Limit Cycles for a Discontinuous Quintic Polynomial Differential System

Abstract

In this article, we study the maximum number of limit cycles for a discontinuous quintic differential system. Using the first-order averaging method, we explain how limit cycles can bifurcate from the period annulus around the center of the considered system when it is perturbed inside a class of discontinuous quintic polynomial differential systems. Our results show that the lower bound and the upper bound of the number of limit cycles, 8 and 10 respectively, that can bifurcate from the period annulus around the center.

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Acknowledgements

I wish to thank Xiuli Cen for her helpful suggestions and to Dongming Wang for his profound concern and encouragement. I also wish to thank the reviewers for their valuable comments that have helped to improve the presentation of the paper.

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Correspondence to Bo Huang.

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The project was supported by China Scholarship Council, No. 201806020128.

Appendix

Appendix

$$\begin{aligned} W_{61}(\omega )=&\,2\Big (-(2-\omega ^2)(15{\omega }^{12}-515{\omega }^{10}+4307{\omega }^{8}-14{,}496{\omega }^{6}+24{,}528{\omega }^{4}\\&-20{,}736{\omega }^{2}+6912)\sqrt{1-\omega ^2}+{\omega }^{16}-113{\omega }^{14}+1983{\omega }^{12}-13{,}052{\omega }^{10}\\&+43{,}630{\omega }^{8}-83{,}232{\omega }^{6}+92{,}256{\omega }^{4}-55{,}296 {\omega }^{2}+13{,}824\Big ),\\ W_{62}(\omega )=&\,2\Big (15{\omega }^{12}-445{\omega }^{10}+2701{\omega }^{8}-6816{\omega } ^{6}+9168{\omega }^{4}-6912{\omega }^{2}\\&+2304\Big )\sqrt{1-\omega ^2}-(2-\omega ^2)({\omega }^{12}-217{\omega }^{10}+2185{\omega }^{8}-6240{\omega }^{6}\\&+8880{\omega }^{4}-6912{\omega }^{2}+2304),\\ W_{71}(\omega )=&\,-(1-\omega ^2)\Big (2\Big (55{\omega }^{20}-3965{\omega }^{18}+34{,}706{\omega }^{16}-87{,}929{ \omega }^{14}\\&-70{,}542{\omega }^{12}+838{,}995{\omega }^{10}-2{,}036{,}497{\omega }^{8}+2{,}680{,}896{\omega }^{6}-2{,}139{,}024{\omega }^{4}\\&+979{,}200{\omega }^{2}-195{,}840\Big )\sqrt{1-\omega ^2}+(2-\omega ^2)\Big (4{\omega }^{20}+1415{\omega }^{18}\\&-25{,}573{\omega }^{16}+92{,}751{\omega }^{14}-1319{\omega }^{12}-687{,}111{\omega }^{10}+1{,}871{,}629{\omega }^{8}\\&-2{,}582{,}976{\omega }^{6}+2{,}114{,}544{\omega }^{4}-979{,}200{\omega }^{2}+195{,}840\Big )\Big ), \end{aligned}$$
$$\begin{aligned} W_{72}(\omega )=&\,4\Big ((2-\omega ^2)(15{\omega }^{20}-560{\omega }^{18}+1315{\omega }^{16}+18{,}712{ \omega }^{14}-133{,}543{\omega }^{12}\\&+424{,}314{\omega }^{10}-807{,}614{\omega }^{8}+985{,}728{\omega }^{6}-764{,}832{\omega }^{4}+345{,}600{\omega }^{2}\\&-69{,}120)\sqrt{1-\omega ^2}-2(1-\omega ^2)\Big (67{\omega }^{20}-751{\omega }^{18}-514{\omega }^{16}\\&+31{,}416{\omega }^{14}-170{,}195{\omega }^{12}+484{,}866{\omega }^{10}-868{,}118{\omega }^{8}+1{,}020{,}288{\omega }^{6}\\&-773{,}472{\omega }^{4}+345{,}600{\omega }^{2}-69{,}120\Big )\Big ),\\ W_{73}(\omega )=&\,2\Big (-15{\omega }^{20}+535{\omega }^{18}-1400{\omega }^{16}-20{,}261{ \omega }^{14}+155{,}972{\omega }^{12}\\&-510{,}069{\omega }^{10}+970{,}375{\omega }^{8}-1{,}169{,}856{\omega }^{6}+897{,}264{\omega }^{4}-403{,}200{\omega }^{2}\\&+80{,}640\Big )\sqrt{1-\omega ^2}+(1-\omega ^2)(2-\omega ^2)\Big (269{\omega }^{16}-1878{\omega }^{14}\\&-10{,}645{\omega }^{12}+105{,}830{\omega }^{10}-335{,}515{\omega }^{8}+564{,}912{\omega }^{6}-564{,}624{\omega }^{4}\\&+322{,}560{\omega }^{2}-80{,}640\Big ),\\ W_{81}(\omega )=&\,(1-\omega ^2)\Big (2(2-\omega ^2)\Big (15{\omega }^{28}-3805{\omega }^{26}+15{,}445{\omega }^{24}+423{,}144{ \omega }^{22}\\&-3{,}256{,}879{\omega }^{20}+10{,}877{,}937{\omega }^{18}-20{,}260{,}675 {\omega }^{16}+19{,}667{,}742{\omega }^{14}\\&+236{,}463{\omega }^{12}-30{,}298{,}655{\omega }^{10}+47{,}913{,}763{\omega }^{8}-41{,}908{,}896{\omega }^{6}\\&+22{,}884{,}336{\omega }^{4}-7{,}338{,}240{\omega }^{2}+1{,}048{,}320\Big )\sqrt{1-\omega ^2}\\&-(1-\omega ^2)\Big (166{\omega }^{28}-30{,}635{\omega }^{26}-114{,}628{\omega }^{24}+2{,}981{,}161{\omega }^{22}\\&-16{,}868{,}075{\omega }^{20}+49{,}560{,}193{\omega }^{18}-85{,}016{,}753{\omega }^{16}+75{,}241{,}204{\omega }^{14}\\&+11{,}894{,}813{\omega }^{12}-133{,}688{,}060{\omega }^{10}+199{,}919{,}500{\omega }^{8}-170{,}780{,}544{\omega }^{6}\\&+92{,}061{,}504{\omega }^{4}-29{,}352{,}960{\omega }^{2}+4{,}193{,}280\Big )\Big ),\\ W_{82}(\omega )=&\,-3(1-\omega ^2)\Big (4\Big (375{\omega }^{28}+2350{\omega }^{26}-80{,}645{\omega }^{24}+612{,}991{ \omega }^{22}\\&-2{,}378{,}834{\omega }^{20}+5{,}284{,}716{\omega }^{18}-5{,}787{,}406{ \omega }^{16}-2{,}489{,}222{\omega }^{14}\\&+21{,}368{,}654{\omega }^{12}-41{,}457{,}795{\omega }^{10}+48{,}346{,}871{\omega }^{8}-37{,}666{,}336{\omega }^{6}\\&+19{,}459{,}376{\omega }^{4}-6{,}083{,}840{\omega }^{2}+869{,}120\Big )\sqrt{1-\omega ^2}+(2-\omega ^2)\Big (40{\omega }^{28}\\&+2843{\omega }^{26}+86{,}650{\omega }^{24}-877{,}807{ \omega }^{22}+3{,}952{,}955{\omega }^{20}-9{,}806{,}021{\omega }^{18}\\&+12{,}310{,}193{\omega }^{16}+1{,}445{,}216{\omega }^{14}-36{,}961{,}541{\omega }^{12}+77{,}263{,}870{\omega }^{10}\\&-93{,}173{,}318{\omega }^{8}+74{,}028{,}992{\omega }^{6}-38{,}701{,}472 {\omega }^{4}+12{,}167{,}680{\omega }^{2}\\&-1{,}738{,}240\Big )\Big ), \end{aligned}$$
$$\begin{aligned} W_{83}(\omega )=&\,3\Big (2(2-\omega ^2)(150{\omega }^{28}+1635{\omega }^{26}-30{,}525{\omega }^{24}+222{,}020{ \omega }^{22}\nonumber \\&-751{,}166{\omega }^{20}+1{,}008{,}949{\omega }^{18}+1{,}539{,}615{ \omega }^{16}-10{,}218{,}942{\omega }^{14}\nonumber \\&+24{,}688{,}074{\omega }^{12}-37{,}922{,}615{\omega }^{10}+40{,}617{,}435{\omega }^{8}-30{,}574{,}496{\omega }^{6}\nonumber \\&+15{,}559{,}536{\omega }^{4}-4{,}829{,}440{\omega }^{2}+689{,}920 )\sqrt{1-\omega ^2}+(1-\omega ^2)(40{\omega }^{28}\nonumber \\&-16{,}931{\omega }^{26}+214{,}548{\omega }^{24}-1{,}175{,}771{\omega }^{22}+3{,}287{,}235{\omega }^{20}\nonumber \\&-3{,}211{,}611{\omega }^{18}-9{,}971{,}385{\omega }^{16}+48{,}766{,}876{\omega }^{14}-109{,}237{,}393{\omega }^{12}\nonumber \\&+161{,}159{,}900{\omega }^{10}-168{,}158{,}188{\omega }^{8}+124{,}367{,}744{\omega }^{6}-62{,}583{,}104{\omega }^{4}\nonumber \\&+19{,}317{,}760{\omega }^{2}-2{,}759{,}680)\Big ),\nonumber \\ W_{84}(\omega )=&\,2\Big ((-225{\omega }^{28}-2760{\omega }^{26}+46{,}180{\omega }^{24}-263{,}710{ \omega }^{22}+704{,}483{\omega }^{20}\nonumber \\&-263{,}096{\omega }^{18}-4{,}685{,}954{ \omega }^{16}+18{,}586{,}360{\omega }^{14}-40{,}396{,}567{\omega }^{12}\nonumber \\&+59{,}375{,}110{\omega }^{10}-62{,}023{,}406{\omega }^{8}+45{,}943{,}872{\omega }^{6}-23{,}149{,}152 {\omega }^{4}\nonumber \\&+7{,}150{,}080{\omega }^{2}-1{,}021{,}440 )\sqrt{1-\omega ^2}-(1-\omega ^2)(2-\omega ^2)(15{\omega }^{24}\nonumber \\&-13{,}694{\omega }^{22}+88{,}328{\omega }^{20}-246{,}412{ \omega }^{18}+65{,}615{\omega }^{16}+1{,}695{,}988{\omega }^{14}\nonumber \\&-6{,}144{,}362{\omega }^{12}+12{,}122{,}284{\omega }^{10}-15{,}817{,}411{\omega }^{8}+14{,}142{,}480{\omega }^{6}\nonumber \\&-8{,}446{,}416{\omega }^{4}+3{,}064{,}320{\omega }^{2}-510{,}720)\Big ). \end{aligned}$$
(A.1)
$$\begin{aligned} {\overline{R}}_5(s)=&\,2{,}941{,}225{s}^{144}+579{,}421{,}325{s}^{140}+47{,}227{,}910{,}100{s}^{136}\nonumber \\&+1{,}926{,}032{,}559{,}325{s}^{132}+37{,}660{,}303{,}681{,}105{s}^{128}\nonumber \\&+246{,}136{,}259{,}034{,}120{s}^{124}-1{,}823{,}983{,}697{,}116{,}000{s}^{120}\nonumber \\&-31{,}213{,}705{,}481{,}805{,}336{s}^{116}-67{,}111{,}899{,}806{,}222{,}348{s}^{112}\nonumber \\&+458{,}988{,}474{,}275{,}583{,}420{s}^{108}-542{,}168{,}545{,}377{,}471{,}824{s}^{104}\nonumber \\&-10{,}701{,}062{,}367{,}993{,}919{,}908{s}^{100}+37{,}889{,}550{,}367{,}024{,}958{,}340{s}^{96}\nonumber \\&+377{,}847{,}678{,}557{,}396{,}528{,}280{s}^{92}+2{,}773{,}832{,}017{,}881{,}298{,}541{,}664{s}^{88}\nonumber \\&+8{,}428{,}988{,}256{,}294{,}203{,}968{,}248{s}^{84}-24{,}544{,}609{,}741{,}261{,}925{,}789{,}698{s}^{80}\nonumber \\&-102{,}167{,}600{,}587{,}970{,}326{,}132{,}642{s}^{76}+27{,}554{,}822{,}107{,}811{,}495{,}115{,}256{s}^{72}\nonumber \\&+409{,}002{,}187{,}750{,}390{,}867{,}930{,}014{s}^{68}-234{,}586{,}698{,}910{,}862{,}607{,}724{,}386{s}^{64}\nonumber \\&-789{,}956{,}047{,}449{,}543{,}547{,}297{,}288{s}^{60}+2{,}239{,}015{,}322{,}570{,}858{,}864{,}250{,}464{s}^{56}\nonumber \\&+2{,}750{,}837{,}821{,}133{,}545{,}249{,}466{,}776{s}^{52}-6{,}276{,}621{,}919{,}879{,}816{,}057{,}033{,}276{s}^{48}\nonumber \\&-6{,}266{,}251{,}256{,}585{,}257{,}436{,}149{,}796{s}^{44}+5{,}405{,}386{,}853{,}541{,}122{,}104{,}297{,}648{s}^{40}\nonumber \\&+2{,}423{,}125{,}568{,}081{,}097{,}333{,}018{,}684{s}^{36}-2{,}164{,}583{,}789{,}856{,}807{,}700{,}884{,}300{s}^{32}\nonumber \\&+698{,}517{,}087{,}417{,}367{,}164{,}419{,}816{s}^{28}-56{,}252{,}042{,}857{,}985{,}960{,}444{,}256{s}^{24}\nonumber \\&-22{,}837{,}292{,}333{,}731{,}650{,}878{,}712{s}^{20}+9{,}876{,}705{,}429{,}478{,}534{,}369{,}569{s}^{16}\nonumber \\&-1{,}833{,}722{,}742{,}234{,}923{,}676{,}675{s}^{12}+221{,}276{,}265{,}598{,}049{,}133{,}780{s}^{8}\nonumber \\&-15{,}471{,}983{,}904{,}966{,}371{,}475{s}^{4}+678{,}361{,}492{,}782{,}891{,}225. \end{aligned}$$
(A.2)
$$\begin{aligned} e_0(s)=&\,-(s^2+1)^4(79{,}695{s}^{40}-15{,}960{s}^{38}+838{,}872{s}^{36}-201{,}288{s}^{34}\nonumber \\&-2{,}616{,}123{s}^{32}+487{,}072{s}^{30}-55{,}859{,}248{s}^{28}+17{,}699{,}424{s}^{26}\nonumber \\&-88{,}304{,}106{s}^{24}+49{,}708{,}464{s}^{22}-70{,}523{,}328{s}^{20}+65{,}079{,}056{s}^{18}\nonumber \\&+848{,}553{,}226{s}^{16}-301{,}408{,}224{s}^{14}-62{,}704{,}848{s}^{12}-351{,}147{,}552{s}^{10}\nonumber \\&-651{,}317{,}877{s}^{8}+53{,}955{,}048{s}^{6}+94{,}296{,}552{s}^{4}-6{,}015{,}240{s}^{2}\nonumber \\&-12{,}442{,}815),\nonumber \\ e_1(s)=&\,12(s^2+1)^2(26{,}600{s}^{44}-5390{s}^{42}+264{,}319{s}^{40}-63{,}832{s}^{38}\nonumber \\&-962{,}464{s}^{36}+253{,}902{s}^{34}-28{,}179{,}723{s}^{32}+8{,}822{,}464{s}^{30}\nonumber \\&-36{,}661{,}856{s}^{28}+21{,}718{,}164{s}^{26}+50{,}531{,}446{s}^{24}+295{,}120{s}^{22}\nonumber \\&-5{,}467{,}504{s}^{20}-23{,}416{,}964{s}^{18}+424{,}147{,}946{s}^{16}-208{,}591{,}744{s}^{14}\nonumber \\&-154{,}529{,}928{s}^{12}-146{,}158{,}502{s}^{10}-300{,}630{,}533{s}^{8}+40{,}869{,}192{s}^{6}\nonumber \\&+57{,}307{,}152{s}^{4}-8{,}295{,}210{s}^{2}-5{,}845{,}455),\nonumber \\ e_2(s)=&\,6(1-s^4)(64{,}505{s}^{44}-13{,}580{s}^{42}+354{,}377{s}^{40}-98{,}264{s}^{38}\nonumber \\&-12{,}400{,}449{s}^{36}+3{,}260{,}308{s}^{34}-96{,}433{,}881{s}^{32}+33{,}863{,}776{s}^{30}\nonumber \\&-5{,}486{,}150{s}^{28}+40{,}567{,}272{s}^{26}+367{,}029{,}434{s}^{24}-113{,}360{,}784{s}^{22}\nonumber \\&+974{,}759{,}326{s}^{20}-585{,}098{,}808{s}^{18}-1{,}973{,}504{,}242{s}^{16}+309{,}305{,}696{s}^{14}\nonumber \\&-100{,}864{,}067{s}^{12}+531{,}765{,}668{s}^{10}+1{,}085{,}558{,}957{s}^{8}-166{,}478{,}424{s}^{6}\nonumber \\&-219{,}764{,}685{s}^{4}+30{,}173{,}220{s}^{2}+22{,}629{,}915),\nonumber \\ e_3(s)=&\,4(s^2+1)(s^2-1)^2(33{,}810{s}^{42}-42{,}000{s}^{40}-27{,}363{s}^{38}+25{,}131{s}^{36}\nonumber \\&-14{,}559{,}057{s}^{34}+18{,}767{,}503{s}^{32}-69{,}564{,}044{s}^{30}+94{,}875{,}724{s}^{28}\nonumber \\&+192{,}702{,}600{s}^{26}-257{,}244{,}372{s}^{24}+579{,}728{,}646{s}^{22}- 856{,}279{,}286{s}^{20}\nonumber \\&-789{,}453{,}014{s}^{18}+1{,}115{,}190{,}706{s}^{16}-288{,}657{,}564 {s}^{14}+612{,}375{,}324{s}^{12}\nonumber \\&+566{,}578{,}110{s}^{10}-1{,}056{,}132{,}756{s}^{8}-97{,}955{,}067{s}^{6}+240{,}057{,}699{s}^{4}\nonumber \\&+5{,}845{,}455{s}^{2}-27{,}723{,}465),\nonumber \\ e_4(s)=&\,3(s^2-1)^4(4095{s}^{40}+1116{s}^{36}-2{,}104{,}223{s}^{32}-12{,}655{,}840{s}^{28}\nonumber \\&+32{,}270{,}886{s}^{24}+138{,}275{,}320{s}^{20}-162{,}868{,}846{s}^{16}-161{,}858{,}880{s}^{12}\nonumber \\&+244{,}777{,}323{s}^{8}-71{,}051{,}316{s}^{4}+10{,}939{,}005). \end{aligned}$$
(A.3)
$$\begin{aligned} {\overline{R}}_6(s)=&\,4095{s}^{40}+1116{s}^{36}-2{,}104{,}223{s}^{32}-12{,}655{,}840{s}^{28}+ 32{,}270{,}886{s}^{24}\nonumber \\&+138{,}275{,}320{s}^{20}-162{,}868{,}846{s}^{16}-161{,}858{,}880 {s}^{12}+244{,}777{,}323{s}^{8}\nonumber \\&-71{,}051{,}316{s}^{4}+10{,}939{,}005,\nonumber \\ {\overline{R}}_7(s)=&\,1715{s}^{60}+42{,}875{s}^{56}-546{,}245{s}^{52}-2{,}0297{,}949{s}^{48}- 147{,}278{,}321{s}^{44}\nonumber \\&-315{,}876{,}681{s}^{40}-601{,}421{,}497{s}^{36}+4{,}168{,}696{,}335{s}^{32}\nonumber \\&+26{,}747{,}311{,}673{s}^{28}+9{,}133{,}930{,}257{s}^{24}-126{,}618{,}219{,}791{s}^{20}\nonumber \\&-87{,}973{,}596{,}375{s}^{16}+100{,}648{,}007{,}621{s}^{12}-43{,}177{,}388{,}163{s}^{8}\nonumber \\&+7{,}311{,}111{,}885{s}^{4}-823{,}627{,}035,\nonumber \\ {\overline{R}}_8(s)=&\,18{,}907{,}875{s}^{132}-1{,}272{,}229{,}875{s}^{128}-367{,}280{,}970{,}000{s}^{124}\nonumber \\&-8{,}402{,}945{,}300{,}400{s}^{120}+302{,}875{,}523{,}517{,}000{s}^{116}\nonumber \\&+1{,}073{,}179{,}283{,}636{,}600{s}^{112}-23{,}385{,}311{,}222{,}383{,}600{s}^{108}\nonumber \\&-378{,}877{,}883{,}529{,}465{,}040{s}^{104}-3{,}036{,}210{,}018{,}795{,}329{,}740{s}^{100}\nonumber \\&-3{,}930{,}131{,}622{,}728{,}001{,}460{s}^{96}+17{,}771{,}179{,}533{,}292{,}826{,}480{s}^{92}\nonumber \\&-136{,}910{,}752{,}996{,}471{,}952{,}176{s}^{88}-559{,}090{,}161{,}627{,}967{,}791{,}304{s}^{84}\nonumber \\&+1{,}192{,}444{,}509{,}254{,}152{,}751{,}240{s}^{80}+3{,}564{,}834{,}377{,}135{,}010{,}561{,}040{s}^{76}\nonumber \\&-28{,}573{,}332{,}758{,}236{,}001{,}727{,}440{s}^{72}+58{,}544{,}479{,}926{,}871{,}099{,}855{,}858{s}^{68}\nonumber \\&-161{,}247{,}515{,}767{,}368{,}120{,}376{,}338{s}^{64}+241{,}479{,}484{,}792{,}277{,}279{,}975{,}760{s}^{60}\nonumber \\&-127{,}216{,}223{,}013{,}187{,}862{,}735{,}760{s}^{56}+368{,}552{,}348{,}471{,}240{,}996{,}093{,}560{s}^{52}\nonumber \\&-359{,}441{,}894{,}506{,}586{,}390{,}307{,}128{s}^{48}-201{,}269{,}314{,}484{,}307{,}038{,}913{,}872{s}^{44}\nonumber \\&-16{,}351{,}303{,}318{,}552{,}845{,}993{,}200{s}^{40}+54{,}040{,}256{,}823{,}508{,}990{,}124{,}020{s}^{36}\nonumber \\&+3{,}953{,}552{,}508{,}620{,}315{,}027{,}980{s}^{32}+53{,}798{,}571{,}199{,}144{,}791{,}586{,}896{s}^{28}\nonumber \\&-26{,}980{,}369{,}081{,}417{,}936{,}081{,}296{s}^{24}+4{,}246{,}521{,}581{,}708{,}650{,}172{,}040{s}^{20}\nonumber \\&-424{,}695{,}380{,}941{,}145{,}933{,}640{s}^{16}+299{,}094{,}958{,}378{,}505{,}520{s}^{12}\nonumber \\&+1{,}046{,}551{,}688{,}501{,}734{,}800{s}^{8}-153{,}981{,}468{,}243{,}481{,}725{s}^{4}\nonumber \\&+2{,}097{,}592{,}742{,}062{,}125. \end{aligned}$$
(A.4)

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Huang, B. Limit Cycles for a Discontinuous Quintic Polynomial Differential System. Qual. Theory Dyn. Syst. 18, 769–792 (2019). https://doi.org/10.1007/s12346-018-00312-6

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Keywords

  • Averaging method
  • Center
  • Discontinuous quintic system
  • Limit cycle
  • Period annulus

Mathematics Subject Classification

  • 34C05
  • 34C07