On the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems

Transcription

1 Internationa Journa of Bifurcation and Chaos Vo. 25 No pages c Word Scientific Pubishing Company DOI: /S X On the Number of Limit Cyces for Discontinuous Generaized Liénard Poynomia Differentia Systems Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. Fangfang Jiang Department of Mathematics Tongi University Shanghai P. R. China iangfangfang87@126.com Junping Shi Department of Mathematics Coege of Wiiam and Mary Wiiamsburg VA USA shi@math.wm.edu Jitao Sun Department of Mathematics Tongi University Shanghai P. R. China sunt@sh163.net Received November ; Revised March In this paper we investigate the number of imit cyces for a cass of discontinuous panar differentia systems with mutipe sectors separated by many rays originating from the origin. In each sector it is a smooth generaized Liénard poynomia differentia system x = y g 1 xf 1 xy and y = x g 2 xf 2 xy wheref i x andg i x fori =1 2arepoynomias of variabe x with any given degree. By the averaging theory of first-order for discontinuous differentia systems we provide the criteria on the maximum number of medium ampitude imit cyces for the discontinuous generaized Liénard poynomia differentia systems. The upper bound for the number of medium ampitude imit cyces can be attained by specific exampes. Keywords: Discontinuous panar system; Liénard poynomia system; averaging theory; number of imit cyces. 1. Introduction One of main topics of the quaitative theory of panar differentia systems is the number and reative position of imit cyces. This probem restricted to continuous panar poynomia differentia systems is the we known Hibert s 16th probem see for exampe Li 2003]. Up to now there have been many achievements concerning the existence uniqueness and the number of imit cyces see for exampe García et a. 201; Justino & Jorge 2012; Li & Libre 2012; Libre 2010; Libre & Mereu ; Libre et a. 2015; Libre & Vas a 2013b; Loyd & Lynch 1988; Martins & Mereu 201; Shen & Han 2013; Sun 1992; Xiong & Zhong 2013] and references therein. A imit cyce bifurcating from a singe degenerate singuar point is caed a sma ampitude imit cyce and the one bifurcating from periodic orbits of a inear center is caed a medium ampitude imit cyce. In Libre & Vas 2012] the authors studied the number Author for correspondence

2 Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. F. Jiang et a. of medium ampitude imit cyces for a cass of poynomia differentia systems of the form x = y g 1 x f 1 xy 1 y = x g 2 x f 2 xy where f i x and g i x i = 1 2 are poynomias of a given degree. For system 1 when f 1 x = g 1 x = 0 it is the generaized Liénard poynomia differentia system. Note that the cassica generaized Liénard poynomia differentia equation is x fxx gx = 0 or equivaenty x = y F x y = gx 2 where F x = x 0 fsds fx andgx arepoynomias in the variabe x. For system 2 when poynomias F x and gx have degrees n and m respectivey the generaization to discontinuous generaized Liénard poynomia differentia systems was studied in Libre & Mereu 2013]. To the best of our knowedge there are very few resuts concerning the number of imit cyces which bifurcate from a continuum of periodic orbits of differentia systems perturbed by discontinuous Liénard poynomia differentia systems. So we restrict our attention to a cass of discontinuous and piecewise smooth panar poynomia differentia systems with mutipe sectors separated by rays starting from the origin. In this paper we investigate the maximum number of medium ampitude imit cyces for a discontinuous generaized Liénard poynomia differentia system with even number sectors separated by /2 straight ines passing through the origin 0 0 where the discontinuous set is of the form /2 1 x y :y =tan α 2kπ } x. k=0 We first consider the case where two different smooth generaized Liénard poynomia differentia systems are defined aternativey in every two neighboring sectors on the pane. By appying the averaging theory of first-order for discontinuous differentia systems we obtain a ower bound for the maximum number of imit cyces and we show that the upper bound for the number of medium ampitude imit cyces can be attained by some constructed exampes see Exampe.1 for an exampe and Matab numerica simuation. The obtained imit cyces bifurcate from the periodic orbits of the inear center x = y y = x. Moreover we give the discussion on the resut if there are different vector fieds in different sectors respectivey. The paper is organized as foows. In the next section we present some reevant preiminaries for the discontinuous generaized Liénard poynomia differentia system. In Sec. 3 we first present the averaging theory of first-order for discontinuous differentia systems and then we prove our main resuts on the maximum number of medium ampitude imit cyces. In Sec. we give two exampes to iustrate the proved resuts. Concusion is presented in Sec Preiminaries Consider the foowing discontinuous generaized Liénard poynomia differentia system x = y εg 11 xf 11 xy y = x εg 12 xf 12 xy hx y > 0 x = y εg 21 xf 21 xy y = x εg 22 xf 22 xy hx y < 0 3 where ε is a sufficienty sma parameter and g i x and f i x fori =1 2 are poynomias of variabe x with degrees m and n respectivey. Let a function h : R 2 R be given by 2 1 hx y = k=0 y tan α 2kπ x with >1 being an even integer and α R. Then h 1 0 = 2 1 k=0 x y :y =tan α 2kπ } x 5 is a discontinuous set consisting of /2 straight ines passing through the origin O0 0 which divides the pane R 2 into sectors with ange 2π/. When ε = 0 the system 3 becomes a panar inear center as foows x = y y = x. 6 Note that the system 6 has a continuum of periodic orbits surrounding the origin and the orbits are countercockwise oriented in the phase pane. In this paper by using the averaging theory of first-order for discontinuous differentia systems we investigate the maximum number of medium ampitude imit cyces for the discontinuous generaized

3 The Number of Limit Cyces for Discontinuous Systems Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. Liénard poynomia differentia system 3 with ε > 0 sufficienty sma. 3. Main Resuts First we reca the foowing resut concerning the existence of imit cyces from the averaging theory of first-order for discontinuous differentia systems. Lemma 3.1 Libre & Mereu 201]. Consider the foowing discontinuous differentia equation x t =εht xε 2 Gt x ε 7 where ε is a sufficienty sma rea number functions Ht xgt x ε are given by Ht x =H 1 t xsgnht xh 2 t x Gt x ε =G 1 t x εsgnht xg 2 t x ε functions H 1 H 2 : R D R n G 1 G 2 : R D R R n and h : R D R are continuous T- periodic in the variabe t and D R n is an open subset. Let h be a C 1 function having 0 as a reguar vaue. Denote by M = h 1 0 N = 0} D M and K = N\M. Moreover define the averaging function d : D R n of 7 to be Assume that dx = T 0 Ht xdt. i functions H 1 H 2 G 1 G 2 and h are ocay Lipschitz with respect to the variabe x; ii for z 0 K with dz 0 = 0 there exists a neighborhood V of z 0 such that dz 0for every z V \z 0 } and d B d V z 0 0 where d B d V z 0 denotes the Brouwer degree of the function d with respect to the bounded open subset V D and the fixed point z 0 ; iii if h t t 0z 0 =0for some t 0 z 0 M then x h H 1 ] 2 x h H 2 t 0 z 0 > 0 where A B] denotes the inner product of the vectors A and B x h denotes the gradient of ht x with respect to the variabe x. Then for 0 < ε sufficienty sma there exists a T -periodic soution x ε of 7 such that x0ε z 0 as ε 0. By taking the ange θ as the new independent variabe and using Tayor series expansion the sys- Our main resut of this paper is as foows. tem 3 becomes m dρ dθ = ε a 1i ρ i cos i1 θ c 1i ρ i1 cos i1 θ sin θ b 1i ρ i cos i θ sin θ Oε 2 Theorem 3.2. Consider the system 3 with ε > 0 sufficienty sma. Assume that the poynomias g i x and f i x i=1 2 have degrees m 1 and n 1 respectivey then the maximum number of medium ampitude imit cyces bifurcating from the periodic orbits of system 6 is λ =maxm n 1}. Proof. Let x = ρ cos θ y = ρ sin θ with ρ>0 then the system 3 can be transformed into dρ dt = εcos θg 11ρ cos θf 11 ρ cos θρ sin θ] sinθg 12 ρ cos θf 12 ρ cos θρ sin θ]} dθ dt =1ε ρ cos θg 12ρ cos θf 12 ρ cos θρ sin θ] sin θg 11 ρ cos θf 11 ρ cos θρ sin θ]} for θ α kπ α k2π k = and dρ dt = εcos θg 21ρ cos θf 21 ρ cos θρ sin θ] sinθg 22 ρ cos θf 22 ρ cos θρ sin θ]} dθ dt =1ε ρ cos θg 22ρ cos θf 22 ρ cos θρ sin θ] sin θg 21 ρ cos θf 21 ρ cos θρ sin θ]} for θ α k2π α k1π k = Denote by g 11 x = g 21 x = f 11 x = f 21 x = a 1i x i g 12 x = a 2i x i g 22 x = c 1i x i f 12 x = c 2i x i f 22 x = b 1i x i b 2i x i d 1i x i d 2i x i. d 1i ρ i1 cos i θ sin 2 θ

4 F. Jiang et a. for θ α kπ α k2π k = and m dρ dθ = ε a 2i ρ i cos i1 θ c 2i ρ i1 cos i1 θ sin θ b 2i ρ i cos i θ sin θ d 2i ρ i1 cos i θ sin 2 θ Oε 2 9 Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. for θ α k2π h 1 0 = h θ ρ = θ α k1π 2 1 k=0 2 1 p=0 k = From and 5 then θ ρ :sinθ =cosθtan α 2kπ } ρ 2 cos θ sinθtan α 2pπ ] 2 1 k p sin θ cos θ tan α 2kπ ]. By some tedious but simpe computations one has that h θ θ ρ 0foranyθ ρ h 1 0. On the other hand for the purpose of the averaging function we need to use the foowing formuas Gradshteyn & Ryshik 199] cos 2m θdθ = 1 2 2m cos 2m1 θdθ = 1 2 2m 2m θ 1 m 2 2m 1 m 1 2m sin2m 2θ 10 2m 2 2m 1 sin2m 2 1θ. 11 2m 2 1 Then the averaging function dρ of 8 and 9 is as foows where d 1 ρ = d 2 ρ = d 3 ρ = d ρ = 2 2 k=0 2 2 k=0 2 2 k=0 2 2 k=0 α k2π α kπ α k2π α kπ α k2π α kπ α k2π α kπ dρ =d 1 ρd 2 ρd 3 ρd ρ Substituting 10 and 11 into the above equaities then k1π α a 1i ρ i cos i1 θdθ a 2i ρ i cos i1 θdθ α k2π α k1π c 1i ρ i1 cos i1 θ sin θdθ c 2i ρ i1 cos i1 θ sin θdθ α k2π k1π α b 1i ρ i cos i θ sin θdθ b 2i ρ i cos i θ sin θdθ α k2π α k1π d 1i ρ i1 cos i θ sin 2 θdθ d 2i ρ i1 cos i θ sin 2 θdθ. α k2π d 1 ρ = 2 2 k=0 ρ i 2 i i i 1 a 1i A ik a 2i A ik m 1 a 12i1 a 22i1 ρ2i1 2 2i2 2i 2 i 1 2π

5 with A ik = A ik = sin i 2 1 α sin i 2 1 α The Number of Limit Cyces for Discontinuous Systems ] sin i 2 1 α kπ k 2π i 2 1 ] k 1π sin i 2 1 i 2 1 α ] ] k 2π Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. with d 2 ρ = d 3 ρ = 2 2 k=0 2 2 k=0 B ik = B ik = C ik = C ik = c 1i ρ i1 B ik c 2i ρ i1 B ik b 1i ρ i C ik b 2i ρ i C ik cos i2 α cos i2 α cos i1 α cos i1 α k 2π i 2 k 1π k 1π cos i2 α kπ cos i2 α k 2π i 2 k 2π cos i1 α kπ i 1 cos i1 α i 1 n 1 2i d ρ = d 12i d 22i ρ 2i1 2 2i 1 i 2 2i2 2 2 k=0 2 2 k=0 ρ i1 2 i 1 ρ i1 2 i1 i i 1 i 2 i1 k 2π 2i 2 i 1 d 1i D ik d 2i D ik d 1i E ik d 2i E ik ] 2π with D ik = sin i 2 α k 2π i 2 ] sin i 2 α kπ ]

6 Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. F. Jiang et a. E ik = D ik = E ik = sin i 2 2 α sin i 2 α Therefore we obtain that dρ = m 1 sin i 2 2 α 2 2 ] sin i 2 2 α kπ k 2π i 2 2 ] sin i 2 k 1π a 12i1 a 22i1 ρ 2i1 2i 2 2 2i2 i 1 k=0 2 2 k=0 2 2 k=0 ρ i 2 i i i 1 ρ i1 2 i 1 ρ i1 2 i1 i i 1 i 2 i1 From the above equaity we observe that the function dρ is a poynomia of variabe ρ with degree maxm n 1} so the poynomia dρ has at most λ = maxm n 1} positive roots that is the maximum number of bifurcating medium ampitude imit cyces is λ. Moreover if there exists some ρ 0 > 0 such that dρ 0 =0andd ρ 0 0 then the Brouwer degree d B d V ρ 0 0forsome sma open neighborhood V of ρ 0. Therefore by Lemma 3.1 for 0 < ε sufficienty sma the discontinuous system 3 has at most λ medium ampitude imit cyces bifurcating from the periodic orbits of the inear center x = y y = x. This competes the proof of Theorem 3.2. Remark 3.3. It shoud be noted that for some given degree m 1andn 1 there exist discontinuous generaized Liénard poynomia differentia systems having exacty λ =maxm n 1} medium ampitude imit cyces see Exampe.1 for an exampe with m = n =1. When α =0and =2wehave A ik = A ik =0 D ik = D ik =0 α i 2 ] k 1π sin i 2 2 i 2 2 2π n a 1i A ik a 2i A ik ] k 2π α d 12i d 22i ρ 2i1 2 2i1 i k=0 2 2 d 1i D ik d 2i D ik d 1i E ik d 2i E ik. k=0 ] ] k 2π 2i i 2π c 1i B ik c 2i B ik ρ i1 b 1i C ik b 2i C ik ρ i and E ik = E ik =0. In this case the function h : R 2 R is of the form hx y = y and the discontinuous generaized Liénard poynomia differentia system with two zones separated by the straight ine y = 0} is as foows x = y εg 11 xf 11 xy y = x εg 12 xf 12 xy y > 0 x = y εg 21 xf 21 xy y = x εg 22 xf 22 xy y < 0. We have the foowing resut:. 12 Coroary 3.. Consider the system 12 with 0 < ε sufficienty sma then the maximum number of medium ampitude imit cyces is λ =maxm n 1}. Proof. In the same way as Theorem 3.2 then the averaging function dρ isoftheform

7 dρ = = π 0 m a 1i ρ i cos i1 θ c 1i ρ i1 cos i1 θ sin θ The Number of Limit Cyces for Discontinuous Systems b 1i ρ i cos i θ sin θ 2π m d 1i ρ i1 cos i θ sin 2 θ dθ a 2i ρ i cos i1 θ π b 2i ρ i cos i θ sin θ d 2i ρ i1 cos i θ sin 2 θ dθ m 1 n 1 2i1 π2i 1!! a 12i1 a 22i1 ρ 2i 2!! c 2i ρ i1 cos i1 θ sin θ c 12i1 c 22i1 ρ 2i2 2 2i 3 Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. m b 12i b 22i ρ 2i 2 2i 1 2i1 π2i 1!! d 12i d 22i ρ 2i 2!!. n Therefore we obtain that for ε > 0 sufficienty sma the discontinuous system 12 has at most λ = maxm n 1} that bifurcate medium ampitude imit cyces and then the concusion hods. When there are different vector fieds in the different sectors respectivey we consider the foowing discontinuous generaized Liénard poynomia differentia system of the form x = y εg i1 xf i1 xy y = x εg i2 xf i2 xy x y S i 13 where S i denotes the ith sector separated by /2 straight ines passing through the origin is a positive even integer and the poynomias g i x and f i x fori = =1 2are g i1 x = a i x g i2 x = b i x f i1 x = c i x f i2 x = d i x 1 satisfying m 1n 1. Then we have the foowing resut. Theorem 3.5. Consider the system 13 with 0 < ε sufficienty sma then the maximum number of medium ampitude imit cyces bifurcating from the periodic orbits of 6 is λ =maxm n 1}. Proof. Let x = ρ cos θ y = ρ sin θ with ρ>0 then the system 13 is transformed into dρ dθ = ε a i ρ cos 1 θ c i ρ 1 cos 1 θ sin θ b i ρ cos θ sin θ d i ρ 1 cos θ sin 2 θ Oε 2 for θ α 2i 2π α 2iπ i =1 2...With the same proof as Theorem 3.2 it foows from Lemma 3.1 that the concusion hods. Remark 3.6. For system 13 with 1 if S i i = is the ith sector separated by rays starting from the origin then the pane R 2 is divided by the discontinuous rays into sectors. In this case we have the same concusion as Theorem 3.5. Remark 3.7. Note that the maximum number of medium ampitude imit cyces is independent of the number of sectors but it depends on the degree of the poynomias g i x andf i x.. Exampes Exampe.1. Consider the foowing discontinuous generaized Liénard poynomia differentia system with two zones separated by the straight ine

8 F. Jiang et a. y =0}: 2 x = y ε π x 3 2 xy y = x εx 1 y > 0 2 x = y ε π x 1y y = x ε x 2π y xy y < y From the system 15 we observe that g 11 x = 2 π x g 12x =x x Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. g 21 x = 2 π x 1 g 22x =x f 11 x = 3 2 x f 12x =0 f 21 x =1 f 22 x = 2 π x so the functions g i x f i x fori =1 2have degrees m =1andn = 1 respectivey. By Lemma 3.1 the averaging function dρ of 15 is as foows π 2 dρ = cos θ 0 π ρ cos θ 3 2 ρ2 sin θ cos θ ] sinθρcos θ 1 dθ 2π π 2 cos θ ρ cos θ 1ρsin θ π sinθ ρ cos θ 2 π ρ sin θ ] ρ 2 sin θ cos θ dθ = ρ 2 3ρ 2= ρ 2ρ 1. It is easy to see that the zeros of dρ areρ 1 =2 and ρ 2 = 1 satisfying d 1 > 0andd 2 < 0. Therefore by Theorem 3.2 we obtain that for ε > 0 sufficienty sma the discontinuous system 15 has two medium ampitude imit cyces. One is a stabe imit cyce bifurcating from the periodic orbit of radius ρ 1 = 2 of the inear center x = y y = x and the other is an unstabe imit cyce bifurcating from the periodic orbit of radius ρ 2 =1.ByMatab we see that the discontinuous Fig. 1. Two medium ampitude imit cyces of 15 with ε =0.01. system 15 has indeed two medium ampitude imit cyces see Fig. 1 the bigger one is a stabe imit cyce and the smaer one is an unstabe imit cyce. Exampe.2. Let = and α = π and we consider the foowing discontinuous generaized Liénard poynomia differentia system x = y ε2x 2 xy y = x ε 1x x 2 8 π x2 y hx y > 0 x = y ε 13x 2 x π 2 y x2 y y = x εx 2 xy hx y < It foows that the scaar function h : R 2 R is defined as hx y =y xy x andthediscontinuity set is of the form h 1 0 = x y :y = x} x y :y = x}. Obviousy g 11 x =2x 2 g 12 x =1x x 2 g 21 x =13x 2 f 11 x =x f 12 x = 8 π x2 f 21 x =x 2 1 x π 2 g 22x =x 2 f 22 x =x then the poynomias g i x f i x fori =1 2 have degrees m =2andn = 2 respectivey

9 By Lemma 3.1 the averaging function dρ of 16 is as foows dρ = 3π π 5π 3π 7π 5π 9π 7π The Number of Limit Cyces for Discontinuous Systems cos θg 11 ρ cos θf 11 ρ cos θρ sin θsinθg 12 ρ cos θf 12 ρ cos θρ sin θ]dθ cos θg 21 ρ cos θf 21 ρ cos θρ sin θsinθg 22 ρ cos θf 22 ρ cos θρ sin θ]dθ cos θg 11 ρ cos θf 11 ρ cos θρ sin θsinθg 12 ρ cos θf 12 ρ cos θρ sin θ]dθ cos θg 21 ρ cos θf 21 ρ cos θρ sin θsinθg 22 ρ cos θf 22 ρ cos θρ sin θ]dθ Int. J. Bifurcation Chaos Downoaded from by CITY UNIVERSITY OF HONG KONG on 09/29/15. For persona use ony. = ρ 3 3ρ 2 2ρ = ρρ 2ρ 1. Therefore for ε > 0 sufficienty sma the discontinuous system 16 has two medium ampitude imit cyces. One is bifurcating from the periodic orbit of radius 2 centered at the origin and it is stabe. The other is bifurcating from the periodic orbit of radius 1 and it is unstabe. Remark.3. Note that for smooth generaized Liénard poynomia differentia system x = y εg 1 x f 1 xy y = x εg 2 x f 2 xy where f i x and g i x i = 1 2 are poynomias with degrees n and m respectivey. Then the maximum number of medium ampitude imit cyces by using the averaging theory of first-order is max m 1 n 2 ]}. So roughy speaking Theorem 3.2 in this paper shows that the discontinuous generaized Liénard poynomia differentia systems can have at east one more imit cyce than the corresponding smooth ones see for instance Exampes.1 and Concusion In this paper we have investigated the maximum number of medium ampitude imit cyces for a cass of discontinuous generaized Liénard poynomia differentia systems with sectors separated by /2 straight ines passing through the origin of coordinates. The discontinuous system consists of two different smooth generaized Liénard poynomia differentia systems of the form x = y g 1 x f 1 xy y = x g 2 x f 2 xy ocated aternativey in every two neighboring sectors where the poynomias f i x andg i x fori =1 2 have degrees n and m respectivey. By using the averaging theory of first-order for discontinuous differentia systems we have obtained that for ε > 0 sufficienty sma the discontinuous generaized Liénard poynomia differentia system has at most maxm n 1} medium ampitude imit cyces and the upper bound maxm n 1} of the number of medium ampitude imit cyces can be attained for some vaues m and n. Moreover we have aso obtained the same resut if there are different vector fieds in different sectors respectivey. Acknowedgments The authors are gratefu to the referee for his/her constructive comments. This work was revised when the first author visited Coege of Wiiam and Mary in Spring She expresses her gratitude to Department of Mathematics Coege of Wiiam and Mary for warm hospitaity and is aso very gratefu to Professor Shi for vauabe guidance. This work is supported by the NSF of China under grant References García B. Libre J. & Pérez de Río J. S. 201] Limit cyces of generaized Liénard poynomia differentia systems via averaging theory Chaos Soit. Fract Gradshteyn I. S. & Ryshik I. M. 199] Tabe of Integras Series and Products 5th edition ed. Jeffrey A. Academic Press NY. Justino A. R. & Jorge L. L. 2012] On the maximum number of imit cyces of a cass of generaized Liénard differentia systems Int. J. Bifurcation and Chaos

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