Massachusetts
Institute
of
Technology
Department
of
Electrical
Engineering
and
Computer
Science
6.243j
(Fall
2003):
DYNAMICS
OF
NONLINEAR
SYSTEMS
by
A.
Megretski
Lecture
8:
Local
Behavior
at
Eqilibria
1
This
lecture
presents
results
which
describe
local
behavior
of
autonomous
systems
in
terms
of
Taylor
series
expansions
of
system
equations
in
a
neigborhood
of
an
equilibrium.
8.1
First
order
conditions
This
section
describes
the
relation
between
eigenvalues
of
a
Jacobian
a
(¯
x
0
)
and
behavior
of
ODE
x
˙(
t
)
=
a
(
x
(
t
))
(8.1)
or
a
diﬀerence
equation
x
(
t
+
1)
=
a
(
x
(
t
))
(8.2)
in
a
neigborhood
of
equilibrium
¯
x
0
.
In
the
statements
below,
it
is
assumed
that
a
:
X
≥
R
n
is
a
continuous
function
deﬁned
on
an
open
subset
X
±
R
n
.
It
is
further
assumed
that
¯
x
0
⊂
X
,
and
there
exists
an
n
by
n
matrix
A
such
that

a
(¯
x
0
)
−
Aλ

x
0
+
λ
)
−
a
(¯
0
as

λ

0
.
(8.3)

λ

If
derivatives
da
k
/dx
i
of
each
component
a
k
of
a
with
respect
to
each
cpomponent
x
i
of
x
exist
at
¯
x
0
,
A
is
the
matrix
with
coeﬃcients
da
k
/dx
i
,
i.e.
the
Jacobian
of
the
system.
However,
diﬀerentiability
at
a
single
point
¯
x
0
does
not
guarantee
that
(8.3)
holds.
On
the
other
hand,
(8.3)
follows
from
continuous
diﬀerentiability
of
a
in
a
neigborhood
of
¯
x
0
.
1
Version
of
October
3,
2003
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2
Example
8.1
Function
a
:
R
2
≥
R
2
,
deﬁned
by
¯
¯
2
¯
2
−
(¯
2
x
2
)
2
±
¯
x
1
x
1
x
2
x
1
−
¯
2
x
1
²
a
=
¯
2
¯
2
x
2
¯
¯
x
2
x
1
x
2
+
(¯
2
−
¯
2
)
2
x
2
x
1
for
x
≤
x
1
and
¯
¯ = 0,
and
by
a
(0)
=
0,
is
diﬀerentiable
with
respect
to
¯
x
2
at
every
point
¯
x
⊂
R
2
,
and
its
Jacobian
a
(0)
=
A
equals
minus
identity
matrix.
However,
condition
(8.3)
is
not
satisﬁed
(note
that
a
(¯
x
=
0).
x
)
is
not
continuous
at
¯
8.1.1
The
continuous
time
case
Let
us
call
an
equilibrium
¯
x
0
of
(8.1)
exponentially
stable
if
there
exist
positive
real
num
bers
σ, r, C
such
that
every
solution
x
:
[0
, T
]
≥
X
with

x
(0)
−
¯
x
0

<
σ
satisﬁes
x
0
 ∀
Ce
−
rt

x
(0)
−
¯

x
(
t
)
−
¯
x
0
 ±
t
→
0
.
The
following
theorem
can
be
attributed
directly
to
Lyapunov.
Theorem
8.1
Assume
that
a
(¯
x
0
)
=
0
and
condition
(8.3)
is
satisﬁed.
Then
(a)
if
A
=
a
(¯
x
0
)
is
a
Hurwitz
matrix
(i.e.
if
all
eigenvalues
of
A
have
negative
real
part)
then
¯
x
0
is
a
(locally)
exponentially
stable
equilibrium
of
(8.1);
x
0
)
has
an
eigenvalue
with
a
nonnegative
real
part
then
¯
(b)
if
A
=
a
(¯
x
0
is
not
an
exponentially
stable
equilibrium
of
(8.1);
x
0
)
has
an
eigenvalue
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 Spring '09
 AlexandreMegretski
 Dynamics, Stability theory, Dynamical systems, exponentially stable equilibrium

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