Biology Reference
In-Depth Information
4.2 Definitions of the Notions of Stability and Robustness
There exist several definitions of the stability of a dynamical system and we will
give hereafter the most useful. We define a trajectory
x
(
a
,
t
) in a state space
E
n
as
the set of all states observed as the time goes from initial value 0, corresponding to
the state
x
(
a
,0)
R
a
, to infinity. The set of states visited when
t
tends to infinity is
called the limit set of the trajectory starting in
a
, and is denoted
L
(
a
). If the initial
states lie in a set
A
, then
L
(
A
) is the union of all limit sets
L
(
a
), for a belonging to
A
.
Conversely,
B
(
A
), called the attraction basin of
A
is the set of all initial conditions
outside
A
, whose limit set
L
(
a
) is included in
A
. We will call attractor in the
following a set
A
such as (1)
A
¼
¼
L
(
B
(
A
)), (2)
A
is not contained in a wider set
B
,
such as d(
A
,
B
\
A
)
0 and verifying (1), and (3) A does not
contain a strictly smaller subset
C
verifying (1) and (2). Such an attractor
A
associated to its basin
B
(
A
) is the exact set of the states “attracting” the trajectories
coming from
B
(
A
) outside
A
.
¼
inf
a2A
,
b2B
\
A
d(
a
,
b
)
¼
4.2.1 Definition of the Lyapunov (or Trajectorial) Stability
Atrajectory
x
(
a
,
t
) is called Lyapunov stable, if no perturbation at any time
t
should be
amplified: if
b
¼
x
(
a
,
t
)+
ε
denotes the perturbed state, then for any
s
>
t
,d(
x
(
a
,
s
),
x
(
b
,
s
-
t
))
ε
.
4.2.2 Definition of the Asymptotic Stability
A trajectory
x
(
a
,
t
) is called asymptotically stable, if any perturbation at any time
t
is asymptotically damped: if
b
¼
x
(
a
,
t
)+
ε
denotes the perturbed state, then
lim
s
!1
d(
x
(
a
,
s
),
x
(
b
,
s
-
t
))
¼
0.
4.2.3 Definition of the Structural Stability with Respect
to a Parameter
μ
A dynamical system whose trajectories
x
p
(
a
,
t
) depend on a parameter
p
is called
structurally stable with respect to the parameter
p
, if no perturbation of
p
at any time
can provoke a change in number or nature (fixed attractor, called steady state, or
periodic attractor, called limit cycle) of its attractors.
p
may parametrize the state
transition rule of the system or its architectural characteristics (number of elements,
intensity of interactions).
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