{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 27 "The Chua Electrical Cir cuit" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "R estart Maple and load the necessary packages." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Define the system. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p:=x->m1*x+(m0-m1)/2*(abs(x+1)-abs(x-1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f:=c1*(y-x-p(x)):g:=c2*(x-y+z):h:=- c3*y:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We need to make assumptions about the parameters for the \+ maths to work." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "assume(c1>0,c2>0,c3>0,m0<-1,m1<0, m1>-1):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Fi nd the equilibrium points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "eps:=solve(\{f,g,h\},\{x,y,z\});" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 " Now carry out a linearization." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "A:=matrix(3,3,[diff(f,x),di ff(f,y),diff(f,z),diff(g,x),diff(g,y),diff(g,z),diff(h,x),diff(h,y),di ff(h,z)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A0:=map(simpl ify,subs(eps[2],evalm(A)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A1:=map(simplify,subs(eps[3],evalm(A)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A2:=map(simplify,subs(eps[1],evalm(A)));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Find the characteristic polynomial s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "cp0:=collect(charpoly(A0,lambda),lambda);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " } {XPPEDIT 18 0 "m0<-1" "6#2%#m0G,$\"\"\"!\"\"" }{TEXT -1 99 " the const ant term is negative and hence there must be a positive real root. Thu s E.P. is unstable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " cp1:=collect(charpoly(A1,lambda),lambda) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 229 "With conditions on parameters polynomial has positive coefficient s and constant term. Hence no positive real root. Proceed as in accomp anying booklet to show there can only be a bifurcation when complex ro ots have zero real part." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "trA:=trace(A1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "c3c:=solve(subs(lambda=trA,cp1),c3);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "No te that stability is lost as the parameter decreases through the criti cal value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Choose parameter values." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c1:=15.6:c2:=1:m0:=-8/ 7:m1:=-5/7:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 21 "Obtain critical value" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "c3c := -c1*m1/c2*(c2+c1*m 1+c1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Now vary the value of " }{XPPEDIT 18 0 "c3" "6#%#c3G" } {TEXT -1 70 ". Note the use of two i.c.s. Necessary to obtain the comp lete picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c3:=65:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de1:=diff(x(t),t)=f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de2:=diff(y(t),t)=g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de3:=diff(z(t),t)=h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ic s:=[[x(0)=2,y(0)=0,z(0)=0],[x(0)=-2,y(0)=0,z(0)=0]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot3d(\{de1,de2,de3\},[x(t),y(t),z(t)] ,t=50..100,ics,stepsize=0.05,linecolour=[blue,red]);" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 15 "Two stable foci" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c3:=45:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " de1:=diff(x(t),t)=f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de2 :=diff(y(t),t)=g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de3:=d iff(z(t),t)=h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ics:=[[x( 0)=2,y(0)=0,z(0)=0],[x(0)=-2,y(0)=0,z(0)=0]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot3d(\{de1,de2,de3\},[x(t),y(t),z(t)],t=50.. 100,ics,stepsize=0.05,linecolour=[blue,red]);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 16 "Two limit cycles" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c3:=35:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 " de1:=diff(x(t),t)=f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de2 :=diff(y(t),t)=g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de3:=d iff(z(t),t)=h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ics:=[[x( 0)=2,y(0)=0,z(0)=0],[x(0)=-2,y(0)=0,z(0)=0]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot3d(\{de1,de2,de3\},[x(t),y(t),z(t)],t=50.. 100,ics,stepsize=0.05,linecolour=[blue,red]);" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 15 "Period Doubling" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "c3:=33.5:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de1:=diff(x(t),t)=f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de2:=diff(y(t),t)=g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de 3:=diff(z(t),t)=h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ics:= [[x(0)=2,y(0)=0,z(0)=0],[x(0)=-2,y(0)=0,z(0)=0]]:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 91 "DEplot3d(\{de1,de2,de3\},[x(t),y(t),z(t)],t= 50..100,ics,stepsize=0.05,linecolour=[blue,red]);" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 23 "Another period doubling" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c3:=32:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de1:=diff(x(t),t)=f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de2:=diff(y(t),t)=g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de3:=diff(z(t),t)=h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ics:=[[x(0)=2,y(0)=0,z(0)=0],[x(0)=-2,y(0)=0,z(0)=0]]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot3d(\{de1,de2,de3\},[x(t),y(t) ,z(t)],t=50..100,ics,stepsize=0.05,linecolour=[blue,red]);" }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 31 "Two separate strange attractors" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c3:=30:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de1:=diff(x(t),t)=f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de2:=diff(y(t),t)=g:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "de3:=diff(z(t),t)=h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ics:=[[x(0)=2,y(0)=0,z(0)=0],[x(0)=-2,y(0)=0,z(0)=0]] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot3d(\{de1,de2,de3 \},[x(t),y(t),z(t)],t=50..100,ics,stepsize=0.05,linecolour=[blue,red]) ;" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 16 "Homoclinic orbit" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c3:=28:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de1:=diff(x(t),t)=f:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de2:=diff(y(t),t)=g:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 20 "de3:=diff(z(t),t)=h:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "ics:=[[x(0)=2,y(0)=0,z(0)=0],[x(0)=-2,y(0)=0,z(0)=0]] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "DEplot3d(\{de1,de2,de3 \},[x(t),y(t),z(t)],t=50..200,ics,stepsize=0.05,linecolour=[blue,red]) ;" }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 23 "Double scroll attractor" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Now plot the time series for two slightly different initial conditions." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ics:=[[x(0)=2,y(0)=0,z(0)=0],[x(0)=2.001,y(0)=0,z(0)=0]]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "DEplot(\{de1,de2,de3\},[x(t ),y(t),z(t)],t=100..150,ics,stepsize=0.05,linecolour=[blue,red],scene= [t,x(t)]);" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 42 "Sensitive depende nce on initial conditions" }}}}{MARK "48 0 0" 14 }{VIEWOPTS 0 0 0 1 1 1803 }