We study persistence in coupled circle map lattices at the onset of
spatiotemporal intermittency, an onset which marks a continuous transition, in
the universality class of directed percolation, to a unique absorbing state. We
obtain a local persistence exponent of theta_l = 1.49 +- 0.02 at this
transition, a value which closely matches values for theta_l obtained in
stochastic models of directed percolation. This result constitutes suggestive
evidence for the universality of persistence exponents at the directed
percolation transition. Given that many experimental systems are modelled
accurately by coupled map lattices, experimental measurements of this
persistence exponent may be feasible.Comment: 7 pages, Latex, 6 Postscript figures, Europhysics Letters (to appear

Bray A. J.

Daviaud F.

Derrida B.

G. I Menon

Janaki T. M.

Kaneko K. (ed)

Majumdar S. N.

Mohanty P. K.

Oerding K.

P Ray

Rupp P.

S Sinha

English

arXiv

We study persistence in coupled circle map lattices at the onset of spatio-temporal intermittency, an onset which marks a continuous transition, in the universality class of directed percolation, to a unique absorbing state. We obtain a local persistence exponent of θl=1.49±0.02 at this transition, a value which closely matches values for θl obtained in stochastic models of directed percolation. This result constitutes suggestive evidence for the universality of persistence exponents at the directed percolation transition. Given that many experimental systems are modelled accurately by coupled map lattices, experimental measurements of this persistence exponent may be feasible

Menon, G. I.

Sinha, S.

Ray, P.

Indian Academy of Sciences

Persistence at the onset of spatio-temporal intermittency in coupled map lattices

Crossref

Publications of the IAS Fellows

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Europhysics Letters PREPRINT
Persistence at the onset of spatiotemporal intermittency
in coupled map lattices
Gautam I. Menon(∗), Sudeshna Sinha(∗∗) and Purusattam Ray(∗∗∗)
The Institute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai 600 113,
India
PACS. 05.45.-a – Nonlinear dynamics and nonlinear dynamical systems.
PACS. 02.50.Ey – Stochastic Processes.
PACS. 05.45.Ra – Coupled Map lattices.
Abstract. – We study persistence in coupled circle map lattices at the onset of spatiotemporal
intermittency, an onset which marks a continuous transition, in the universality class of directed
percolation, to a unique absorbing state. We obtain a local persistence exponent of θl =
1.49± 0.02 at this transition, a value which closely matches values for θl obtained in stochastic
models of directed percolation. This result constitutes suggestive evidence for the universality
of persistence exponents at the directed percolation transition. Given that many experimental
systems are modelled accurately by coupled map lattices, experimental measurements of this
persistence exponent may be feasible.
The study of extended dynamical systems is relevant to the understanding of many phe-
nomena in condensed matter physics, such as pattern formation and non-equilibrium phase
transitions in coupled chemical reactions, charge density waves and Josephson junction ar-
rays [1]. A simple model which captures much of the underlying complexity of these systems
is the coupled map lattice, defined as a collection of elements on a lattice which locally exhibit
chaotic dynamics, together with a diffusive local coupling between these elements [2]. Coupled
map lattices exhibit a remarkable variety of behaviour, ranging from periodic spatio-temporal
structure to intermittency and chaos.
Spatially extended dynamical systems exhibiting spatio-temporal intermittency possess
both laminar and turbulent phases. The laminar phase is characterized by periodic or even
weakly chaotic dynamics, while no spatio-temporally regular structure exists in the turbulent
regime. Spatio-temporal intermittency refers to the properties of the state which marks the
transition between laminar and turbulent phases. A typical spatial pattern arising in spatio-
temporal intermittency consists of fluctuating domains of laminar regions interspersed amidst
turbulent ones. Such intermittency is often the precursor of fully developed chaos [2].
An initially laminar site becomes turbulent only if at least one of its neighbours was
turbulent at the previous time. A turbulent site can either relax spontaneously to a laminar
(∗) E-mail:menon@imsc.ernet.in
(∗∗) E-mail:sudeshna@imsc.ernet.in
(∗∗∗) E-mail:ray@imsc.ernet.in
c© EDP Sciences
2 EUROPHYSICS LETTERS
state or contaminate its neighbours at the next time step [3]. In some regimes of parameter
space, turbulent states percolate in space-time, leading to an active phase. In other regimes,
turbulent states die out, following which the system is confined to the laminar (equivalently,
inactive) state for all time. This laminar state is thus a unique absorbing state.
It has been conjectured that critical exponents associated with the onset of spatiotemporal
intermittency belong generically to the universality class of directed percolation [4]. Underly-
ing this conjectured correspondence is the following more general question: Are transitions in
deterministic chaotic systems governed by the universality classes of stochastic models [5]? To
address this issue, we study persistence and persistence exponents in coupled map lattices at
the onset of spatiotemporal intermittency. Persistence exponents for stochastic systems have
attracted attention recently as they appear to be a new class of exponents, not derivable in
general from the other static and dynamic exponents [6]. Systems which exhibit a non-trivial
persistence exponent include Ising and Potts models in one and higher dimensions, the diffu-
sion equation, the random velocity and random acceleration problems and several models for
interface growth [7].
Our principal result is the following: We find that the local persistence exponent θl at
the onset of spatiotemporal intermittency exhibits a remarkable universality at the critical
points we have examined. The value of the persistence exponent is θl = 1.49± 0.02, in close
agreement with values obtained for stochastic models with discrete states in one dimension,
at the directed percolation transition [9,8,10]. Our model, however, differs from these earlier
models in that it is not stochastic but deterministic. Another important difference is that the
local variable in our case is a continuum variable, unlike in the models studied earlier.
This result is unusual for the following reason: The persistence exponent is known to be far
less universal than the other three exponents which typically characterize scaling behaviour
in interacting stochastic systems, the static exponents η and ν and the dynamical exponent
z. The reason is that persistence probes the full, in general non-Markovian, time evolution of
a local fluctuating variable, such as a spin or density field, from its initial state [7]. Knowing
the asymptotic properties of the evolution kernel governing this time evolution is insufficient
to obtain the persistence exponent, although in many cases progress can be made through
controlled expansions about Markov processes [7]. Thus, such universality of the persistence
exponent at the directed percolation transition, in particular for systems with deterministic
chaos, is a priori unexpected.
We study the dynamics of a 1-dimensional coupled map lattice, with on-site circle maps
coupled diffusively to nearest neighbours [2, 11]:
xi,t+1 = f(xi,t) +
ǫ
2
(f(xi−1,t) + f(xi+1,t)− 2f(xi,t)) mod 1 (1)
where t is the discrete time index, and i is the site index: i = 1, . . . L, with L the system size.
The parameter ǫ measures the strength of the diffusive coupling between site i and its two
neighbours. The on-site map is chosen to be
f(x) = x+ ω −
k
2π
sin(2πx) (2)
where the parameter k denotes the nonlinearity. All sites are updated in parallel.
The synchronized spatiotemporal fixed point x⋆ corresponds to the unique absorbing state.
Turbulent sites are those which have local variables which differ from x⋆. The fixed point
solution x⋆ is easily obtained:
x⋆ =
1
2π
sin−1(
2πω
k
). (3)
G.I. Menon, S. Sinha and P. Ray: Persistence in Coupled Map Lattices 3
0 20 40 60 80 100
0
200
400
600
800
1000
0 20 40 60 80 100
0
200
400
600
800
1000
Fig. 1 – Time evolution of the coupled circle map lattice defined through eqns. 1-2, in a system of
size L = 100, for parameters k = 1 and ǫ = 0.63775. Figure 1(a) is for ω = 0.05 (laminar region)
while fig. 1(b) is for ω = 0.1 (turbulent region). The horizontal axis is the site index i = 1, . . . L; the
vertical axis denotes discrete time t.
Figures 1(a),1(b), and 2(a) show space-time density plots of xi for three choices of parameter
values. The site index is plotted along the x axis and time evolves along the y direction.
Laminar regions are dark in this representation. The first of these plots, fig. 1(a), is for
parameter values for which the final state is laminar, while the second, fig. 1(b), is for
parameter values in the chaotic regime. Fig. 2(a) marks the transition between laminar and
chaotic regimes.
The critical exponents of this coupled map lattice at the onset of spatio-temporal inter-
mittency have been obtained numerically; these match directed percolation values to high
accuracy [12]. This model, at the critical points studied in this paper, constitutes the only
known example of a coupled map lattice with a unique absorbing state, whose transition falls
cleanly in the universality class of directed percolation [12,13]. Finding direct transitions to a
unique absorbing state from turbulent states in coupled map lattices is nontrivial; transitions
into spatio-temporally periodic patterns are far more common. As a consequence, even though
the past decade has seen considerable activity in this field, we know of no other examples of
such critical points in the literature [14].
A further advantage of this model system is the absence of any additional special spatio-
temporal structures, a feature which commonly disrupts the analogy to DP in extended dy-
namical systems [15]. Operationally, this means that no asynchronicity in updates need be
introduced here to destroy “solitonic” behaviour [15], Thus, coupled circle maps are clean
model systems for checking DP universality.
Persistence in the context of stochastic processes is defined as the probability P (t) that a
stochastically fluctuating variable has not crossed a threshold value upto time t [6]. For a spin
system with discrete states, such as Ising or Potts spins, persistence is defined in terms of the
probability that a given spin has not flipped out of its initial state upto time t. A power-law
4 EUROPHYSICS LETTERS
0 100 200 300 400 500
0
100
200
300
400
500
0 100 200 300 400 500
0
20
40
60
80
100
Fig. 2 – Time evolution of the coupled circle map lattice defined through eqns. 1-2, in a system of
size L = 500 at the critical point (k = 1, ω = 0.068, ǫ = 0.63775). The horizontal axis is the site
index i = 1, . . . L and the vertical axis denotes discrete time t. Fig. 2(a) is obtained from a density
plot of the actual xi,t values (the absorbing regions appear dark). Fig. 2(b) is a plot of the persistent
sites (marked in white) vs. time.
tail to P (t), i.e.
P (t) ∼ 1/tθ, (4)
defines the persistence exponent θ, with P (t) averaged over an ensemble of random initial
conditions.
What is the appropriate description of persistence in coupled map lattices? A natural
generalization of the ideas above defines local persistence in terms of the probability that a
local state variable xi,t does not cross the fixed point value x
⋆ up to time t. With this definition,
we study persistence in the coupled map lattice defined through Eqn. (1) via simulations on
systems of linear size ranging from L = 10 to L = 105. We start with the local variable xi,0
distributed uniformly in the interval [0, 1], update all sites synchronously, and measure the
normalized fraction of persistent sites, defined as sites for which xi,t has not crossed x
⋆ up to
time t. Thus, sites for which the sign of (xi,t− x
⋆) has not changed upto time t are persistent
at time t.
Fig. 2(b) shows the persistent sites at the critical point corresponding to the configu-
ration of fig. 2(a). The inset to Fig. 3 shows plots of P (t) at three different parameter
values, illustrating the following points: (a) P (t) in the laminar region saturates to a time-
independent value P∞ at large times, with P∞ becoming smaller and smaller as the transition
is approached; (b) Precisely at the transition between chaotic and laminar regimes, P∞ is zero
and P (t) decays as a power law, thereby defining the local persistence exponent θl and finally;
(c) In the chaotic regime P (t) decays exponentially. These curves are analogous to data ob-
tained for persistence in the Domany-Kinzel cellular automaton [9], with the identification of
the turbulent phase here with the active phase of the Domany-Kinzel model and the laminar
phase with the inactive one.
Figure 3 (main panel) shows our most accurate determination of the local persistence
G.I. Menon, S. Sinha and P. Ray: Persistence in Coupled Map Lattices 5
Fig. 3 – Local persistence distribution Pl(t) vs time t for the critical point at: k = 1, ω = 0.068,
ǫ = 0.63775. Inset shows the log-log plot of Pl(t) vs t for parameters k = 1, ǫ = 0.63775 and (a)
ω = 0.066 (below critical point); (b) ω = 0.068 (at the critical point displaying power law scaling)
and (c) ω = 0.07 (above critical point).
probability distribution Pl(t) at a critical point, on a system of size L = 10
5. We work
at the two critical points known to high accuracy (k = 1.0, ω = 0.068, ǫ = 0.63775) and
(k = 1.0, ω = 0.064, ǫ = 0.73277), averaging over 104 initial conditions. To within the accuracy
of our simulations, the values of θl at both these critical points are the same, consistent with
a persistence exponent of θl = 1.49± 0.02 [16].
If the diverging length (and time) scales at the continuous directed percolation transition
determine the scaling properties of local persistence, we expect a scaling form of the type
Pl(t, L, δ) ∼ t
−θlF±(δ
ν‖ t, L−zt). (5)
Here z = ν‖/ν⊥, with ν⊥ the standard 1+1 dimensional DP exponent governing the decay
of correlation lengths in the spatial direction and ν‖ the analogous exponent for the time
direction (ν‖ = 1.733847, ν⊥ = 1.096854) [9, 17]. F± is a universal scaling function (the
subscript ± refers to the direction of approach to the critical point), while δ measures the
distance from the critical point. At δ = 0, F± behaves asymptotically as F±(0, 0) ∼ const;
F±(x, 0) ∼ x
θl , x → ∞; F±(0, y) = const, y << 1 and F±(0, y) ∼ y
θl , y > 1. This
scaling function is similar to one proposed in the study of the spatial scaling properties of
persistence [18]. Fig. 5 shows this scaling function at criticality (δ = 0), for various system
sizes L. The best data collapse is obtained for z = 1.58 and θl = 1.485. The quality of
the data collapse verifies that the scaling forms are fully consistent with directed percolation
exponents [8, 9].
6 EUROPHYSICS LETTERS
Fig. 4 – Log-log plot of scaled persistence probability Pl(t)t
θl versus scaled time tL−z for different
lattice sizes at the critical point: k = 1, ω = 0.068, ǫ = 0.63775. The data collapses onto a single
curve for L = 50, 100, 300, 500 and 1000.
Our result for the local persistence exponent in coupled circle maps at the laminar-
turbulent transition agrees well with the best estimates for θ in the Domany-Kinzel automa-
ton at the directed percolation transition in 1+1 dimensions, where Hinrichsen and Koduvely
found θ = 1.5 ± 0.02 [9]. A recent study by Albano and Mun˜oz finds θ = 1.5 ± 0.01 for the
contact process at its critical point in 1+1 dimensions [10], a value in agreement with our sim-
ulations. These papers also study global persistence properties [19], drawing on calculations of
the global persistence exponent θg for phase ordering in Ising and Potts systems, where such
exponents differ from their local values. However, the properties of θg remain controversial for
DP-related problems [20]. Given the somewhat contrived definition of global persistence in
directed percolation, we believe that it would certainly be far harder to access this exponent
experimentally than the local exponent.
In summary, this Letter presents suggestive numerical evidence for the universality of the
persistence exponent at the onset of spatio-temporal intermittency in coupled map lattices.
We obtain a persistence exponent whose value is numerically identical to values obtained
at the directed percolation transition in 1+1 dimensions in a variety of systems, including
the Domany-Kinzel automaton and the one-dimensional contact process [21]. However, the
model we study here is both deterministic and possesses a continuum degree of freedom at
each lattice site, unlike these other models.
The implications of these results for experiments is particularly noteworthy. Spatio-
temporal intermittency is a common phenomena of many extended systems. It is seen in
experiments on convection [22] and in the “printers instability” [23]. The state variables of
coupled map lattices can often be identified with physical quantities such as voltages, cur-
G.I. Menon, S. Sinha and P. Ray: Persistence in Coupled Map Lattices 7
rents, pressures, temperatures, concentrations or velocities. A recent experiment by Rupp,
Richter and Rehberg [24] obtains directed percolation exponents at the transition to spatio-
temporal intermittency in a one-dimensional system of ferrofluid spikes, driven by an external
oscillating magnetic field. We believe that obtaining the local persistence exponent from an
experiment such as this should be fairly straightforward. We hope this Letter stimulates
further work towards identifying new candidates for measuring persistence exponents in real
world phenomena.
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the literature.
[4] Pomeau Y., Physica D, 23 (1986) 3.
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[12] Janaki T.M.,Sinha S. and Gupte N.M., nlin.CD/0210034, (2002) .
[13] There are examples of transitions to non-unique absorbing states in a class of systems proposed
by Chate´ and Manneville (Chate´ H. and Manneville P., Physica D, 32 (1998) 409) and its
variations [15].
[14] We also point out that the location of the critical point must be known to high accuracy for a
precise evaluation of θ and the standard DP exponents.
[15] Rolf J., Bohr T. and Jensen M.H., Phys. Rev. E, 57 (1998) R2503; Bohr T.et. al., Phys.
Rev. Lett , 86 (2001) 5482
[16] In one specific limit, our model is related to another for which persistence properties have been
studied extensively. If f(x) ≡ x; we obtain the space (and time) discretized one-dimensional
diffusion equation with random initial conditions i.e. xi,t+1 = xi,t + a
∑
j
[xj,t − xi,t] , where the
sum is over the neighbours of site i, and a = 1/2 in one dimension. The persistence exponent
for this system is θl ∼ 0.12050 (See Newman T.J. and Toroczkai Z., Phys. Rev. E, 58 (1998)
R2685), a value considerably different from the θl ∼ 1.5 we obtain indicating that non-trivial
on-site dynamics changes the character of persistence in diffusive systems.
[17] Hinrichsen H., Braz. J. Phys, 30 (2000) 69.
[18] Manoj G. and Ray P., Phys. Rev. E, 62 (2000) 7755.
[19] S.N. Majumdar et. al., Phys. Rev. Lett, 77 (1996) 2867.
[20] Oerding K. and van Wijland F., J. Phys. A, 31 (1988) 7011;
[21] It has recently been suggested that sandpile models incorporating “stickiness” of grains should
generically flow to the directed percolation fixed point(Mohanty P.K. and Dhar. D,cond-
mat/0202345, (2002) ). It would be interesting to see if an appropriate definition of persistence
in sandpile models might yield the same universality we address in this paper.
[22] Ciliberto S. and Bigazzi P., Phys. Rev. Lett, 60 (1988) 286; Daviaud F., Dubois M. and
Berge P., Europhys. Lett., 9 (1989) 441
[23] Rabaud M., Michalland S. and Couder Y., Phys. Rev. Lett, 64 (1990) 184; Michalland
S. and Rabaud M., Physica, D 61 (1992) 197
[24] Rupp P., Richter R. and Rehberg R., cond-mat/0201308, (2002)


We study persistence in coupled circle map lattices at the onset
of spatio-temporal intermittency, an onset which marks a continuous
transition, in the universality class of directed percolation, to
a unique absorbing state. We obtain a local persistence exponent
of $\theta_\ab{l}= 1.49 \pm 0.02$ at this transition, a value which
closely matches values for $\theta_\ab{l}$ obtained in stochastic
models of directed percolation. This result constitutes suggestive
evidence for the universality of persistence exponents at the
directed percolation transition. Given that many experimental
systems are modelled accurately by coupled map lattices,
experimental measurements of this persistence exponent may be
feasible

G. I. Menon

S. Sinha

P. Ray

EDP Sciences OAI-PMH repository (1.2.0)

Persistence at the onset of spatio-temporal 
intermittency in coupled map lattices

Persistence at the onset of spatiotemporal intermittency in coupled map
  lattices

Persistence at the onset of spatiotemporal intermittency in coupled map lattices

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