<s>
In	O
mathematics	O
and	O
particularly	O
in	O
dynamic	O
systems	O
,	O
an	O
initial	B-Algorithm
condition	I-Algorithm
,	O
in	O
some	O
contexts	O
called	O
a	O
seed	B-Algorithm
value	I-Algorithm
,	O
is	O
a	O
value	O
of	O
an	O
evolving	O
variable	O
at	O
some	O
point	O
in	O
time	O
designated	O
as	O
the	O
initial	O
time	O
(	O
typically	O
denoted	O
t	O
=	O
0	O
)	O
.	O
</s>
<s>
For	O
a	O
system	O
of	O
order	O
k	O
(	O
the	O
number	O
of	O
time	O
lags	O
in	O
discrete	O
time	O
,	O
or	O
the	O
order	O
of	O
the	O
largest	O
derivative	O
in	O
continuous	O
time	O
)	O
and	O
dimension	O
n	O
(	O
that	O
is	O
,	O
with	O
n	O
different	O
evolving	O
variables	O
,	O
which	O
together	O
can	O
be	O
denoted	O
by	O
an	O
n-dimensional	O
coordinate	O
vector	O
)	O
,	O
generally	O
nk	O
initial	B-Algorithm
conditions	I-Algorithm
are	O
needed	O
in	O
order	O
to	O
trace	O
the	O
system	O
's	O
variables	O
forward	O
through	O
time	O
.	O
</s>
<s>
In	O
both	O
differential	O
equations	O
in	O
continuous	O
time	O
and	O
difference	O
equations	O
in	O
discrete	O
time	O
,	O
initial	B-Algorithm
conditions	I-Algorithm
affect	O
the	O
value	O
of	O
the	O
dynamic	O
variables	O
(	O
state	O
variables	O
)	O
at	O
any	O
future	O
time	O
.	O
</s>
<s>
In	O
continuous	O
time	O
,	O
the	O
problem	O
of	O
finding	O
a	O
closed	O
form	O
solution	O
for	O
the	O
state	O
variables	O
as	O
a	O
function	O
of	O
time	O
and	O
of	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
is	O
called	O
the	O
initial	O
value	O
problem	O
.	O
</s>
<s>
A	O
linear	O
matrix	B-Algorithm
difference	I-Algorithm
equation	I-Algorithm
of	O
the	O
homogeneous	O
(	O
having	O
no	O
constant	O
term	O
)	O
form	O
has	O
closed	O
form	O
solution	O
predicated	O
on	O
the	O
vector	O
of	O
initial	B-Algorithm
conditions	I-Algorithm
on	O
the	O
individual	O
variables	O
that	O
are	O
stacked	O
into	O
the	O
vector	O
;	O
is	O
called	O
the	O
vector	O
of	O
initial	B-Algorithm
conditions	I-Algorithm
or	O
simply	O
the	O
initial	B-Algorithm
condition	I-Algorithm
,	O
and	O
contains	O
nk	O
pieces	O
of	O
information	O
,	O
n	O
being	O
the	O
dimension	O
of	O
the	O
vector	O
X	O
and	O
k	O
=	O
1	O
being	O
the	O
number	O
of	O
time	O
lags	O
in	O
the	O
system	O
.	O
</s>
<s>
The	O
initial	B-Algorithm
conditions	I-Algorithm
in	O
this	O
linear	O
system	O
do	O
not	O
affect	O
the	O
qualitative	O
nature	O
of	O
the	O
future	O
behavior	O
of	O
the	O
state	O
variable	O
X	O
;	O
that	O
behavior	O
is	O
stable	O
or	O
unstable	O
based	O
on	O
the	O
eigenvalues	O
of	O
the	O
matrix	O
A	O
but	O
not	O
based	O
on	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
.	O
</s>
<s>
Here	O
the	O
dimension	O
is	O
n	O
=	O
1	O
and	O
the	O
order	O
is	O
k	O
,	O
so	O
the	O
necessary	O
number	O
of	O
initial	B-Algorithm
conditions	I-Algorithm
to	O
trace	O
the	O
system	O
through	O
time	O
,	O
either	O
iteratively	O
or	O
via	O
closed	O
form	O
solution	O
,	O
is	O
nk	O
=	O
k	O
.	O
</s>
<s>
Again	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
do	O
not	O
affect	O
the	O
qualitative	O
nature	O
of	O
the	O
variable	O
's	O
long-term	O
evolution	O
.	O
</s>
<s>
Here	O
the	O
constants	O
are	O
found	O
by	O
solving	O
a	O
system	O
of	O
k	O
different	O
equations	O
based	O
on	O
this	O
equation	O
,	O
each	O
using	O
one	O
of	O
k	O
different	O
values	O
of	O
t	O
for	O
which	O
the	O
specific	O
initial	B-Algorithm
condition	I-Algorithm
Is	O
known	O
.	O
</s>
<s>
Its	O
behavior	O
through	O
time	O
can	O
be	O
traced	O
with	O
a	O
closed	O
form	O
solution	O
conditional	O
on	O
an	O
initial	B-Algorithm
condition	I-Algorithm
vector	O
.	O
</s>
<s>
The	O
number	O
of	O
required	O
initial	O
pieces	O
of	O
information	O
is	O
the	O
dimension	O
n	O
of	O
the	O
system	O
times	O
the	O
order	O
k	O
=	O
1	O
of	O
the	O
system	O
,	O
or	O
n	O
.	O
The	O
initial	B-Algorithm
conditions	I-Algorithm
do	O
not	O
affect	O
the	O
qualitative	O
behavior	O
(	O
stable	O
or	O
unstable	O
)	O
of	O
the	O
system	O
.	O
</s>
<s>
Here	O
the	O
number	O
of	O
initial	B-Algorithm
conditions	I-Algorithm
necessary	O
for	O
obtaining	O
a	O
closed	O
form	O
solution	O
is	O
the	O
dimension	O
n	O
=	O
1	O
times	O
the	O
order	O
k	O
,	O
or	O
simply	O
k	O
.	O
In	O
this	O
case	O
the	O
k	O
initial	O
pieces	O
of	O
information	O
will	O
typically	O
not	O
be	O
different	O
values	O
of	O
the	O
variable	O
x	O
at	O
different	O
points	O
in	O
time	O
,	O
but	O
rather	O
the	O
values	O
of	O
x	O
and	O
its	O
first	O
k	O
–	O
1	O
derivatives	O
,	O
all	O
at	O
some	O
point	O
in	O
time	O
such	O
as	O
time	O
zero	O
.	O
</s>
<s>
The	O
initial	B-Algorithm
conditions	I-Algorithm
do	O
not	O
affect	O
the	O
qualitative	O
nature	O
of	O
the	O
system	O
's	O
behavior	O
.	O
</s>
<s>
This	O
equation	O
and	O
its	O
first	O
k	O
–	O
1	O
derivatives	O
form	O
a	O
system	O
of	O
k	O
equations	O
that	O
can	O
be	O
solved	O
for	O
the	O
k	O
parameters	O
given	O
the	O
known	O
initial	B-Algorithm
conditions	I-Algorithm
on	O
x	O
and	O
its	O
k	O
–	O
1	O
derivatives	O
 '	O
values	O
at	O
some	O
time	O
t	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
can	O
affect	O
whether	O
the	O
system	O
diverges	O
to	O
infinity	O
or	O
whether	O
it	O
converges	B-Algorithm
to	O
one	O
or	O
another	O
attractor	O
of	O
the	O
system	O
.	O
</s>
<s>
Each	O
attractor	O
,	O
a	O
(	O
possibly	O
disconnected	O
)	O
region	O
of	O
values	O
that	O
some	O
dynamic	O
paths	O
approach	O
but	O
never	O
leave	O
,	O
has	O
a	O
(	O
possibly	O
disconnected	O
)	O
basin	O
of	O
attraction	O
such	O
that	O
state	O
variables	O
with	O
initial	B-Algorithm
conditions	I-Algorithm
in	O
that	O
basin	O
(	O
and	O
nowhere	O
else	O
)	O
will	O
evolve	O
toward	O
that	O
attractor	O
.	O
</s>
<s>
Even	O
nearby	O
initial	B-Algorithm
conditions	I-Algorithm
could	O
be	O
in	O
basins	O
of	O
attraction	O
of	O
different	O
attractors	O
(	O
see	O
for	O
example	O
Newton	O
's	O
method	O
#Basins	O
of	O
attraction	O
)	O
.	O
</s>
<s>
Moreover	O
,	O
in	O
those	O
nonlinear	O
systems	O
showing	O
chaotic	O
behavior	O
,	O
the	O
evolution	O
of	O
the	O
variables	O
exhibits	O
sensitive	O
dependence	O
on	O
initial	B-Algorithm
conditions	I-Algorithm
:	O
the	O
iterated	O
values	O
of	O
any	O
two	O
very	O
nearby	O
points	O
on	O
the	O
same	O
strange	O
attractor	O
,	O
while	O
each	O
remaining	O
on	O
the	O
attractor	O
,	O
will	O
diverge	O
from	O
each	O
other	O
over	O
time	O
.	O
</s>
<s>
Thus	O
even	O
on	O
a	O
single	O
attractor	O
the	O
precise	O
values	O
of	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
make	O
a	O
substantial	O
difference	O
for	O
the	O
future	O
positions	O
of	O
the	O
iterates	O
.	O
</s>
<s>
This	O
feature	O
makes	O
accurate	O
simulation	O
of	O
future	O
values	O
difficult	O
,	O
and	O
impossible	O
over	O
long	O
horizons	O
,	O
because	O
stating	O
the	O
initial	B-Algorithm
conditions	I-Algorithm
with	O
exact	O
precision	O
is	O
seldom	O
possible	O
and	O
because	O
rounding	O
error	O
is	O
inevitable	O
after	O
even	O
only	O
a	O
few	O
iterations	O
from	O
an	O
exact	O
initial	B-Algorithm
condition	I-Algorithm
.	O
</s>
