<s>
A	O
phase	B-Application
portrait	I-Application
is	O
a	O
geometric	O
representation	O
of	O
the	O
trajectories	O
of	O
a	O
dynamical	O
system	O
in	O
the	O
phase	O
plane	O
.	O
</s>
<s>
Phase	B-Application
portraits	I-Application
are	O
an	O
invaluable	O
tool	O
in	O
studying	O
dynamical	O
systems	O
.	O
</s>
<s>
They	O
consist	O
of	O
a	O
plot	B-Application
of	O
typical	O
trajectories	O
in	O
the	O
state	O
space	O
.	O
</s>
<s>
The	O
concept	O
of	O
topological	O
equivalence	O
is	O
important	O
in	O
classifying	O
the	O
behaviour	O
of	O
systems	O
by	O
specifying	O
when	O
two	O
different	O
phase	B-Application
portraits	I-Application
represent	O
the	O
same	O
qualitative	O
dynamic	O
behavior	O
.	O
</s>
<s>
A	O
phase	B-Application
portrait	I-Application
graph	O
of	O
a	O
dynamical	O
system	O
depicts	O
the	O
system	O
's	O
trajectories	O
(	O
with	O
arrows	O
)	O
and	O
stable	O
steady	O
states	O
(	O
with	O
dots	O
)	O
and	O
unstable	O
steady	O
states	O
(	O
with	O
circles	O
)	O
in	O
a	O
state	O
space	O
.	O
</s>
<s>
Simple	O
harmonic	O
oscillator	O
where	O
the	O
phase	B-Application
portrait	I-Application
is	O
made	O
up	O
of	O
ellipses	O
centred	O
at	O
the	O
origin	O
,	O
which	O
is	O
a	O
fixed	O
point	O
.	O
</s>
<s>
A	O
phase	B-Application
portrait	I-Application
represents	O
the	O
directional	O
behavior	O
of	O
a	O
system	O
of	O
ordinary	O
differential	O
equations	O
(	O
ODEs	O
)	O
.	O
</s>
<s>
The	O
phase	B-Application
portrait	I-Application
can	O
indicate	O
the	O
stability	O
of	O
the	O
system	O
.	O
</s>
<s>
The	O
phase	B-Application
portrait	I-Application
behavior	O
of	O
a	O
system	O
of	O
ODEs	O
can	O
be	O
determined	O
by	O
the	O
eigenvalues	O
or	O
the	O
trace	O
and	O
determinant	O
(	O
trace	O
=	O
λ1	O
+	O
λ2	O
,	O
determinant	O
=	O
λ1	O
x	O
λ2	O
)	O
of	O
the	O
system	O
.	O
</s>
