Chaos in Neural Networks:
Chaotic Solutions
A. Crisanti∗
H.-J. Sommers†
H. Sompolinsky‡
September 11, 2008
Abstract
A continuous-time dynamic model of a network of N nonlinear elements interacting via random fully asymmetric couplings is studied. The long time (and large
N ) behaviour is analyzed by means of a self-consistent dynamic mean-field theory,
which is exact in the limit N → ∞. The theory predicts a sharp transition from
a steady state to a chaotic flow at a critical value of the nonlinearity. The maximal Lyapunov exponent of the flow can be obtained from the fluctuations about
the mean-field theory solution. This is calculated analytically in the limit case of
high nonlinearity, and when the transition to chaos is approached from the chaotic
regime.
PACS number: 05.45.+b, 05.20.-y, 47.20.Tg, 87.10.+e ????
∗
Institut de Physique Théorique, B.S.P. Université de Lausanne, CH–1015 Lausanne, Switzerland
Fachbereich Physick, Universität – Gesamnthochschule Essen, D–4300 Essen, Federal Republic of
Germany
‡
Racah Institute of Physics, The Hebrew University, 91904 Jerusalem, Israel
†
1
1
Introduction
Chaos, in the sense that neighboring orbits separate exponentially in time, is a feature
common to a wide class of dynamical systems [1]. From a physical point of view this has
important consequences since, due to some initial uncertainty, the information about
the original state of the system is lost in a finite amount of time and so the system is
effectively unpredictable. This phenomenon, called deterministic chaos, is essentially
due to a sensitive dependence on initial conditions and has a great relevance in the
description of systems whose dynamics is described by ordinary differential equations
or maps [2, 3].
The study of the onset and of the nature of chaos in deterministic dynamical systems
has focused mainly on systems with few degrees of freedom. In these systems quite often
the chaotic behaviour is achieved, by varying a control parameter, through a sequence
of bifurcations leading to an increasing complexity of the flow [2, 3].
Very little is, however, known on the nature of chaotic flows in deterministic dynamical systems described by a large number of degrees of freedom. In this paper we study
the dynamical behaviour of a deterministic nonlinear system which was introduced in
the context of neural networks. We show that in the limit of a large number (infinite)
of degrees of freedom there is a sharp transition from a steady state to a chaotic flow,
as the degree of nonlinearity is varied. The statistical properties of the flow, as well as
the transition to chaos, are well described by a time-dependent mean-field theory. The
maximal Lyapunov exponent of the flow can be calculated from the fluctuations about
the mean field solution.
This paper is devoted only to the nature of the transition to chaos. In a second one
it will be presented the study of the phase diagram of the model. The paper is organized
as follows. In Section 2 we define the model, which was first introduced by Amari in the
context of neural networks models. Here we also review the solution derived by Amari
from a ‘naive’ mean-field theory, and its limits. In Section 3 a systematic dynamical
mean-field theory is presented. This is based on the dynamical mean-field theory developed in Reference [4] for spin glass models with random asymmetric couplings. The
stability of the mean-field solutions is studied in Section 4, where the fluctuations about
the saddle point describing the dynamical mean-field theory are calculated. It is found
that all but the chaotic solution are unstable. In Section 5 we show how the maximal Lyapunov exponent λ of the flow can be obtained form the Gaussian fluctuations
about the dynamical mean-field solution. The exponent λ is calculated analytically
in the limit case of high nonlinearity, and when the transition to chaos is approached
from the chaotic regime. The results obtained for numerical simulations are presented
in Section 6. Finally Section 7 is devoted to conclusions and discussion.
Part of these results have been already published in Reference [5].
2
2
2.1
The Model
The Model
The model was introduced by Amari [6] to describe the dynamics of a network of N
‘formal neurons’ coupled by a synaptic matrix J which defines the ‘geometry’ of the net.
As usual, the ‘neurons’ are modelled by N localized continuous variables {Si (t)}i=1,N ,
with −1 ≤ Si ≤ 1. At each time t the state of the i-th ‘neuron’ is determined by its
‘post-synaptic’ potential hi (t) through the relationship Si (t) = φ (ghi (t)), where φ(x) is
the non-linear gain function which defines the input (hi ) - output (Si ) characteristics of
the neurons [7]. In the biological context, hi may be related to the membrane potential
of the nerve cell and Si to its electrical activity (e.g., its firing rate).
The function φ(x) can be any function with a sigmoid shape, i.e., φ(±∞) = ±1,
φ(−x) = −φ(x) and φ0 (x) > 0 for any x. For concreteness we may choose the function,
φ(x) = tanh(x).
(1)
The non negative parameter g measures the level of nonlinearity in the system in fact,
if g ∼ 0 then φ(x) ∼ x, whereas if g → ∞ the Ising limit Si = ±1 is recovered.
The dynamics of the network is given by N coupled first-order differential equations,
(‘circuit’ equations) [6, 7]
∂t hi = −hi +
N
X
Jij Sj = −hi +
j=1
N
X
Jij φ (ghj ) .
(2)
j=1
Here Jij is the synaptic efficacy which couples the output of the (presynaptic) j-th
neuron to the input of the (postsynaptic) i-th neuron, and Jii = 0. In electrical terms
Eqs. (2) are Kirchhoff equations in which the l.h.s represents the current leakage due to
the membrane capacitance. The first term in the r.h.s. represents the current through
the membrane resistance and the last term denotes the input current flowing to the cell
due to the activity of the other cells. For simplicity the microscopic time constant has
been set equal to unity.
We consider here networks with random synaptic couplings. Each of the Jij ’s is an
independent random variable which can be assumed for convenience to have a Gaussian
distribution. The mean of Jij is Jo /N whereas the variance is Jij2 − (Jij )2 = 1/N . The
diagonal elements are zero. With this normalization the intensive parameter is the
dimensionless gain parameter g.
If the synaptic matrix Jij were symmetric then Eqs. (2) would describe a relaxation of a global energy function [8] which, for random couplings, is just a spin glass
Hamiltonian, so that the system always flows to stable fixed points which are the local
minima of the spin glass energy [9].
Here we study synaptic matrices where Jij and Jji are uncorrelated [6] in which
case the dynamics in general is non relaxational. In this case the long time behavior
may depend on the particular realization of the Jij ’s.
3
2.2
The Amari Solution
The first systematic study of the macroscopic properties of this model was done by
Amari [6], who developed a ‘naive’ mean field theory to describe the long time (and
large N ) behaviour. The Amari solution is based on the following two Hypotheses on
the probability distribution of the macroscopic state,
Hypothesis 1 Provided that N is sufficiently large, all hi (t) are stochastically independent and have a common Gaussian distribution.
Hypothesis 2 For large N and t,
N Z t
N
1 X
1 X
0
dt0 e−(t−t ) Si (t)Si (t0 ) '
S 2 (t) = q.
N i=1 −∞
N i=1 i
By means of the first hypothesis, every macroscopic quantity can be calculated from
the microscopic state of the system. Let, in fact, f (h) be an arbitrary function and F
the mean of f (h) over all spins, i.e.,
F =
N
1 X
f (hi ).
N i=1
(3)
Then when N is sufficiently large, for almost all realizations of Jij , F is given by,
(h − ho )2
dh
f
(h)
exp
−
,
(4)
F = p
2σo2
2πσo2
√
except for small fluctuations of order O(1/ N ). The parameters ho and σo2 are the
average and the variance of hi ,
(
Z
ho =
)
N
N
1 X
1 X
hi ; σo2 =
(hi − ho )2 .
N i=1
N i=1
(5)
The macroscopic state equations which determine the time evolution of ho and σo2 ,
are obtained by taking the time derivative of Eqs. (5), and using the Hypothesis 2 to
obtain a closed system of equations.
These equations of motion lead to the static solution ho = Jo m and σo2 = q, where
P
m is the average local magnetization m = N −1 N
i=1 Si (t). The statics of the system is
then described by the self-consistent mean field equations,
Z
m=
Z
q=
dh −h2 /2
√
√
e
tanh g ( q h + Jo m) ,
2π
(6)
dh −h2 /2
√
√
e
tanh2 g ( q h + Jo m) .
2π
(7)
4
It is easy to recognize in Eqs. (6),(7) the replica symmetric solution of the symmetric
SK model [10] with g = T −1 , where T is the temperature. Therefore, from this mean
field theory it follows that for any g the system always flows to time-independent fixed
points.
This solution is correct in the limit g → 0, as can be seen by taking the g → 0
limit of Eq. (2). It is, however, certainly wrong in the Ising limit, g → ∞, where one
expects a time-dependent solution [4, 11]. The following simple arguments shows that
that solution is also chaotic, i.e., it has a positive maximal Lyapunov exponent.
In the Ising limit the autocorrelation functions of the spins (or of the fields), decay
exponentially in time, with a finite characteristic time τa [11]. Then if we take ∆t ∼ τa
and discretize the time, the evolution of the model is described by the N -dimensional
map,
hi (n + 1) = (1 − ∆t)hi (n) + ∆t
N
X
Jij S(hj (n)).
(8)
j=1
The maximal Lyapunov exponent λ is obtained from the time-evolution of the tangent
vector [2, 3, 12],
(9)
ξ(n + 1) = A(n)ξ(n),
Aij (n) = (1 − ∆t)δij + g∆tJij cosh−2 (ghj (n)).
(10)
When g 1, the leading contribution to A(n) comes from |hj | < 1/g. Since ∆t ∼ τa ,
we can assume that h(n) and h(n0 ) are uncorrelated if n 6= n0 . Thus we can replace in
Eq. (10) cosh−2 by a constant, so that ξ(n) is given by a product of independent N × N
random matrices. In the limit N 1 the diagonal part of A(n) does not contribute
and one has [13, 14],


 ln Aij , if Aij 6= 0;
λ∼
(11)

 ln A2 , if A = 0;
ij
ij
where (·) means averaging over the different realizations of Jij ’s. Therefore,
λ ∼ ln g, g 1.
(12)
Before discussing a more systematic mean-field theory, let us spend a few words to
underline the main difficulties of the Amari mean field theory.
i) Stability of the static solution. Within the derivation of the Amari mean field theory
it is not possible to analyze the stability of the solution (6)-(7).
ii) Replica symmetry breaking. The symmetric SK model below the de Almeida Thouless line [15] shows the well known structure of the replica symmetry breaking [16]. Is this true also for our model? If it is so, the Hypothesis 1 cannot
obviously be true in the replica symmetry breaking phase [17].
5
iii) Time dependent solutions. The main deficiency of Amari’s approach is that time
dependent solutions cannot be self-consistently described by this mean field theory. It is easy to realize that the Hypothesis 2 does not hold for time dependent
solutions.
3
Dynamical Mean Field Theory
In this Section we study the large N and long time behavior of the model by means of
the dynamical mean-field theory developed to describe the dynamics of spin systems
with random couplings [11, 18]. We will consider here only the case of Jo = 0. The
Jo 6= 0 case will be discussed in a separate paper.
3.1
The Self-Consistent Equation of Motion
To treat the equation of motion (2) we use a functional integral formalism which is
very convenient for the discussion of quenched random systems. Following the standard procedure, we define a generating functional for dynamic correlation and response
functions,
Z
Z=
b exp
Dh Dh

X 
b i (t) + L[h, h]
b ,
li (t) hi (t) + ibli (t) ih
b =
L[h, h]
(13)

i,t

X



b i (t) (1 + ∂t )hi (t) −
−ih
i,t
X
Jij Sj (t) ,
(14)
j
P R
where i,t = i dt. We extended the the time integration to to → ∞, forgetting
about initial conditions. The operator ∂t should be intended as ∂t + δ (δ → 0+ ) to
ensure causality.
The generating functional can be averaged over the random matrices Jij . This
generates an effective action with non-local four-body interactions. By introducing
b t0 ), the non-local interactions can be decoupled by
two auxiliary fields C(t, t0 ) and C(t,
Gaussian transformations. This leads to the following generating functional,
P
Z
Z=
b =
Ω(C, C)
N
2
Z
n
b t0 ) C(t, t0 ) − ln
dt dt0 iC(t,
b C, C]
b =
Le [h, h,
X
o
b ,
DC DCb exp −Ω(C, C)
(15)
Z
b exp{Le [h, h,
b C, C]}
b
Dh Dh
(16)
b i (t)(1 + ∂t )hi (t) + li (t) hi (t) + ib
b i (t) +
−ih
li (t) ih
i,t
1X b
b i (t) ih
b i (t0 ) .
iC(t, t0 ) Si (t) Si (t0 ) + C(t, t0 ) ih
2 i,tt0
6
(17)
Note that unlike the symmetric case, only two auxiliary fields suffice for decoupling the
non-local interactions. As a consequence, the randomness in the couplings introduces
only an effective self-consistent random field in the single-spin dynamics, and does not
renormalize the bare propagator [4, 11, 19].
The system is fully connected, thus in the N → ∞ limit we expect that a saddle
point integration leads to the correct value of the integral. For a saddle point integration, however, we need homogeneous sources. This is achieved by replacing li and bli by
b i . With
homogeneous field l and bl, which generate the uniform correlations of Si and h
P
this change Le becomes diagonal in the site index, i.e., Le = i Li , and the functional
b factorizes into N identical integrals. In the limit N → ∞ the
integral over h and h
b
saddle point integration leads to the self-consistent equations for C and C,
C(t, t0 ) = < S(t) S(t0 ) >,
0
0
b
b ) >,
b t ) = < ih(t)
iC(t,
ih(t
(18)
(19)
where the average is taken with the weight exp{L1 } evaluated at the saddle point. The
b t0 ) ≡ 0 for b
l = 0, leading to the effective local
normalization of Z implies that C(t,
generating functional,
Z
Z=
Z
b =
L1 [h, h]
−
b e−L1 [h,h] ,
Dh Dh
b
1
b
dt ih(t)(1
+ ∂t )h(t) −
2
Z
Z
(20)
b
b 0)
dtdt0 ih(t)
C(t − t0 ) ih(t
b
dt[l(t) h(t) + ibl(t) ih(t)].
(21)
Provided that l and bl are zero, at the saddle point C(t, t0 ) is, by assumption of time
translational invariance of the saddle point, function only of the time difference. We
expect the averaging over Jij being equivalent to averaging over initial conditions.
b we finally get the self-consistent
Carrying out the functional integral over h and h,
single-spin equation of motion,
∂t h(t) = −h(t) + η(t),
(22)
where η(t) is a time-dependent Gaussian field with zero mean and variance,
< η(t) η(t0 ) >η = C(t − t0 ).
(23)
The spin correlation function C(t) has to be determined self-consistently through,
C(t − t0 ) =< S(t)S(t0 ) >η =< tanh(gh(t)) tanh(gh(t0 )) >η .
(24)
Note that the average over η is equivalent to average with the weight exp{L1 } (with
l = bl = 0), thus we can drop the subscript η. Equations (22)-(24) are the self-consistent
equations of motion describing the behavior of the system (2) at long time and large
N.
7
3.2
Solutions of the Self-Consistent Equation of Motion
In this section we discuss the solutions of Eqs. (22)-(24). Let us start with the timeindependent solutions, given by h(t) = const.. In this case C(t) is independent of
time, C(t) = q, so that η becomes a static Gaussian field. Therefore the solution of
Eqs. (22)-(24) reads,
Z
dη
2
e−η /2q tanh2 (gη).
(25)
q= √
2πq
which is nothing but the Amari solution (c.f.r. Eq. (7) with m = 0). Note that q = 0
for g < 1.
We now turn to the time-dependent solutions. The self-consistent equation for C(t)
is not local in time, we thus introduce the local field autocorrelation function,
∆(t) =< h(t0 + t) h(t0 ) > .
(26)
It is easy to verify that ∆(t) obeys the differential equation,
∆(t) − ∂t2 ∆(t) = C(t).
(27)
From Eq. (24) follows that C(t) is function only of ∆(t) and ∆o = ∆(t = 0). This can
be seen, e.g., by expanding the r.h.s. of Eq. (24) in powers of h. Therefore Eq. (27)
can be conveniently written as,
∂
d2
∆(t) = −
V (∆)
2
dt
∂∆
(28)
where
∆
∆2
e C(∆;
e ∆o )
V (∆) = −
d∆
+
2
0
∆2
= −
− g −2 [Φ]2∆o
2
Z
+ g −2
Z
Z
Dz
q
q
2
Dx Φ xg ∆o − |∆| + zg |∆|
(29)
with Φ(x) = ln cosh(x) (see Appendix), and we have introduced the short-hand notation,
Z
p
dx −z 2 /2
Dx = √
e
[f ]∆o = Dx f (xg ∆o );
.
(30)
2π
We have scaled V (∆) so that V (∆ = 0) = 0. Once ∆(t) is known, C(t) is obtained
through Eq. (27).
Equation (28) can be viewed as the equation of motion of one classical particle
in the one-dimensional classical potential V (∆). Note that V (∆) is a self-consistent
potential. It depends parametrically on ∆o , the value of which has to be consistent
with the solution of Eqs. (28)-(29).
8
The motion of the classical particle must satisfy very important boundary conditions
related to the nature of ∆(t) and C(t). First of all, since ∆(t) is an autocorrelation
function, it must be bounded, i.e., |∆(t)| ≤ ∆o . As a consequence only bounded
classical orbits are allowed. From Eq. (27) one has,
1
∆(t) =
2
Z +∞
0
dt0 e−|t−t | C(t0 ),
(31)
−∞
˙ = 0) = 0,
thus ∆(t) must be a differentiable even function of t. This implies that ∆(t
so that the classical orbits must have zero initial kinetic energy. We observe that there
is only one initial condition, namely the initial velocity of the classical particle, thus
we may expect many solutions corresponding to all the allowed values of the classical
˙ 2 /2 + V (∆) = V (∆o ).
energy Ec = ∆
Since all the orbits must be bounded, a qualitative analysis of the possible solutions,
can be obtained just studying the form of the potential V (∆) near the origin ∆ = 0.
Since V (∆) is an even function of ∆, we will assume that ∆ > 0. The ∆ < 0 branch is
obtained then by reflection.
If we expand V (∆) around the origin we have (see Appendix),
V (∆) = (−1 + g 2 [φ0 ]2∆o )
∆4
∆2
+ g 6 [φ000 ]2∆o
+ ···,
2
24
(32)
where,
dn
φ(x).
(33)
dxn
The potential V (∆) has either a single-well shape or a double-well shape, depending
on the sign of the coefficient of ∆2 . Therefore, taking into account the condition
|∆(t)| ≤ ∆o , the possible solutions are,
φ(n) (x) =
a)
−1 + g 2 [φ0 ]2∆o > 0,
(34)
quasi-harmonic oscillatory solutions around ∆ = 0 and positive classical energy;
b)
−1 + g 2 [φ0 ]2∆o = 0,
(35)
oscillatory solutions around ∆ = 0 and positive classical energy.
In the case,
−1 + g 2 [φ0 ]2∆o < 0,
(36)
there are four different classes of allowed solutions depending on the value of the energy,
i.e., of ∆o .
9
c)
∆2o
+ g −2 [Φ2 ]∆o − g −2 [Φ]2∆o > 0,
(37)
2
harmonic oscillatory solutions around ∆ = 0 and positive classical energy;
V (∆o ) = −
d)
∆2o
+ g −2 [Φ2 ]∆o − g −2 [Φ]2∆o = 0,
2
decaying solution, ∆(t) → 0 when t → ∞, and zero classical energy;
−
(38)
e)


∆2

 − o + g −2 [Φ2 ]∆o − g −2 [Φ]2∆o < 0,
2


 V 0 (∆ ) > 0,
(39)
o
non harmonic oscillatory solutions around ∆ 6= 0 and negative classical energy;
f)
V 0 (∆o ) = −∆o + [φ2 ]∆o = 0,
(40)
static solution, ∆(t) = const., corresponding to the minimum of the classical
energy.
We note that, independently of the value of g, there is always the solution ∆ ≡
∆o = 0.
Plotting the marginal curves (35), (38) and (40) in the plane (∆o , 1/g) we obtain
the “phase diagram” shown in Figure 1. From this we see that above the line (40),
indicated by f, there is only the static solution, Eq. 25, whereas each point below it
corresponds to a time-dependent solution. For g → ∞ the three curves reach the values
2/π, 2(1 − 2/π) and 1, respectively.
We then conclude that for g < 1 the only solution is the static solution ∆(t) ≡ 0.
The vanishing of the steady state equal time autocorrelation implies that for (almost)
all initial conditions the system flows to the zero fixed point {hi ≡ 0}. The stability
of this solution below g = 1 can be deduced by linearizing Eq. (2) and noting that the
maximum real part of the eigenvalues of J is 1 [20].
When g > 1 there are different classes of possible solutions. The form of V (∆) is
not unique but depends on the assumed value of ∆o . Furthermore, for any given shape
of V (∆) there is a continuum of allowed solutions corresponding to the continuum of
allowed values of the classical energy Ec . In general, there exists a value ∆1 (g) such that
for 0 < ∆o < ∆1 the potential has the shape shown in Figure 2(a). The solutions are,
therefore, periodic orbits, implying the system converges to limited cycles. For ∆o > ∆1
the potential V (∆) has the double-well shape shown in Figure 2(b), so that in this case
there are four different types of solutions. The solution with the lowest classical energy
10
[f] is the static solution, ∆(t) ≡ ∆ > 0, found by Amari, and corresponds to a non zero
fixed point of the system characterized by a non trivial distribution of hi ’s. This fixed
point would be analogous to a spin glass freezing which occurs in systems with random
symmetric Jij ’s. Solutions with Ec < 0 [e] represents oscillatory solutions with non
zero average, whereas solutions with Ec > 0 [c] oscillatory solutions with zero average.
Finally there is a decaying solution, which corresponds to Ec = 0 [d] for which ∆(t)
decreases monotonically to zero as t → ∞. The decay of the correlations between two
points along the flow implies that the flow is chaotic.
4
Stability of the Dynamic Mean Field Solutions
The large variety of allowed solutions for g above 1 rises the question of the stability of
such solutions. In other words, is the saddle point representing the mean field theory
stable for all these solutions? To answer this question we have studied the statistical
fluctuations about the mean field theory. This is done by going to a replicated system,
where the propagators of the fluctuations can be expressed by simple averages over
quantities corresponding to different replicas.
The analysis shows that for 1-spin system all the discussed solutions have a stable
mean field saddle point. This means simply that there exist fix points, orbits, attractors
with translational invariant correlation function. The question is if all these solutions
represent stable attractors. Going to a replicated system means that we take identically
preparated systems with almost equal initial conditions and look if the assumption of
translationally invariant correlation function leads to a stable mean field saddle point.
If not, the solution does not represent a stable ergodic attractor.
4.1
The Self-Consistent Equation of Motion of the Replicated System
The equations which describe the time evolution of the replicated system are,
∂t hαi (t) = −hαi (t) +
N
X
Jij Sjα (t)
j=1
= −hαi (t) +
N
X
Jij φ ghαj (t) .
(41)
j=1
The replica index α runs from 1 to the natural number n. As done for the 1-spin
system, we can use the functional integral method to perform the average over the
Jij ’s. Following the same steps of Section 3.1 we arrive at the effective generating
functional,
Z
Zn =
n
o
b ,
DC DCb exp −N Ωn (C, C)
11
(42)
b = −
Ωn (C, C)
1 X b αβ αβ
αβ b αβ
iCab +
iCbab
iC C
2 αβ,ab ab ab
αβ,ab
X
− ln
b
b
L(n)
e [h, h, C, C] = −
Z
b exp{L(n) [h, h,
b C, C]}
b
Dh Dh
e
X
b α (1 + ∂a )hα + lα hα + ib
bα
ih
laα ih
a
a
a a
a
(43)
α,a
+
1 X b αβ α β
αβ b α b β
iha ihb .
iCab Sa Sb + Cab
2 αβ,ab
(44)
We have introduced the short-hand notation,
Saα ≡ S α (ta ),
X
α,a
≡
XZ
αβ
≡ C αβ (ta , tb ),
Cab
dta , ∂a ≡ ∂ta + δ
(δ → 0+ ).
α
αβ
βα
αβ
βα
We may restrict ourselves to symmetric variables, Cab
= Cba
and Cbab
= Cbba
. In
order to have well defined integrals we have used the following integral representation
of the δ-function,
Z
dy −y 2 /2 − iyx
δ(x) =
e
, → 0+ .
(45)
2π
This form is very useful since it does not introduce extra instability.
αβ
αβ
In the N → ∞ limit the functional integrals over Cab
and Cbab
can be performed
by saddle point integration. This leads to the self-consistent equation,
αβ
αβ
Cab
= −iCbab
+ < Saα Sbβ >,
αβ
iCbab
= <
b α ih
bβ
ih
a
b
>,
(46)
(47)
(n)
where the average < · · · > is taken with the weight exp{Le } evaluated at the saddle
point. From the invariance of the saddle point under time translations and exchange
αβ
αβ
of replica, we expect that Cab
and Cbab
are symmetric functions of the replica indexes
and even functions of the time difference.
αβ
The conservation of the normalization of Z n leads to Cbab
≡ 0. Substituting this
b and h we
solution into Eqs. (42)-(44), and performing the functional integral over h
obtain the self-consistent single-spin equation of motion,
∂a hαa = −hαa + ηaα ,
(48)
where ηaα is a time-dependent Gaussian field with zero mean and variance,
αβ
< ηaα ηbβ >= Cab
.
12
(49)
αβ
has to be determined self-consistently through,
The correlation function Cab
αβ
= < Saα Sbβ >
Cab
= < tanh(ghαa ) tanh(ghβb ) > .
(50)
As done for the 1-spin system it is useful to introduce the local field autocorrelation
α β
function ∆αβ
ab =< ha hb > which obeys the differential equation,
αβ
(1 + ∂a )(1 + ∂b )∆αβ
ab = Cab .
(51)
Note that, since the correlations of hαa are Gaussian, the l.h.s. of Eq. (50) is function
ββ
αα
only of ∆αβ
ab , ∆aa and ∆bb . The solutions found in Section 3.2 correspond to the replica
symmetric solutions of Eqs. (48)-(51).
4.2
Fluctuations About the Saddle Point
To study the stability of the replica symmetric solutions, we consider the Gaussian
fluctuations about the saddle point representing the dynamic mean field theory. For
αβ
this purpose we expand Ωn around the saddle point (46)-(47) with Cbab
= 0. Keeping
αβ
αβ
b
only the second order terms, and indicating by Qab and iQab the fluctuations, we find,
Zn
Z
=
(o)
b
Ω(2)
n (Q, Q) = Ωn −
n
o
b exp −N Ω(2) (Q, Q)
b ,
DQ DQ
n
(52)
X
X
b αβ )2 + 1
b αβ Qαβ
(iQ
iQ
ab
4 αβ,ab
2 αβ,ab ab ab
−
1 X
b αβ [< S α S β S γ S δ > − < S α S β >< S γ S δ >]iQ
b γδ
iQ
a b c d
a b
c d
cd
8 αβγδ,abcd ab
−
1 X
γδ
b γ ih
bδ
b αβ < S α S β ih
iQ
a b
c d > Qcd ,
4 αβγδ,abcd ab
(53)
(o)
where Ωn is the value of Ωn at the saddle point.
To calculate the averages we introduce Ψαβ
ab defined as,
αβ
(1 + ∂a )(1 + ∂b )Ψαβ
ab = Qab .
(54)
Eq. (53) can be rewritten as,
(o)
b
Ω(2)
n (Ψ, Q) = Ωn −
X
1 X
b αβ M αβ,γδ iQ
b γδ + 1
b αβ B Ψαβ ,
iQ
iQ
ab
ab,cd
cd
ab
8 αβγδ,abcd
2 αβ,ab ab
(55)
where,
αβ,γδ
αβ,γδ
Mab,cd
= δab,cd
+ < Saα Sbβ Scγ Sdδ > − < Saα Sbβ >< Scγ Sdδ >,
13
(56)
with
αβ,γδ
= δαγ δ(ta − tc ) δβδ δ(tb − td ) + δαδ δ(ta − td ) δβγ δ(tb − tc )
δab,cd
(57)
and the operator B is given by,
BΨαβ
ab
= (1 + ∂a )(1 +
−
∂b )Ψαβ
ab
Ψαβ
ab
αβ
∂∆ab
< Saα Sbβ > ββ
Ψbb .
∂∆ββ
bb
−
∂ < Saα Sbβ > αα ∂
Ψaa −
∂∆αα
aa
∂ < Saα Sbβ >
(58)
αβ
b is well defined.
is positive semidefinite (for > 0) the integration over Q
Since Cab
+
Therefore the fluctuations are stable if the operator B B, where B+ is the adjoint
of B, is positive definite. This implies that B must not have zero eigenvalues. From
Eq. (58) we have the spectral equation of B with eigenvalue Λ,
αβ
BΨαβ
ab = ΛΨab .
4.3
(59)
Stability of the Dynamic Mean Field Solutions
To study the stability of the fluctuations around a given solution, we have to substitute
to < Saα Sbβ > its expression in the given solution and look for non-trivial solution of
αβ
βα
αβ
βα
= Cba
and Cbab
= Cbba
, so we may
Eq. (59) with Λ = 0. At the saddle point Cab
restrict to fluctuations which are symmetric in the replica and time indexes. There are,
then, only two types of fluctuations,
(
and
Ψαβ (ta , tb ) = ΨS (ta , tb ),
ΨS (tb , ta ) = ΨS (ta , tb ),

αβ
αβ

 Ψ (ta , tb ) = ΨA (ta , tb ),
Ψ (t , t ) = −ΨA (ta , tb ),
A b a

 βα = −αβ
(60)
(61)
We will call the fluctuations (60) “symmetric fluctuations”, whereas the fluctuations
(61) will be called “antisymmetric fluctuations”.
Let us consider first the time-independent solution. For g < 1 Eq. (59) becomes,
h
i
αβ
(1 + ∂a )(1 + ∂b ) − g 2 Ψαβ
ab = ΛΨab ,
(62)
for both symmetric and antisymmetric solutions. Taking the Fourier transform with
respect to ta and tb and substituting Λ = 0 we find the stability equation,
(1 − iωa + δ)(1 − iωb + δ) − g 2 = 0, δ → 0+ .
14
(63)
Since g is real Eq. (63) may have solution only if ωb = −ωa and,
(1 + δ)2 − g 2 + ωa2 = 0.
(64)
Equation (64) does not have solution for 1 − g 2 ≥ 0, so that the static solution q = 0
is stable for g < 1.
For g > 1 the stability equation for the static solution reads,
h
i
(1 + ∂a )(1 + ∂b ) − g 2 [φ0 ]q Ψαβ
ab −
g2
αβ
ββ
[φφ00 ]q (Ψαα
aa + Ψbb ) = ΛΨab ,
2
(65)
where [φ(n) ]q is given by Eq. (30) with ∆o = q. In this case we must distinguish between
the symmetric and the antisymmetric fluctuations. For the former Eq. (65) reads,
h
g2
[φφ00 ]q (ΨS (ta , ta ) + ΨS (tb , tb ))
2
= ΛΨS (ta , tb ).
(66)
i
(1 + ∂a )(1 + ∂b ) − g 2 [φ02 ]q ΨS (ta , tb ) −
Since Λ would be complex if ΨS (ta , tb ) does not depend on τ = ta −tb , we will restrict to
fluctuations which depend only on τ . Then taking the Fourier transform with respect
to τ we have the stability equation,
(1 − iω + δ) − g 2 ([φ02 ]q + [φφ00 ]q ) = Λ, δ → 0+ ,
(67)
which, for ω = 0 and Λ = 0, correspond to the second eigenvalue found by de Almeida
and Thouless for symmetric SK model [15]. Since this is non-zero for all g > 1, symmetric fluctuations around the static solution are stable also for g > 1.
For antisymmetric fluctuations the stability equation reads,
(1 − iωa + δ)(1 − iωb + δ) − g 2 [φ02 ]q = Λ, δ → 0+ .
(68)
ββ
since Ψαα
aa = Ψbb = 0. Again non-trivial solutions with Λ = 0 can be obtained only for
ωb = −ωa so that the stability equation reads,
(1 + δ)2 − g 2 [φ02 ]q + ωa2 = 0,
(69)
which leads to an instability since for the static solution,
1 − g 2 [φ02 ]q < 0,
g > 1.
(70)
We observe that Eq. (70) corresponds to the instability condition of the replica symmetric solution of the SK model found by de Almeida and Thouless [15].
In conclusion, therefore, the static solution (25), i.e., the Amari solution, is stable
only below g = 1. For g > 1 we have to look for time-dependent solutions. We then
turn our attention to the stability of these solutions.
15
For time-dependent (replica symmetric) solutions the stability equation reads (See
Eqs. (27) and (28)),
"
#
∂2
V (∆, ∆o ) Ψαβ
∂a + ∂b + ∂a ∂b −
ab
2
∂∆
−
h
i
1 ∂2
ββ
V (∆, ∆o ) Ψαα
= ΛΨαβ
aa + Ψbb
ab .
2 ∂∆∂∆o
(71)
Again, since Λ would be complex if Ψαβ
ab does not depend only on τ = ta − tb , we will
limit ourselves to this case. With this assumption Eq. (71) becomes,
"
#
2δ −
−
∂τ2
∂2
−
V (∆, ∆o ) Ψαβ (τ )
∂∆2
h
i
1 ∂2
V (∆, ∆o ) Ψαα (0) + Ψββ (0) = ΛΨαβ (τ ).
2 ∂∆∂∆o
(72)
As found for the static solution, the critical fluctuations are the antisymmetric ones.
In fact, one can see that the term ∝ ∂ 2 V /∂∆∂∆o stabilizes symmetric fluctuations.
For antisymmetric fluctuations the second term in the l.h.s. of Eq. (72) is zero, and
the stability equation reduces to a ‘sort’ of quantum mechanical problem associated to
the classical equation of motion (28)-(29),


 HΨ(τ ) =


!
∂2
∂2
V (∆, ∆o ) Ψ(τ ) = Ψ(τ )
− 2−
∂τ
∂∆2
(73)
Λ = + 2δ
˙ ).
From Eq. (28) it follows that the solution of Eq. (73) with = 0 is Ψ(τ ) = ∆(τ
˙
For the oscillatory solutions ∆(τ ) is a periodic function which changes sign within
one period To , and vanishes at τ = 0. Thus from elementary one-dimensional quantum
mechanics we know that there is exactly one periodic eigenfunction with o < 0 which
vanishes only at τ = 0, To . However, for periodic solutions the quantum mechanical
potential ∂ 2 V /∂∆2 is periodic itself. Thus the Bloch theorem implies that for such
potentials the states are not isolated, but organized in continuous bands. Therefore
there will be a continuum of solutions with close to 0− as we want, so that Λ passes
continuously through zero whatever small δ is. As a consequence all the oscillatory
solutions are unstable.
The quantum mechanical potential generated by the chaotic solution, on the other
hand, is not periodic and the states with = 0 and o < 0 are isolated. This implies
that the solution is (marginally) stable since Λ = 2δ + o cannot be zero and Λ = 2δ
is marginally positive (for δ → 0+ ). We then conclude that for g > 1 the only stable
solution is the chaotic one.
16
5
Maximal Lyapunov Exponent
For the chaotic solution we are able to calculate the Gaussian fluctuations around the
saddle point. As one might expect those fluctuations involving two replica are related
to the growth of the distance of two nearby starting trajectories in the vicinity of the
chaotic attractor.
5.1
Maximal Lyapunov Exponent from Fluctuations
Assuming that a small change δhoi of initial conditions at time to induces a small change
δhi (t) at time t, we linearize Eq. (2),
∂t δhi (t) = −δhi (t) + g
N
X
Jij φ0 (ghj ) δhj (t).
(74)
j=1
The solution of this linear equation can be written with the help of a susceptibility
eij (t, to ),
function χ
δhi (t) =
N
X
eij (t, to )δhoj .
χ
(75)
j=1
The Lyapunov exponents describe the asymptotic growth for large times. More precisely, the Lyapunov exponents are half of the growth rates of the eigenvalues of the
symmetric matrix,
1 X
eki χ
ekj
χ
(76)
N k
for t → ∞ [3]. The system is chaotic if the maximal Lyapunov exponent, λ, is positive.
The exponent λ can be obtained directly from the trace of the matrix (76),


1
1 X 2
e (t, to )
λ = lim ln 
χ
t→∞ 2t
N ij ij
(77)
In what follows the trace will be related to the Gaussian fluctuations around the saddle
point. We are, therefore, able to calculate its configurational average or, equivalently,
P
e2ij /N
its limits for N → ∞. Hence λ is the exponent that dominates the sum χ2 = ij χ
in the limit N → ∞. Fortunately this quantity is selfaveraging, so that we obtain the
average maximal Lyapunov exponent in the limit N → ∞. In fact, in principle the
average over Jij should be performed on ln χ2 . However, in the N → ∞ limit χ2 is
selfaveraging, which implies that the system is not intermittent and the average and
eij in Eq. (77) may
the logarithm can be interchanged [14]. Under these conditions χ
also be replaced by the usual susceptibility χij ,
χij (t, to ) =
δSi (t)
eij (t, to ).
= gφ0 (ghi ) χ
δhj (to )
17
(78)
Starting from the generating functional (42) we obtain by partial integration with
γδ
αβ
the following identity (for → 0),
and Cbcd
respect to Cab
D
E
E
1 X D α
αβ,γδ
αβ b γδ
b δ (t ) .
b γ (tc ) ih
Si (ta ) Siβ (tb ) ih
+
iCcd = δab,cd
N Cab
d
j
j
N ij
(79)
The angular brackets mean the functional integral with the full weight from Eqs. (42)(44). The second term of the r.h.s. of Eq. (79) evaluated for α = γ 6= β = δ, ta = tb = t,
tc = td = to , is just the trace we need for the maximal Lyapunov exponent λ.
For large N we calculate the fluctuations in the Gaussian approximation around
the saddlepoint . This means we have to replace C and iCb in the l.h.s. of Eq. (79) by
b and evaluate the functional integral with (52)-(53). Instead of Qαβ one may
Q and iQ,
ab
again introduce Ψαβ
ab by Eq. (54). For N → ∞ this is equivalent to passing from local
spin variables Si to local field variables hi . We find the fluctuation equation,
"
∂ < Saα Sbβ > D
#
(1 + ∂a )(1 + ∂b ) −
∂∆αβ
ab
α β
E
D
E
b γδ − ∂ < Sa Sb > Ψαα iQ
b γδ
Ψαβ
i
Q
aa
ab
cd
cd
∂∆αα
aa
−
∂ < Saα Sbβ > D
∂∆ββ
bb
E
1 αβ,γδ
b γδ
Ψββ
δ
,
bb iQcd =
N ab,cd
(80)
corresponding to the eigenvalue equation (59). Existence of mean field fluctuations,
invertibility of the operator B and stability turn out to be equivalent. For the Lyapunov
exponent λ, equation (77), we need
D
E
b αβ
χ2 = N Ψαβ
ab iQab ,
(81)
with ta = t and tb = to .
We turn our attention to the case α = γ 6= β = δ. Setting,
τ = ta − tb ,
τ 0 = tc − td ,
(82)
T = ta + tb ,
T 0 = tc + td ,
(83)
and
we obtain the equation,
!
D
E
∂
∂2
2
αβ b αβ
2
+
+
H
Ψ
=
δ(τ − τ 0 ) δ(T − T 0 ),
i
Q
ab
cd
2
∂T
∂T
N
(84)
where H is the hamiltonian of Eq. (73). We may expand in (real) orthonormalized
eigenfunctions ϕn of H with eigenvalue n ,
D
E
X
b αβ = 2
gn (T, T 0 ) ϕn (τ ) ϕn (τ 0 ),
Ψαβ
i
Q
ab
cd
N n
18
(85)
where
!
∂
∂2
+ n gn (T, T 0 ) = δ(T − T 0 ).
2
+
∂T
∂T 2
The ansatz gn ∼ exp(λn T ) for T > T 0 yields,
√
λn = −1 ± 1 − n .
(86)
(87)
The solution gn (T, T 0 ) of Eq. (86) that vanishes for T < T 0 is given by,
√
θ(T − T 0 ) −(T −T 0 )
gn (T, T 0 ) = √
e
sinh{(T − T 0 ) 1 − n }.
1 − n
We conclude that the maximal Lyapunov exponent is given by,
√
λ = λo = −1 ± 1 − o .
(88)
(89)
where o is the ground state energy of Eq. (73). This implies an exponential increasing
solution if o < 0. The averaged quantity χ2ij is given by,
χ2ij = 2
X
√
√
e−2t sinh 2t 1 − n ϕ2n (0)/ 1 − n ,
(90)
n
for t > 0. The sum over n may include a continuum part of the spectrum of H. Odd
˙ do not contribute to the sum. Of
functions ϕn , like the zero eigenvalue solution ∆,
particular interest is the contribution of the ground state o . Note, that the r.h.s. of
Eq. (90) is finite for N → ∞.
To solve Eq. (73) for an arbitrary g > 1 it is not an easy task. However, in the
limit cases of g → 1+ and g → ∞ this can be done, and will be presented below.
Here we note that from the discussion at the end of Section 4.3, we conclude that the
dynamically stable mean field solution for g > 1 gives rise to an Hamiltonian H with a
negative ground state energy o . Therefore the solution has λ > 0 for all g > 1, i.e., it
is chaotic.
5.2
Maximal Lyapunov Exponent from a Ground State: g → 1+
In the limit g → 1+ , |∆(t)| ≤ ∆o 1 for all t. Expanding V (∆) in powers of ∆ and
∆o (See Eq. (32)), and keeping only the terms up to the fourth order, one finds,
V (∆) = −1 + g 2 − 2g 4 ∆o + 5g 6 ∆2o
∆2
2
+ g6
∆4
.
6
(91)
The value of ∆o is given by the non-trivial solution of the equation V (∆o ) = 0 (See
Eq. (38)), and reads,
4
∆o = σ − σ 2 + O(σ 3 ),
(92)
3
19
where σ = (g 2 − 1)/2. If this expression of ∆o is substituted back into Eq. (91) one
has,
σ2
1
V (∆) ' − ∆2 + ∆4 ,
(93)
6
6
form which is valid for σ 1. Note that the coefficient of ∆2 is negative, ensuring the
double-well shape of the potential. Indeed in this limit ∆1 = σ − 32 σ 2 + O(σ 3 ) < ∆o .
The chaotic solution has zero classical energy Ec , so that,
√ Z
Z ∆
3 ∆o /σ
dx
d∆
p
√
=
.
(94)
t=−
σ ∆/σ x 1 − x2
−2V (∆)
∆o
Evaluating the integral and retaining only the leading terms in σ we finally have,
√ ∆(t) ' σ cosh−1 σt/ 3
(95)
Note that as g → 1+ the amplitude of the autocorrelation function vanishes as ∆o ∼ σ,
whereas the characteristic relaxation time diverges as τa ∼ σ −1 .
To find the maximal Lyapunov exponent λ we have to solve the quantum equation
(73). Inserting the expression (93) of V (∆), with ∆(t) given by Eq. (95), and keeping
only the lowest order in σ, the quantum equation takes the form,
√ ∂2
σ2
2
−2
3 Ψ(τ ) = 0.
Ψ(τ
)
+
−
+
2σ
cosh
σt/
∂τ 2
3
#
"
(96)
The solutions of this differential equation are the generalized Legendre functions with
energy levels [21],
i
σ2 h
n = −
(2 − n)2 − 1 .
(97)
3
The ground state energy is therefore o = −σ 2 so that,
λ'
σ2
,
2
σ 1.
(98)
Note that the divergence of the characteristic time λ−1 as g → 1+ is stronger than
that of the autocorrelation function τa .
5.3
Maximal Lyapunov Exponent from a Ground State: g → ∞
It turns out that in the limit g → ∞ only times t < 1/g are important. Therefore only
the small time behaviour of the g = ∞ chaotic solution enters. The classical equation
for g = ∞ reads,
d2
∂
2
∆(t)
∆(t) = −
V (∆) = ∆(t) − ∆o arcsin
,
2
dt
∂∆
π
∆o
20
(99)
with ∆o = 2(1 − 2/π) for the chaotic solution. Expanding near t = 0 we have,
t2
+ ...
(100)
2
Inserting only the lowest order terms, we obtain for g → ∞ the quantum equation (73)
in the form,
!
∂2
(101)
− 2 − gf (gτ ) Ψ(τ ) = Ψ(τ ),
∂τ
∆(t) = ∆o + (∆o − 1)
with the scaling function,
Z
Z
f (y) =
−2
Dz
q
p
xy (1 − ∆o )/2 + z ∆o
Dx cosh
2
,
(102)
This function has for g → ∞ the asymptotic form,
f (y) =' C/y,
(103)
with
2 q
/ ∆o (1 − ∆o ) = 1.4286
π
We may rescale the quantum equation,
C=
/g 2 → E,
gτ → y,
(104)
(105)
!
−
∂2
1
− f (y) Ψ(y) = EΨ(y).
2
∂y
g
(106)
From usual quantum mechanics we know,
dE
1
= 2
dg
g
R∞
2
0 R dyΨ(y) f (y)
∞
(107)
dyΨ(y)2
√
The form of Ψ(y) for large y is Ψ(y) ' exp −|y| −E , which mainly contributes to
the integrals for large g. We obtain asymptotically,
√
dE
2C √
= − 2 −E log −E,
(108)
dg
g
with the asymptotic solution,
√
C
−E ' log g.
(109)
g
Hence the maximal Lyapunov exponent for g → ∞ is given by,
√
λ ' −o ' C log g,
(110)
0
in agreement with the result obtained form the random matrices approach of Section 2.2, Eq. (12).
On the other hand the characteristic relaxation time of the autocorrelation function
is finite in the limit g → ∞. In fact substituting
p ∆(t) ∼ ∆o exp(−t/τa ) into Eq. (99)
−1
and taking the limit t → ∞, we have τa = 1 − 2/π in agreement with the value
found in Reference [11] for the fully asymmetric Ising spin SG model.
21
6
Numerical Results
In this section we rapidly review the results of the numerical simulations of the model
(1) and (2). The natural question which at this point is to what extent the above
results, i.e., the results of the mean field theory, apply to a large but finite system. Tho
study this point we have solved numerically (1) and (2) up to system size N = 1000,
with several different realizations of the couplings Jij and initial conditions. We limited
ourselves to analyse the possible different types of flows. If the mean field solution is
valid also for large but finite systems we expect that, depending on the value of g,
the typical behaviour should be either stationary or chaotic, but nothing else. This,
however, does not happen in a finite system. Indeed the most important new feature is
the appearance of an intermediate regime separating the steady and the chaotic phases.
We may summarise the results of numerical simulations as follows. If g < 1, for
almost realizations of couplings and/or initial conditions, the solution rapidly decays
to a zero fixed point. On the other side, for g sufficiently larger than one, we observe
the expected chaotic behaviour. This are the steady and chaotic solutions predicted by
the mean field theory. However, for g just larger than one a new behaviour emerges.
In this region we observe either non zero fixed points or, more often, limit cycles. The
appearance of one or of the other type of solution depends on the realization of Jij ’s
but not on the initial condition. When g is increased the number of runs which leads
to a non zero fixed point decreases, whereas the limit cycles become more and more
complex.
The fraction of runs which leads to a non zero fixed point is always very small.
Thus we can draw the the following picture for the typical flow. Systems with large but
finite N undergo a transition from the stationary state to chaos through a sequence of
bifurcations which increase the number of fundamental frequencies of the limit cycles.
Varying g then the system goes from a steady state to a limit cycle with one frequency,
then with two, and so on till the motion becomes chaotic.
One may be tempted to conclude that the limit cycles (and the non zero fixed
points) are the unstable solutions of the mean field theory. However, most probably
the two are not directly related since, for fixed g, the frequencies of the limit cycles
vary strongly from one realization of Jij ’s to another. Actually they are determined
only by the Jij ’s. Systems with the same couplings, but different initial conditions flow
to the same limit cycle. We note that most probably also the sequence of bifurcations
leading to the cahotic behaviour is not universal, but depends on Jij . We, however, did
not study this point sistematically.
The range of value of g where this intermediate regime occurs depends on N , and
shrinks to zero as N increases. Therefore, in the limit N → ∞ a sharp transition to
chaos emerges in agreement with the mean field theory. All this can be summarised in
the schematic phase diagram shown in Figure 3.
22
7
Conclusions
We have studied the dynamic behaviour of the deterministic nonlinear system (2) as
function of the nonlinearity g. In the limit N → ∞ the dynamics can be studied by
means of the dynamic mean field theory developed for spin glasses [4, 18]. This leads to
a full description of the statistical properties of the flow. By considering the Gaussian
fluctuations about the saddle point describing the mean field theory, we are able to
compute the maximal Lyapunov exponent of the flow. This is done explicitly in the
limit of large nonlinearity and near the transition to chaos. In order to check the results
of the mean field theory we have also solved numerically the model.
The main results is that in the N → ∞ limit the system undergoes a sharp transition
from a steady state (g < 1) to a chaotic flow (g > 1). Near the critical point the
Lyapunov exponent goes as (g − 1)2 , whereas the characteristic relaxation time of the
autocorrelation diverges as (g−1)−1 . For a finite system the two phases are separated by
an intermediate region where the flow converges mainly to limit cycles. The frequencies
of the cycles are not universal but depend strongly from the realization of the coupling
constants Jij . We have then the following picture for the typical behaviour of finite
systems. Systems with large but finite number of degrees of freedom N , undergo a
transition from a steady state to chaos through intermediate stages, where the motion
converges to limit cycles with increasing degree of complexity. However, the range of
g where this intermediate regime occurs shrinks to zero as N increases. Therefore, in
the N → ∞ limit a sharp transition to chaos emerges as predicted by the mean field
theory.
To conclude we would like to stress two points. First of all the above results clearly
shows that the nonlinearity in the equation of motion plays a drammatic role on the
spin glass freezing. We expect that to be true also for partially asymmetric models.
Then for low nonlinearity, the system flows towards fixed point. This agrees with
the result found for partially asymmetric spherical models [4, 19, 22]. On the other
hand for high nonlinearity, the system behaves chaotically (in the thermodynamic limit
N → ∞), and all the autocorrelation functions evaluated along the flow decay to zero
with a finite characteristic time. The second remarks is that to our knowledge this is
the first model with many degrees of freedom where one is able to compute analytically
the maximal Lyapunov exponent.
The phase diagram of the model in an external field, as well as at finite temperature
will be presented in separated papers.
Acknowledgments
A special thanks goes to P. C. Hohenberg for most valuable discussions. We have
also benefit from discussions with D. Ben Simon, B. Derrida, D. S. Fisher, D. Kleinfeld, P. B. Littlewood, I. Procaccia, B. Shraimann, and A. Vulpiani. This research
23
is partially supported by a grant of the U.S.A.-Israel Binational Science Foundation.
One of us (H.J.S.) acknowledges support from the Deutsche Forschungsgemeinschaft,
Sonderforschungsbereich No. 237.
Appendix
In this Appendix we derive Eq. (29) and Eq. (32). To do this it is useful to work with
the Fourier components of φ, defined as,
Z +∞
dk
φ(x) =
−∞
2π
g(k) e−ikx .
(111)
From Eqs. (24),(27) and (28) it follows,
∂V (∆)
∂∆
= −∆ +
= −∆ +
Z
g 2 ∆o 2
dk dk 0
g(k) g(k 0 ) exp{−
(k + k 02 ) − g 2 ∆kk 0 }
2π 2π
2
Z
Z
q
q
2
Dx φ xg ∆o − |∆| + zg |∆|
Dz
(112)
By taking successive derivatives with respect to ∆ one obtains the following expression,
∂ n V (∆)
∂n
=−
n
∂∆
∂∆n
∆2
2
!
+ g 2n−2
Z
Z
Dz
q
q
2
Dx φ(n−1) xg ∆o − |∆| + zg |∆|
(113)
where φ(n) (x) is defined in Eq. (33). Equation (29) is then obtained by taking n = 0
in Eq. (113). To derive Eq. (32) we note that φ(x) is an odd function, so that,
∂n
V (∆)
=0
n
∂∆
∆=0
24
if n = odd.
(114)
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26
Figure Captions
Fig. 1 Plot of the marginals curves (35) [b], (38) [d], (40) [f] (see text).
Fig. 2 Qualitative shape of the classical potential V (∆) for g > 1 in the range
|∆| ≤ ∆o . (a) 0 < ∆o < ∆1 . (b) ∆o > ∆1 (see text). A solution starting at the
point f is the static state. The points c and e are examples of initial conditions
leading to oscillatory solutions with zero and non-zero average, respectively. The
solution starting from d decays to zero.
Fig. 3 Schematic phase diagram in the plane (1/N, g).
27