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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 39 "Stability of equilibria \+
investigation 5" }}{PARA 0 "" 0 "" {TEXT -1 19 "Consider the system" }
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:U^Z3<" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 30 "To find the equilibri
um point." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The equilibrium poin
t is (0,0)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 256 23 "To find the eigenvalues" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 
32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "W
arning, new definition for trace" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "A:=matrix(2,2,[0,1,-1,0]);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 7 "tr(A)=0" }}{PARA 0 "" 0 "" {TEXT -1 8 "det(A)=1" }}
{PARA 0 "" 0 "" {TEXT -1 11 "discrim= -4" }}{PARA 0 "" 0 "" {TEXT -1 
53 "Thus the eigenvalues are complex with real part zero." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Eigenvalues:=eigenvals(A);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "It is unnecessary to find the eig
envectors as they will be complex and therefore there will be no direc
trices." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 
27 "To find the phase portrait." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "P
:=diff(x(t),t)=y;" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "Q:=diff(y(t),t)=-x;" }}{PARA 11 "" 1 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "DEplot([P,Q
],[x(t),y(t)],-2..2,x=-5..5,y=-5..5);" }}{PARA 13 "" 1 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Adding the initial conditions
 gives" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "DEplot([P,Q],[x(
t),y(t)],-4..4,[[x(0)=0,y(0)=1],[x(0)=0,y(0)=2],[x(0)=0,y(0)=3],[x(0)=
0,y(0)=4]],x=-5..5,y=-5..5);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 300 "The trajectories are concentric c
ircles surrounding the equilibrium point. Such closed trajectories are
 characteristic of periodic solutions. No matter at which initial poin
t the trajectory starts it will eventually return to the same point. T
he equilibrium point is called a neutrally stable centre." }}}}{MARK "
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