# Limit Cycles and Invariant Curves in a Class of Switching Systems with Degree Four

Authors: Xinli Li; Huijie Yang; Binghong Wang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, a class of switching systems which have an invariant conicx2+cy2=1,c∈R, is investigated. Half attracting invariant conic x2+cy2=1,c∈R, is found in switching systems. The coexistence of small-amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems is proved.


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## Body

## 1. Introduction

It is well known that the 16th problem stated in 1900 by D. Hilbert is considered to be the most difficult problem in the 23 problems; it is far from being solved. Over past three decades, there have been many good results about this problem. As far as the maximal number of small-amplitude limit cycles which are bifurcated from an elementary center or focus is concerned, the best known result obtained by Bautin in 1952 [1] is M(2)=3, where M(n) denotes the maximal number of small-amplitude limit cycles around a singular point with n being the degree of polynomials in the system. For cubic-degree system, many good results have also been obtained. For example, a cubic system was constructed by Lloyd and Pearson [2] to show 9 limit cycles with the aid of purely symbolic computation. Moreover, Yu and Tian [3] proved that there can exist 12 limit cycles around an elementary center in a planar cubic-degree polynomial system. As far as we know this is the best result obtained so far for cubic-degree polynomial systems with all limit cycles around a single singular point. For n≥4, because of the difficulty of computation of focal values, there are very few results. An example of a quartic system with 8 limit cycles bifurcating from a fine focus [4] was given by Huanget al. Theory of rotated equations and applications to a population model can be found in [5]; they gave a new method to solve the center problem.As far as the maximal number of limit cycles of polynomial systems is concerned, the best results published were given as follows. Articles [6, 7] proved that H(2)⩾4, then [8–10] gave H(3)⩾12 and [11, 12] obtained H(4)⩾16 etc. Here, H(n) denotes the maximal number of limit cycles of polynomial systems. Furthermore, 13 limit cycles bifurcated from Z2-equivariant systems with degree 3 were proved in [13–15], respectively. An improvement on the number of limit cycles bifurcating from a nondegenerate center of homogeneous polynomial systems was given in [16].Center and the coexistence of large and small-amplitude limit cycles problems are two closely related questions of the 16th problem. Algebraic trajectories play an important role in the dynamical behavior of polynomial systems, so it has been an interesting problem in planar polynomial systems. Over the past twenty years, many interesting results were got for quadratic systems; the authors in [17, 18] proved that quadratic systems with a pair of straight lines or an invariant hyperbola, ellipse, can have no limit cycles other than the possible ellipse itself. Furthermore, if there is an invariant line, there can be no more than one limit cycle. The case of parabola was considered in [19]. For cubic systems, there exist different classes of cubic systems in which there may coexist an invariant hyperbola or straight lines with limit cycles (see [20–28]). For a given family of real planar polynomial systems of ordinary differential equations depending on parameters, the problem of how to find the systems in the family which become time-reversible was solved in [29].In modelling many practical problems in science and engineering, switching systems have been widely used recently. The richness of dynamical behavior found in switching systems covers almost all the phenomena discussed in general continuous systems. For example, the maximum number of limit cycles bifurcating from the periodic orbits of the quadratic isochronous centers of switching system was studied in [30]. In [31], limit cycles in a class of continuous and switching cubic polynomial systems were investigated. Bifurcation of limit cycles in switching quadratic systems with two zones was considered in [30]. In [32, 33], the authors considered nonsmooth Hopf bifurcation in switching systems. Limit cycles bifurcating from centers of discontinuous quadratic systems were studied by Chen and Du [34]. Switching Bautin system was also investigated in [35]; they got 10 limit cycles for this system. Z2-equivariant cubic systems were also investigated, and 14 limit cycles were obtained in [36]. Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems was investigated in [37]. Bifurcation theory for finitely smooth planar autonomous differential systems was considered in [38]. All results obtained show that the dynamical behavior of switching systems is more complex than continuous system.About algebraic invariant curves, as far as we know, there are few papers to consider switching system with algebraic invariant curves. In this paper we are concerned with the limit cycle problem and the center problem for a class of degree four polynomial differential systems(1)dxdt=λx+y-1-bx-ey+1+cdx2+c1+2cdy2+bx3+ex2y+bcxy2+cey3-x3+cxy2λ,dydt=λy+x1-cy-1+dx2-c1+2dy2+cx2y+c2y3-λx2y+cy3,dxdt=λ1x+y-1-b1x-e1y+1+cd1x2+c1+2cd1y2+b1x3+e1x2y+b1cxy2+ce1y3-x3+c1xy2λ1,dydt=λ1x+x1-cy-1+d1x2-c1+2d1y2+cx2y+c2y3-λ1x2y+c1y3,which have an invariant conic x2+cy2=1, c∈R, and we prove the coexistence of large elliptic limit cycle that contains at least four small-amplitude limit cycles generated by Hopf bifurcations.The rest of the paper is organized as follows. In the next section, we prove that the switching system (1) has an invariant conic x2+cy2=1, c∈R, and there exists a large limit cycle in switching system (1); half attracting invariant conic x2+cy2=1, c∈R, is found in switching systems. In Section 3, the first eight Lyapunov constants will be computed; bifurcation of limit cycles and center conditions of (1) are investigated. Section 4 is devoted to discuss the number of limit cycles with different parameter c of (1). At last, coexistence of invariant curve and limit cycles of (1) is drawn in Section 5.


## 2. Invariant Curve and Large Limit Cycle of (1)

In this section, we will prove that the switching system (1) has an invariant conic x2+cy2=1, c∈R, and there exists a large limit cycle in switching system (1).Lemma 1.
The conich(x,y)=x2+cy2-1=0, c∈R, is an invariant algebraic curve of system (1). In particular, if c>0 and dd1≠0, this conic is an elliptic hyperbolic limit cycle, attracting if λ>0,λ1>0, a repelling if λ≤0,λ1≤0, and half attracting if λλ1<0.Proof.
It is easy to know that the conich(x,y)=x2+cy2-1, c∈R, is an invariant algebraic curve of systems(2)dxdt=λx+y-1-bx-ey+1+cdx2+c1+2cdy2+bx3+ex2y+bcxy2+cey3-x3+cxy2λ,dydt=λy+x1-cy-1+dx2-c1+2dy2+cx2y+c2y3-λx2y+cy3,and(3)dxdt=λ1x+y-1-b1x-e1y+1+cd1x2+c1+2cd1y2+b1x3+e1x2y+b1cxy2+ce1y3-x3+c1xy2λ1,dydt=λ1x+x1-cy-1+d1x2-c1+2d1y2+cx2y+c2y3-λ1x2y+c1y3,because (4)dhx,ydt=hx,ykix,y,i=1,2,respectively, where (5)k1x,y=-λx2+1-cxy-cλy2+bx2y+c2+exy2,k2x,y=-λ1x2+1-cxy-cλ1y2+b1x2y+c2+e1xy2.In particular, according to Lemma 1 in [39], if c>0 and d≠0, this conic is an elliptic hyperbolic limit cycle of system (2), attracting if λ>0 and a repelling if λ<0. Similarly, if c>0 and d1≠0, this conic is an elliptic hyperbolic limit cycle of system (3), attracting if λ1>0 and a repelling if λ1<0. Especially, if c>0 and dd1≠0 and λλ1<0, the stability of the conic x2+cy2=1, c∈R, is contradict for the upper half system and lower half system.
So, for switching system (1), the conic x2+cy2=1, c∈R, is an invariant algebraic curve. Furthermore, if c>0 and dd1≠0, this conic is an elliptic hyperbolic limit cycle, and (6)attractingλ>0,λ1>0;repellingλ<0,λ1<0;halfattractingλλ1<0.Remark 2.
For planar continuous system, ifc>0 and d≠0, the conic x2+cy2=1, c∈R, is an elliptic hyperbolic limit cycle, attracting if λ>0, a repelling if λ≤0. For switching system, half attracting cases which are different from continuous systems appear. Namely, for the conic x2+cy2=1, c∈R, it is attracting (repelling) for y>0 and repelling (attracting) for y<0. It is an interesting phenomena; see Figure 1.Figure 1
The half attracting conic whenλ>0,λ1<0 or λ<0,λ1>0.


## 3. Bifurcation of Limit Cycle and Center Conditions of (1)

First of all, it is easy to know that the origin of upper half system and lower half system is a fine focus ifλλ1≠0, so we let λ=λ1=0 in order to consider the center conditions and the number of small limit cycles. With the aid of symbolic computation, we obtain the following result.Theorem 3.
For system (1), the first eight Lyapunov constants at the origin are given by(7)μ1=-23b1-b,μ2=bπ8e1+e,μ3=-2b456d+3cd+12c2d-6d1-3cd1-12c2d1+4ce,with two cases: (I) c=±2/2. (8)μ4=bc64+21ce2π2884+c,μ5=0,μ6=bcd1π96-9+b2+7c-6cd1,μ7=bcd1π829440c11763+2434b2-66b4-16486c-3792b2c+544b4c,μ8=bπ978447237120c3-632874756819-1564815308390b2-48058429290b4+31851335688b6-1734059496b8+903172527570c+2206928649816b2c+78231277128b4c-42656920272b6c+1359726496b8c,
(II)c≠±2/2. (9)μ4=bcπ2882+c+4c2-48d1-24cd1+48c3d1+192c4d1+16ce-32c3e+64e2+21ce2,μ5=-bce14175-1+2c22+c+4c2214400-1440b2+12240c-720b2c+8824c2+304c3+1440b2c3-54816c4+5760b2c4-25440c5-36768c6-48256c7-8192c8-1152e2-10764ce2-128130c2e2-74937c3e2-43644c4e2,μ6=-bce2π1658880-1+2c222+c+4c22-938880-740160c+1065264c2-483240c3+2351920c4-1092224c5-3670368c6+8556192c7+4228224c8+2964992c9+409600c10-58752e2+1558128ce2+8882094c2e2+6387798c3e2+3861933c4e2+2842044c5e2+2182200c6e2,μ7=-8bce80372252+c+4c23fc2f1c,μ8=-8bceπ27993602+c+4c23fc3f2c. where (10)fc=-19584+519376c+2960698c2+2129266c3+1287311c4+947348c5+727400c6,f1c=-16642999723622400-533641875920732160c-1422483722108868096c2-6179682878361868032c3-16468630764742211040c4-20952718578375299760c5-47133134356437147504c6-20260317246600069740c7+1872045962529690518c8+124503308483978457972c9+384905600471956478466c10+639925619448235723897c11+995949036122547224578c12+1051854664680123230362c13+989202162935234549752c14+722398370839930377872c15+477074650742338886960c16+266605013954075833728c17+102005297657011481216c18+26101293898978550016c19+1081597406307225600c20-1162211975363604480c21+2379375230976000c22,f2c=117360+92520c+101562c2+245445c3-90866c4+627418c5+277064c6+185312c7+25600c8×-452567167393333248+2718583098083868672c+5921329078274770944c2+37901532550847576832c3+126508437153358233984c4+178634371590216048576c5+516471221184905286240c6+812488329237341411240c7+1323104926777377304180c8+2033464466845502085698c9+1247927693968900807483c10+1013721170261169909103c11+215716197433324981017c12+1064766586075843968365c13+4034487317277427888208c14+5640449144757272315336c15+6613022295691417350356c16+5376059971880816237612c17+4585410563561734908832c18+3103125495812061611648c19+1361862580731165716224c20+378280087173029995008c21+12407010796058419200c22-18830025122086993920c23+39259691311104000c24.Note that in computing the above expressions μk, k=2,…,8, and μ1=μ2=⋯=μk-1=0 have been used.The following proposition follows directly from Theorem3.Proposition 4.
The first eight Lyapunov constants at the origin of system (1) become zero if and only if one of the following conditions is satisfied:(11)λ1=λ=0,b1=b=0;(12)λ1=λ=0,b1=b,e1=-e,d1=d,c=0;(13)λ1=λ=0,b1=b,e1=-e,d1=d,e=0.They are also the center conditions of system (1).Proof.
When the conditions in (11) hold, system (1) can be brought to(14)dxdt=y-1-ey+1+cdx2+c1+2cdy2+ex2y+cey3,dydt=x1-cy-1+dx2-c1+2dy2+cx2y+c2y3,y>0,dxdt=y-1-e1y+1+cd1x2+c1+2cd1y2+e1x2y+ce1y3,dydt=x1-cy-1+d1x2-c1+2d1y2+cx2y+c2y3,y<0.Obviously, the system is symmetric with the y-axis, and so the origin is a center of system (14).
When the conditions in (12) hold, system (1) can be rewritten as(15)dxdt=y-1-bx-ey+x2+bx3+ex2y,dydt=x1-1+dx2,y>0,dxdt=y-1-bx+ey+x2+bx3-ex2y,dydt=x1-1+dx2,y<0,which is symmetric with the x-axis, and so the origin is a center of system (15).
When the conditions in (13) hold, system (1) becomes a continuous system(16)dxdt=y1+bx-1+x2+cy2,dydt=x-1+cy-1+x2+cy2.By elementary integration, the above system in Ω=(x,y)∣x2+cy2<1 is topologically equivalent to the system(17)dxdt=y1+bx,dydt=x-1+cy,which has the first analytic integral (18)Hx,y=bcby-cx+b2ln⁡1-cy+c2ln⁡1+bx.Remark 5.
The phase plane of system (16) can be drawn by Maple; see Figure 2.Figure 2
The phase plane of system (16) when b=1,c=1 or b=1,c=-1.As far as limit cycles are concerned, it follows from Theorem3 that at most 8 limit cycles can bifurcate from the origin of system (1). We have the following theorem.Theorem 6.
If the origin of system (1) is a 8th-order weak focus, then for 0<δ1,δ2≪1, 8 small-amplitude limit cycles can bifurcate from the origin of the perturbed system (1).Proof.
When the origin of system (1) is a 8th-order weak focus, the conditions (19)b1=b,e1=-e,d=6d1+3cd1+12c2d1-4ce32+c+4c2,d1=-e1-16c+32c3+64e1+21ce124-1+2c22+c+4c2,b2=1720-1+2c22+c+4c2-14400-12240c-8824c2-304c3+54816c4+25440c5+36768c6+48256c7+8192c8+1152e2+10764ce2+128130c2e2+74937c3e2+43644c4e2,e2=-83fc-117360-92520c+133158c2-60405c3+293990c4-136528c5-458796c6+1069524c7+528528c8+370624c9+51200c10,c=-2.09067,-1.9427,-0.763201,0.581824should be satisfied. Furthermore, one has the following:
Whenc=-2.09067, (20)∂μ1,μ2,μ3,μ4,μ5,μ6,μ7∂b1,e1,d,d1,b,e,c=-2.23342×106≠0.
Whenc=-1.9427, (21)∂μ1,μ2,μ3,μ4,μ5,μ6,μ7∂b1,e1,d,d1,b,e,c=1.31166×106≠0.
Whenc=-0.763201, (22)∂μ1,μ2,μ3,μ4,μ5,μ6,μ7∂b1,e1,d,d1,b,e,c=-0.23494≠0.
Whenc=0.581824, (23)∂μ1,μ2,μ3,μ4,μ5,μ6,μ7∂b1,e1,d,d1,b,e,c=-0.000330085≠0.
So it implies that8 small-amplitude limit cycles can bifurcate from the origin of the perturbed system (1).


## 4. Number of Limit Cycles with Different Parameterc of (1)

In this section, we devote to discuss the number of limit cycles with different parameterc of (1). The following theorem could be concluded from Theorem 3.Theorem 7.
The number of limit cycles with different parameterc of (1) can be shown in the Table 1.Table 1
The maximum number of small limit cycles around the origin for different parameter c.
Parameter c The maximum number of small limit cycle around the origin ( - ∞ , - 6.14829 ) 6 ( - 6.14829 , - 2.09067 ) 7 -2.09067 8 ( - 2.09067 , - 1.94278 ) 7 -1.94278 8 ( - 1.94278 , - 0.7632018 ) 7 -0.7632018 8 ( - 0.7632018 , - 0.707107 ] 7 ( - 0.707107 , - 0.588861 ] 6 ( - 0.588861 , - 0.44638 ) 7 [ - 0.44638 , - 0.031805 ] 6 ( - 0.031805,0 ) 7 0 3 ( 0,0.031805 ) 7 [ 0.031805,0.129745 ] 6 (0.129745,0.581824) 7 0.581824 8 (0.581824,0.707107] 7 (0.707107,+∞) 6Proof.
Letμ1=μ2=μ3=0, it is easy to obtain that (24)b1=b;e1=-e;d=6d1+3cd1+12c2d1-4ce32+c+4c2.If c=0, it is easy to check that the origin is a three-order weak focus. Furthermore, if c≠0, when c=2/2, the Lyapunov constants in case 1 yield that the origin is a seventh order weak focus.
Whenc≠2/2, the Lyapunov constants μ4=μ5=μ6=μ7=0,μ8≠0 yield that the origin is an eighth-order weak focus if(25)d1=-e1-16c+32c3+64e1+21ce124-1+2c22+c+4c2,b2=f3c720-1+2c22+c+4c2=F3c,e2=-8f4c3fc=F4c, where (26)f3c=-14400-12240c-8824c2-304c3+54816c4+25440c5+36768c6+48256c7+8192c8+1152e2+10764ce2+128130c2e2+74937c3e2+43644c4e2;f4c=-117360-92520c+133158c2-60405c3+293990c4-136528c5-458796c6+1069524c7+528528c8+370624c9+51200c10. and c satisfy that (27)fc=-19584+519376c+2960698c2+2129266c3+1287311c4+947348c5+727400c6≠0.It is easy to conclude that if F3(c)>0,F4(c)>0 and f1(c)=0,f2(c)≠0, there exist 8 limit cycles; namely, (28)c≈-2.09067,-1.9427,-0.763201,0.581824.If F3(c)>0,F4(c)>0, and f1(c)≠0, there exist 7 small limit cycles.
IfF3(c)F4(c)≤0, there exist 6 limit cycles.
Whenc≠2/2,f(c)=0, the Lyapunov constants μ4=μ5=0,μ6≠0 yield that the origin is a sixth-order weak focus. The conclusion can be given in Table 1 for simplify.


## 5. Coexistence of Invariant Curve and Limit Cycles of (1)

From above discussion, we study the coexistence of invariant curve and limit cycles of (1), by perturbation method of small parameters, the following conclusions could be got easily; for example, when c=-2.09067, there exist eight small limit cycles at least and x2-2.09067y2=1 is an invariant algebraic curve. The distribution of limit cycle can be drawn in Figure 3.Figure 3
Whenc=-2.09067, an invariant algebraic curve x2-2.09067y2=1 and eight small limit cycles.Whenc=-2/2, there exist seven small limit cycles at least and x2-(2/2)y2=1 is an invariant algebraic curve. The distribution of limit cycle can be drawn in Figure 4.Figure 4
Whenc=-2/2, an invariant algebraic curve x2-(2/2)y2=1 and seven small limit cycles.Whenc=0, there exist three small limit cycles at least and x=1 and x=-1 are two invariant lines. The distribution of limit cycle can be drawn in Figure 5.Figure 5
Whenc=0, two invariant lines x=±1 and three small limit cycles.Whenc=0.581824, there exist eight small limit cycles and a large limit cycle x2+0.581824y2=1 at the same time, namely, nine limit cycles in total for this system. The distribution of limit cycle can be drawn in Figure 6.Figure 6
Whenc=0.581824, an invariant algebraic curve x2+0.581824y2=1 and eight small limit cycles.Whenc=2/2, there exist seven small limit cycles at least and there is a large limit cycle x2+(2/2)y2=1 at the same time. The distribution of limit cycles can be drawn in Figure 7.Figure 7
Whenc=2/2, an invariant algebraic curve x2+(2/2)y2=1 and seven small limit cycles.Whenc=1, there exist six small limit cycles at least and there is a large limit cycle x2+y2=1 at the same time. The distribution of limit cycles can be drawn in Figure 8.Figure 8
Whenc=1, an invariant algebraic curve x2+y2=1 and six small limit cycles.


## 6. Conclusion

In this paper, a class of switching systems is investigated; the coexistence of small limit cycles and algebraic an invariant curve is proves. An interesting phenomenon that the algebraic invariant curvex2+cy2=1,c>0, can be half attracting is found.



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