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where A, W,
Θ
are, respectively, integer tensor, matrix, and line vector. Let
suppose also that:
8
i
¼
1
;
...
;
n
; Δ
x
i
2
f
1
;
0
;
1
g:
If h denotes the Heaviside function, consider now the potential automaton i
defined by:
x
i
ð
t
þ
1
Þ¼
h
ðΔ
P
=Δ
x
i
þ
x
i
ð
t
ÞÞ;
.
Then, if the tensor A is symmetrical with vanishing diagonal (
i.e.
, if we have the
equalities:
and by the condition x
i
(t + 1)
0, if x
i
(t)
¼
0, such that the flow remains in
Ω
8
i,j,k
¼
1,
,n, a
ijk
¼
a
ikj
¼
a
kij
¼
a
jki
¼
a
jik
¼
a
kji
,
and a
iik
¼
0), and if
...
each sub-matrix (on any subset J of
,n}) of A
k
and W are
non-positive with vanishing diagonal, P decreases on the trajectories of the poten-
tial automaton, for any mode of implementation of the dynamics (sequential, block
sequential, and parallel). Hence, the stable fixed configurations of the automaton
correspond to the minima of its potential P.
indices in {1,
...
Proof.
We have, for a discrete function
P
on
Ω
:
Δ
P
ð
x
Þ=Δ
x
i
¼½
P
ð
x
1
;
...
;
x
i
þ Δ
x
i
;
...
;
x
n
Þ
Px
1
;
...
;
ð
x
i
;
...
;
x
n
Þ=Δ
x
i
and the proof, based on the existence of a Lyapunov function proved in Ben Amor
et al. (
2008
), Demongeot et al. (
2006
,
2008a
), K¨hn (
2010
), Fogelman Souli´
et al. (
1989
), and Cosnard and Goles (
1977
), results from the Proposition 1 of
(Cosnard and Goles
1977
).
∎
Proposition 3.
Let us consider the Hamiltonian getBren which is a circuit with
constant absolute value w for its interaction weights, null threshold
, and temper-
ature T equal to 0, sequentially or synchronously updated, whose Hamiltonian H is
defined by:
Θ
X
i¼
1
;
...
;n
x
i
ð
2
H
ð
x
ð
t
ÞÞ ¼
ð
t
Þ
x
i
ð
t
1
Þ
Þ
=
2
X
i¼
1
;
...
;n
ð
2
¼
h
ð
w
iði
1
Þ
x
i
1
ð
t
1
Þ
x
i
ð
t
1
ÞÞ
=
2
;
then H equals the total discrete kinetic energy and the half of the global dynamic
frustration D(x(t)). The result remains available if the automata network is a circuit
in which transition functions are Boolean identity or negation.
Proof.
It is easy to check that:
H
(
x
(
t
))
¼
D
(
x
(
t
))/2.
∎
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