一类具鞍点-可视折点的平面Filippov系统的分支分析.pdf

Advances in Applied Mathematics A?,2024,13(4),1234-1247Published Online April 2024 in Hans.https:/www.hanspub.org/journal/aamhttps:/doi.org/10.12677/aam.2024.134114aQ:-:?FilippovX?|?nO?,?H vF2024c3?9FF2024c4?8FuF2024c4?15F?k 2 Q:-:?Filippov X?m.k,|5.9w?,?/?.g,y?Q:-:NC3.Q:|:?-E V V1|.AO/,y?3 1 0?m3 1|-.?,?(JL,?A5X?Q:-:?|yu?Filippov X.cFilippovXQ:-:5.|Bifurcation Analysis for a Class of PlanarFilippov Systems with Saddle-Visible FoldSingularityYilin ChenSchool of Mathematics and Statistics,Changsha University of Science and Technology,ChangshaHunanReceived:Mar.9th,2024;accepted:Apr.8th,2024;published:Apr.15th,2024:?.aQ:-:?FilippovX?|J.A?,2024,13(4):1234-1247.DOI:10.12677/aam.2024.134114?AbstractIn this paper,we investigate the generic unfolding of planar Filippov systems withcodimension-2 saddle-visible fold singularity.Firstly,the bifurcation diagrams aregiven completely by means of normal forms and sliding mode dynamics.Secondly,itis proved that there are pseudo boundary saddle bifurcation and collisions of visibletwo-fold singularity V V1bifurcation near saddle-visible fold singularity.In particular,we shown that there exist two codimension-1 bifurcation curves in the parameter spaceof 1 2!)3?+.,?,3X,?fX35Y,UyE,?9?4,5.N5X(PWL)?1?613,uy,?a.?PWL X3U?Filippov X?d,AO3w?|m.u Filippov X?Y?K,14,15?1?,16?Filippov X 2?a.d?,3z 16|5m?2?-?.?11 16?-?(,?mAB?4?.C?17?Filippov X?2?-?,y?B?4d?-:?59?-L.X,uY5?DOI:10.12677/aam.2024.1341141235A?Ek5,du6/?K,(JE,k.u 16 5?L,uQ-:,du3?-QfXk?5,?E,.,?T?Q-:?a.,uy?ko?d?2 Q-:.u Filippov X?Q-?K,8cvk?.u,3?/?d?2 Q-:,=B?2.Q:-:NC?K.LXz5.|w?,uylQ:-:NC?).Q:V V1a.|9?Xk?Q:?-.?/?9?-?L.?|Xe.31 2!,?Filippov X?#?5.31 3!?X?:w?.?(Jy9A?31 4!?.?(?1?o(.2.?5.?e Filippov X x y!=(X(x,y),h(x,y)0,Y(x,y),h(x,y)0?=?(x,y)T|h(x,y)0;(ii)w?:s=p|Xh(p)0;(iii)0,Y h(p).:.AO/,?/:|,?a.?:.-X(x,y)=(X1(x,y),X2(x,y)T,Y(x,y)=(Y1(x,y),Y2(x,y)T?h(x,y)=x,3e,X(2.1)?w?)x=0,y=gs(y)=X1(0,y)Y2(0,y)Y1(0,y)X2(0,y)X1(0,y)Y1(0,y).(2.3)d,3z 10?,e p s g0s(y)0,K:p-?(:;XJ p svg0s(y)0,p ev g0s(y)0(X)2h 0),K p X(2.1)u X?():;e Y h(p)=0 (Y)2h(p)0),K pX(2.1)u Y?():./2.1,keAa.?:.2.2.uX(2.1),:(i)XJ p d.Q:-E/?,o p Q-:;(ii)XJ p d():-E/?,o p ()?-:;(iii)XJ p d:-E/?,o p-?-:.?8I?Q-:?|,d3X(2.1)=(1,2).X(2.1)-?x y!=X1(x,y;)X2(x,y;)!,x 0,Y1(x,y;)Y2(x,y;)!,x.Q:(Y1,Y2)T?:-E/?,=X 1.Figure 1.O s e,the boundary saddle-visiblefold singularitiy of System(2.4)1.O s e,X(2.4)?.Q:-:.?y 2 Q-:?3,7Le:Q:?A?m7L?u,Q:?A?7L.b?(H0)b+1(0)=b+2(0)=b1(0)=0,a+12(0)0,det(J(0)0,b2(0)03?.du b+1(0)=X1(0,0;0)=0 9 a+12(0)=X1(0,0;0)y6=0,Kn3 Cry=1()(2.6)?u?kk=p21+22k 1(0)=0 X1(0,1();)0.z 10 u,eL?E?N?x y!=10a+22()a+12()!xy 1()!(2.7)X(2.4)?z,-()=a+12()b+2(),KX(2.4)?mfXz Li enard 5.:(x=tr(J()x y+O(x2+y2),y=det(J()x ()+O(x2+y2),x 0.(2.8)()=tr(J()2pdet(J(),()=()pdet(J()(2.9)DOI:10.12677/aam.2024.1341141238A?CO(x,y,t)xpdet(J(),y,tpdet(J()!,(2.10)?5.(2.8)=zXe/x=2()x y+O(x2+y2),y=x ()+O(x2+y2),x 0.(2.11)aq/,3?(2.7)C(2.10)?Ce,(2.4)?fX?1z.db?(H0),Y1(0,0;0)=0,Y1(0,0;0)y=a12(0)a+12(0)0,udn,3 Cry=2()(2.12)?2(0)=0 9 Y1(0,2();)0,=X(0,2()T(2.4)?fX?:.b?u()2()Od(2.9)(2.12).?=(1,2)T9=()=(u(),2()T,(2.13)o3Q:-:NC,kXeK?5.K2.3.?(H0),9?X1(0,0;0)1X1(0,0;0)2X2(0,0;0)1X2(0,0;0)2?6=a+12(0)a12(0)?Y1(0,0;0)1Y1(0,0;0)2Y2(0,0;0)1Y2(0,0;0)2?,(2.14),|L1?,K(i)(2.13)?C?;(ii)3(2.7),(2.10)(2.13)?Ce,X(2.4)CXe?d5.x y!=F+(x,y;)G+(x,y;)!,x 0,F(x,y;)G(x,y;)!,x 0,G(0,0;0)0,F(0,2;)0,F(0,0;0)y 0,F(0,0;0)2 0,G(0,0;0)0,F(0,2;)0,F(0,0;0)y=a12(0)a+12(0)0.d,(0,2)Tou|(F,G)T:.,?d F(0,2;)0,3 CrF(x,y;)?F(x,y;)=(y 2)F(x,y;).XF(0,0;0)2=DOI:10.12677/aam.2024.1341141240A?F(0,0;0)y.Q:?|(J,IX(2.15)?:w?,(Jde?K.K3.1.?(H0),uv?kk,X(2.15)e(i)e 1 0,?:O u|(F+,G+)T?:;?e 1 0,K?:O u|(F+,G+)T?:.(ii)?1 0,X(2.15)kJQ:.d?,?1 0,Q:S k-6/-6/,O3(0,ys()T(0,yu()T?,ys()=1?y0s+O?q21+22?,yu()=1?y0u+O?q21+22?,(3.1)y0s=q1+(0)2 0,y0u=q1+(0)2 0.(3.2)y.(i)z 12?O?,ey(ii).dK 2.3(ii)O?F+(0,0;0,0)xF+(0,0;0,0)yG+(0,0;0,0)xG+(0,0;0,0)y?=?20110?=1,9 F+(0,0;0,0)=0,G+(0,0;0,0)=0,un(2.15)?fXkQ:S=(x(1,2),y(1,2)T.F+(0,0;0,2)0 G+(0,0;0,2)0,kx(0,2)0 9 y(0,2)0.VX3 Cr x(1,2)y(1,2)?x(1,2)=1 x(1,2)y(1,2)=1 y(1,2).qd?5?x(0,0)1=1 9y(0,0)1=20,kx()=x(1,2)=1?1+O?q21+22?,y()=y(1,2)=1?20+O?q21+22?.(3.3)Xuv?kk,?1 0 Q:S J?.z 18?n 3.2.1,Q:S kC-6/y=hs(x)C-6/y=hu(x),hs(x()=hu(x()=y(),h0s(x()=y0u,h0u(x()=y0s.DOI:10.12677/aam.2024.1341141241A?dhs(x)=y()y0u(x x()+O?(x x()2?(3.4)hu(x)=y()y0s(x x()+O?(x x()2?,(3.5)-6/-6/O3(0,ys()T(0,yu()T?.?,O3(3.4)(3.5)-x=0,Kkys()=y()+y0ux()+O?(x()2?(3.6)yu()=y()+y0sx()+O?(x()2?.(3.7)(3.3)“?(3.6)(3.7)?ys()=1?y0s+O?q21+22?yu()=1?y0u+O?q21+22?.(3.8)y.?e5,uX(2.15)?w?,XeK.K3.2.uv?kk 26=0,KuX(2.15)ke(i)XJ 2 0,K3B?c=?(0,y)T|0 y 2?,dE c=?(0,y)T|y 2?;(ii)XJ 2 0,K3B?c=?(0,y)T|2 y 0?,?C s=?(0,y)T|y 0,w,?X(2.15)3B?c=?(0,y)T|0 y 2?;e 2 0 KB?C c=?(0,y)T|2 y 0?.nOOw?.Q:|:?-EV V1|.d?,3 1 0,l2=(1,2)|1 0,2=0,l3=(1,2)|1=0,2 0 9 l0=(1,2)|1 0,2 0,R2=(1,2)|1 1(1),R3=(1,2)|1 0,0 2 1(1),R4=(1,2)|1 0,2(1)2 0,R5=(1,2)|1 0,2 0,2.,K3e,-2=i(1)(i 1,2)OA C23 C45.m?Xe 2.Figure 2.Parameter regions,bifurcation curves and bifurcationdiagram of the system(2.15)near the saddlevisible fold singu-larity O 2.X(2.15)3Q:-:O NC?m!|-9?eLAn5?TQ-:a.?(J.n4.1.?|6=0,kk?,ouX(2.15)e?:DOI:10.12677/aam.2024.1341141243A?(i)e 2 0,(1,2)C12,K3 l1=(1,2)|1=0,2 0|-,u).-Q:|;(ii)e 1 0,(1,2)C61,K3 l2=(1,2)|1 0,2=0|-,u):?-E V V1|;(iii)e 2 0,(1,2)C56,K3 l3=(1,2)|1=0,2.-Q:|.y.dK 3.1 K 2.3,u(1,2)|1 0,X(2.15)kQ:S!:(0,2)T.X 1?,3 C12.,Q:S?:O-E/.Q:.?u R1=(1,2)|1 0,2 0,K3:O,O“.Q:.pvkf.d,z 15?,3 l1=(1,2)|1=0,2 0|-u)?.-Q:|,y?(i).ey(ii).uv?kk,e 1 0,KdK 3.1(i)K 2.3(iii),|(F+,G+)Tk:O ,?|(F,G)Tk:(0,2)T.?,3.C61,:-E,=O=(0,2)T=(0,0)T.,?,dK 3.2,3C61.,X(2.15)kw:?w?.?26=0,1 0,:O(0,2)Ty?B?,;l+B?,?1 0,;Cl B?+.d,z 15?,3 l2=(1,2)|1 0,2=0|-u)?:?-E V V1|.=(ii).u(iii),yaqu(i)?y,?.y.?XJ3Q:S,=1 0,K?)?|-.den5,A?X 3 4.Figure 3.Bifurcation diagram of thesystem(2.15)with 1 0 3.1 0 X(2.15)?n4.2.?1 0,ou?kk,3 Cr1(1)=y0u1+O?21?,(4.2)DOI:10.12677/aam.2024.1341141244A?Figure 4.Bifurcation diagram of thesystem(2.15)with 1 0 and 2 0 4.1 0 2 0 X(2.15)?X(2.15)k?Q:S:(0,2)T?.y.?1 0,2=2/1.-yu(1,2)=yu(1,12)/1,(4.3)KdK 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