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1.5,x2=-1.5..1.5,color=grey,grid=[100,100]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "g3_2:=implicitplot((abs(x1))^(3/2)+(abs(x2))^(3 /2)=1,x1=-1.5..1.5,x2=-1.5..1.5,\n color=magenta,grid=[50,50]):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "g2:=implicitplot(x1^2+x2^2=1 ,x1=-1.5..1.5,x2=-1.5..1.5,\n color=red,grid=[50,50]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "g4:=implicitplot(x1^4+x2^4=1,x1=-1. 5..1.5,x2=-1.5..1.5,\n color=blue,grid=[50,50]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "g50:=implicitplot(x1^50+x2^50=1,x1=-1.5..1.5, x2=-1.5..1.5,\n color=black,grid=[100,100]):" }}}{EXCHG }{EXCHG } {EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(g101 _100,g3_2,g2,g4,g50,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "\316\357\356\360\355\340\377 \364\363\355\352\366\350\377 \362\356\367\352\350" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "sf_p:=proc (P)\n unapply(psi1*P[1]+psi2*P[2],psi1,psi2);\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "\316\357\356\360\355\340\377 \364\363\355 \352\366\350\377 \375\353\353\350\357\361\340" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "sf_ell:=proc(R)\n unapply(sqrt(R[1]^2*psi1^2+R[2]^2*p si2^2),psi1,psi2):\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 " \316\357\356\360\355\340\377 \364\363\355\352\366\350\377 \361\363\354 \354\373 \344\342\363\365 \354\355\356\346\345\361\362\342" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "sf_sum:=proc(sf1,sf2)\n unapply(sf1(psi1,ps i2)+sf2(psi1,psi2),psi1,psi2):\nend proc:" }}}{EXCHG }{EXCHG {PARA 0 " " 0 "" {TEXT -1 37 "\316\357\356\360\355\340\377 \364\363\355\352\366 \350\377 \361\343\353\340\346\345\355\355\356\343\356 \354\355\356\346 \345\361\362\342\340" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "sf_smooth: =proc(sf1,sf2,mu)\n unapply((sf1(psi1,psi2)+sf2(psi1,psi2))/2+\n \+ sqrt(mu*(psi1^2+psi2^2)+(sf1(psi1,psi2)-sf2(psi1,psi2))^2/4),\n p si1,psi2):\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 267 "p lot_sf:=proc(sf,N,clr)\n local dc1,dc2:\n dc1:=subs([psi1=cos(alpha),p si2=sin(alpha)],diff(sf(psi1,psi2),psi1));\n dc2:=subs([psi1=cos(alpha ),psi2=sin(alpha)],diff(sf(psi1,psi2),psi2));\n plot([dc1,dc2,alpha=0. .2*Pi],numpoints=N,color=clr,scaling=constrained):\nend proc:" }}} {EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sf_ell_1:=sf_ell([0. 1,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sf_ell_2:=sf_ell( [1,0.1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sf_sum_ells:=s f_sum(sf_ell_1,sf_ell_2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "g1:=plot_sf(sf_ell_1,90,red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g2:=plot_sf(sf_ell_2,90,blue):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "g_12:=plot_sf(sf_sum_ells,90,magenta):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "display(g1,g2,g_12);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=[1,1];\nsf_A:=sf_p(A);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "B:=[-1,-2];\nsf_B:=sf_p(B) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sf_AB_smooth:=sf_smoot h(sf_A,sf_B,0.01);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gAB:= plot_sf(sf_AB_smooth,90,magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "AB:=pointplot([A,B],color=blue,connect=true):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "display(gAB,AB);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C:=[2,-1];\nsf_C:=sf_p(C);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sf_tr:=sf_smooth(sf_AB_smoot h,sf_C,0.01):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "g_tr:=plot _sf(sf_tr,90,red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "g_ABC :=pointplot([A,B,C,A],color=grey,connect=true):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "display(g_tr,gAB,g_ABC);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "c_smooth:=proc(mu,psi1,psi2)\n sqrt(mu*psi1^2 +psi2^2)+sqrt(psi1^2+mu*psi2^2);\nend proc:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "dc_smooth:=proc(mu,psi1,psi2)\n local r1,r2:\n r1:=d iff(c_smooth(mu,psi1,psi2),psi1):\n r2:=diff(c_smooth(mu,psi1,psi2),ps i2):\n return [r1,r2]:\nend proc:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "dc_smooth(mu,psi1,psi2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sub s([psi1=cos(a),psi2=sin(a)],%);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "film:=seq(plot([op(%),a=0..2*Pi]),\n mu=[1,0.8,0.7,0.6,0.5,0.4,0.3,0. 2,0.1,0.05,0.025,0.01,0.001,0.0001,0.00001,0.000001]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "display(film,scaling=constrained,in sequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "plot_li ne:=proc(eq,d)\n local f:\n f:=rhs(isolate(eq,x2)):\n [[-d,eval(f,x1=- d)],[d,eval(f,x1=d)]]:\n pointplot(%,connect=true,color=black):\nend p roc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot_line(x1*2+x2*3 =2,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "plot_set:=proc(s f,N,d)\n local i,beta,res:\n res:=NULL:\n for i from 0 to N do\n b eta:=0.0001+2*Pi/N*i:\n res:=plot_line(x1*cos(beta)+x2*sin(beta)=sf( cos(beta),sin(beta)),d),res;\n end do:\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "s_func:=unapply(c_smooth(0.1,psi1,psi2),psi 1,psi2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "gr_line:=plot_s et(s_func,90,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "gr_smoo th:=plot_sf(s_func,90,magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(gr_smooth,gr_line,view=[-2..2,-2..2]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "gr_line:=plot_set(sf_tr,90,3):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "gr_smooth:=plot_sf(sf_tr,90, magenta):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(gr_smo oth,gr_line,view=[-3..3,-3..3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "c_kr:=(psi1,psi2)->0.2*sqrt(psi1^2+psi2^2)+psi1+psi2; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "c_KR:=(psi1,psi2)->1*sq rt(psi1^2+psi2^2)-1*psi1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "c_Union:=sf_smooth(c_kr,c_KR,0.025);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "gr_Union:=CURVES(op([1,1],plot_sf(c_Union,90,red)),CO LOR(RGB,1,0,0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "gr_kr:= CURVES(op([1,1],plot_sf(c_kr,90,red)),COLOR(RGB,1,0,1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "gr_KR:=CURVES(op([1,1],plot_sf(c_KR ,90,red)),COLOR(RGB,1,0,1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "CURVES(op([1,1],plot_line(x1*cos(1)+x2*sin(1)=c_Union(cos(1),sin (1)),3)),COLOR(RGB,1,1,1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "M:=90;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "d:=2.5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "dc_union:=proc(psi1,psi2)\n [diff(c _Union(psi1,psi2),psi1),diff(c_Union(psi1,psi2),psi2)]:\nend proc:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "R:=proc(p1,p2)\n dc_union( psi1,psi2);\n diff(%[1],psi1)+diff(%[2],psi2):\n evalf(subs(psi1=p1,ps i2=p2,%));\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "'R (-1,0)'=R(-1,0);\n'R(1,0)'=R(1,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 755 "Film:=NULL:\nfor i from 0 to M do\n alpha:=0.000001+ 2*Pi/M*i:\n r:=R(cos(alpha),sin(alpha)):\n text_R:=TEXT([-2,2],\n ca t(\"R = \",convert(r,string)),\n ALIGNRIGHT,ALIGNABOVE,COLOR(RGB,0, 0,1),FONT(TIMES,BOLD,16)):\n ps:=evalf(subs([psi1=cos(alpha),psi2=sin( alpha)],dc_union(psi1,psi2))):\n gr_arc:=CURVES(op([1,1],\n plot([ps [1]+r*(cos(t)-cos(alpha)),ps[2]+r*(sin(t)-sin(alpha)),t=0..2*Pi])),\n \+ COLOR(RGB,0,0,1)):\n s_vector:=CURVES(\n [ps,evalf(ps+0.5*[cos(alpha ),sin(alpha)])],COLOR(RGB,0.2,0.2,0.7),THICKNESS(2)):\n s_set:=POINTS( ps,COLOR(RGB,0,0,1),SYMBOL(CIRCLE,14)):\n s_plane:=\n CURVES(op([1,1] ,plot_line(x1*cos(alpha)+x2*sin(alpha)=c_Union(cos(alpha),sin(alpha)), d))):\n Film:=Film,PLOT(gr_arc,text_R,s_vector,s_set,s_plane,gr_Union, gr_kr,gr_KR):\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 " display(Film,insequence=true,scaling=constrained,view=[-d..d,-d..d],\n title=\"\303\345\356\354\345\362\360\350\377 \357\360\356\366\345\344 \363\360\373 \361\343\353\340\346\350\342\340\355\350\377\",axes=none, titlefont=[TIMES,ROMAN,20]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sf_3_2:=(psi1,psi2)->(abs(psi1)^3+abs(psi2)^3)^(1/3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sf_3_2(psi1,psi2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "G0:=plot_set(sf_3_2,90,3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "G1:=implicitplot(\{abs(x) ^(3/2)+abs(y)^(3/2)=1\},x=-1..1,y=-1..1,grid=[100,100]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(G1,G0,view=[-1.5..1.5,-1.5..1.5]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot3d(sf_3_2(p1,p2),p1=-2.. 2,p2=-2..2,axes=BOXED);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " contourplot(sf_3_2(p1,p2),p1=-2..2,p2=-2..2,axes=BOXED);" }}}{EXCHG } {EXCHG }{EXCHG }{EXCHG }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 " \314\355\356\346\345\361\362 \342\356 \344\356\361\362\350\346\350\354\356\361\362\350" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x10:=1;\nx20:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eq1:=diff(x1(t),t)=a11*x1(t)+a12*x2(t)+u1(psi1(t)+p1,psi2(t)+p2);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "eq2:=diff(x2(t),t)=a21*x1(t)+a22*x2 (t)+u2(psi1(t)+p1,psi2(t)+p2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "e q3:=diff(psi1(t),t)=-a11*(psi1(t)+p1)-a21*(psi2(t)+p2);" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "eq4:=diff(psi2(t),t)=-a12*(psi1(t)+p1)-a22*(ps i2(t)+p2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "u1:=proc(psi 1,psi2)\n if (type(evalf(psi1),'numeric') and type(evalf(psi2),'numeri c')) then\n mu*psi1/sqrt(mu*psi1^2+psi2^2):\n else\n 'procname'(psi1 ,psi2)\n end if:\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 170 "u2:=proc(psi1,psi2)\n if (type(evalf(psi1),'numeric') and type(ev alf(psi1),'numeric')) then\n psi2/sqrt(mu*psi1^2+psi2^2):\n else\n ' procname'(psi1,psi2)\n end if:\nend proc:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 110 "sol:=dsolve(\{eq1,eq2,eq3,eq4,x1(0)=x10,x2(0)=x20, psi1(0)=0,psi2(0)=0\},\n numeric,startinit=true,known=[u1,u2]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "X1:=proc(T,P1,P2,Mu)\n glob al p1,p2,mu:\n p1:=evalf(P1);\n p2:=evalf(P2);\n mu:=Mu;\n subs(sol(T) ,x1(t));\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "X2: =proc(T,P1,P2,Mu)\n global p1,p2,mu:\n p1:=evalf(P1);\n p2:=evalf(P2); \n mu:=Mu;\n subs(sol(T),x2(t));\nend proc:;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "a11:=0; a12:=1;\na21:=0; a22:=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "T:=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(['X1'(T,cos(a),sin(a),0.01),'X2'(T,cos(a),sin(a) ,0.01),a=0..2*Pi]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "plo ts[animate](\n plot,[['X1'(T,cos(a),sin(a),1-Mu),'X2'(T,cos(a),sin(a), 1-Mu),a=0..2*Pi]],\n Mu=0..0.9999,frames=20,\n title=\"\307\340\342 \350\361\350\354\356\361\362\374 \354\355\356\346\345\361\362\342\340 \+ \344\356\361\362\350\346\350\354\356\361\362\350 \356\362 \357\340\360 \340\354\345\362\360\340 mu\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Topt:=subs([a=x10,b=x20],b+2*sqrt(a+b^2/2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(Topt);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 244 "plots[animate](\n plot,[['X1(T,cos(a),sin(a),0.01) ','X2(T,cos(a),sin(a),0.01)',a=0..2*Pi],\n thickness=3,labels=[`x1`,` x2`]],\n 'T'=0..Topt,frames=20,\n background=pointplot([x10,x20],color =blue,symbol=CIRCLE,symbolsize=16),\n scaling=constrained);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "a11:=0; a12:=1;\na21:=-1; a2 2:=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 240 "plots[animate](\n plot,[['X1(T,cos(a),sin(a),0.01)','X2(T,cos(a),sin(a),0.01)',a=0..2*P i],\n thickness=3,labels=[`x1`,`x2`]],\n 'T'=0..2*Pi,frames=21,\n bac kground=pointplot([1,1],color=blue,symbol=CIRCLE,symbolsize=16),\n sca ling=constrained);" }}}{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }}} {MARK "5" 0 }{VIEWOPTS 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