701–2710. MR3312641; url [25] Y. Ye, S. Cai, L. Chen, K. Huang, D. Luo, Z. Ma, E. Wang, M. Wang, X. Yang, Theory of limit cycles, Translations of Mathematical Monographs, Vol. 66, American Mathematical Society, Providence, RI, 1986. MR0854278 [26] X. Zeng, Z. Zhang, S. Gao, On the uniqueness of the limit cycle of the generalized Liénard equation, Bull. London Math. Soc. 26(1994), 213–247. MR1289041; url [27] Z. Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations, Appl. Anal. 23(1986), 63–76. MR0865184; url [28] Z. Zhang, T. Ding, Z. Dong, W. Huang, Qualitative theory of differential equations, Beijing, Science Press, 2006. MR1175631 [29] Z. Zhang, S. Gao, On the uniqueness of limit cycle of one type of nonlinear differential equations, Acta Sci. Nat. Univ. Pekin. 1(1986), 1–13. url.

(15)