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Linear Stability Analysis
Perhaps the most magical moment in a pattern-forming system is when a pattern first appears outof nothing, the genesis of structure. The “nothing” that one starts with is not empty space butsome equilibrium system. As the system is driven further and further out of equilibriumby turning some experimental knob in small successive steps, a point is eventually reached suchthat small perturbations start to grow and a non-uniform spatial structure appears for the first time.Stability of Steady StatesBoth the featureless state of the system and the pattern that arises from it are usually time-independent, and so can be identified with a steady state, or a time-independent solution of theevolution equations. A particular steady state will only be observed, if it is stable. In other words,if perturbed away from that state, the system returns to it after some transient. Otherwise, we willcall that state unstable.Example: (underwater rollercoaster)The motion of the rollercoaster car is governed bythe Newton’s law:
x
dx
x
dU
x x F
x m
where
U
( x
) =
mgh
( x
) is the potential energy of the car and
γ
is the viscous drag force coefficient.
mg
h
( x
)
x
It is easy to see that the point
x
= 0 is a steady state solution,
f (0) = 0, regardless of the value of
r
. Furthermore:
r^
< 0:
x^
= 0 is the only equilibrium and it’s stable.
r > 0:
x^
= 0 is an unstable equilibrium
are two stable equilibria
The value
r
c^
= 0 is the critical point, at which the trivial
state of the system,
x
= 0, becomes unstable and is replaced
by a non-trivial state
. We will call such an event a
bifurcation.
4
2
x
x r
x h
3
(^
x
rx
x f
x^
r
x
We are not interested in the dynamics of the car (the transient), we only want to know what thesteady states (or equilibria) are and determine their stability. For simplicity let’s assume thatwater is very viscous (
γ^
is large) and pick a special height profile
With these simplifications we can ignore the inertia, so theequation of motion reduces toafter rescaling time,
t
new
= (
mg
/γ
) t
old
.
h h
x x x x
f f
Stable
x
= 0
Unstable
x
= 0
x
Example: (real Ginzburg-Landau equation)Consider the following nonlinear PDE (it is just a generalization of our previous example)We can easily see that
u
1
= 0,
, and
are steady states, as
F
[ u
] = 0. In order to i
calculate their stability, we need to determine the dynamics described by this PDE in a smallneighborhood of the steady states. Sufficiently close to these steady states the equation can besimplified by linearizing it. Writing
u
=
u
+i
δ
u
with
δ
u
being “small” we obtain:
We, therefore, see that the Jacobian of the PDE is an operator defined byNote: we could have noticed that
F
[ u
] is a sum of two operators
linear,
and
nonlinear,
We could then simplify the calculations of the Jacobian by only calculating the linearizationof the nonlinear part
N
[ u
] (the Jacobian of
L
[ u
] is
L
[ u
] itself).
].
[
3
2
u F
u
u
ru
u
x
t^
r
u
2
r
u
3
]
[
]
[
2 2 2 3 2 2 2
3
2
3
2
u o u u u u r u u u u u u u r
u
u
ru
u u u u u u r
u F u u F u
i
x
i
i
x
i i x i i i x i
i
i
t
δ δ δ δ δ δ δ δ δ
δ
δ
δ
δ
δ
u u u u r u J i
x
i
δ
δ
δ
δ
2
2
]
[^
u
ru
u L
2 x
]
[^
]
[^
3 u
u N
Example: (nonlinear derivatives)Occasionally the nonlinear terms can involve derivatives, e.g.,To linearize
N
[ u
] we write (with a substitution
)
We therefore obtainThe Jacobian is a differential operator containing
u
and its partial derivatives. What should give us
a pause, however, is the fact that there are partial derivatives of
h
. The smallness of
h
does not
mean that partial derivatives of
h
are also small. For instance,
h
( x
, y
) = 10
cos(
10
xy
)
is small, while any of it partial derivatives is not.In neglecting higher orders in
h
in the above expressions we have made an assumption that not
only
h
, but also all its partial derivatives are small as well. This is a strong assumption justified in
practice by the fact that dissipation in the system usually suppresses large gradients.
( ) ( ) ( ) ( ]
[^
2
2
2
2
2
y
x
y
x^
u u u u u u u u u u N
[^
]^
[^
]^
[^
]
]
[
]
[
2
2
2
2 2 2 2 2 2
2
2
2
2
h o
h
uu
h
uu
h
u
u
hh
hh
h
h u
hh u
hh u
h
uu
h
uu
h
u
u
u u u h h u u h h u u h u u N h u N
y y x x y x
y x y x y y x x y y x x y x
y x y y y y x x x x
h
u
δ
[^
]^
. ) ( 2 ) ( ) ( 2 ) ( ]
[^
2
2
2
h u u u h u u u h u u h
J^
y y
x x
y
x^
For instance, suppose that the evolution equation is equivariant, i.e., is unchanged by a symmetrytransformation
T
for any
u
,
As
T
is the operator acting on the spatial degrees of freedom, it commutes with the time
derivative, so we immediately obtain
F
[
u
] =
T
[ F
[ T
[u]]]. Linearizing the equation about the
steady state
u
u
=
u
δ
u
, we get
where the Jacobian
J
is a linear operator. If the steady state is also symmetric,
T
[ u
*] =
u
*,
the equivariance of
F
[ u
] implies (please check!) that
J
and
T
commute (
J [
δ
u] =
T
[ J
[ T
[δ
u]]] for
any
δ
u
) and so the eigenfunctions of
T
are also the eigenfunctions of
J
.
This is a fact of tremendous importance. For instance,
- for all translationally invariant systems the eigenfunctions are just Fourier modes, and- these eigenfunctions are independent of the specifics of the pattern-forming system!
As the translational invariance is a consequence of the uniformity of the physical space, anydeviations from the exact symmetry are due to either
- the presence of the lateral boundaries or- the gradients imposed to drive the system out of equilibrium.
)]].
[
[
)]
[^
t u T F t u
Tt
x
x
],
[
]
[
*]
[ ] * [ ] *
[^
u J u J u F u u F u u u
t
t
