Intermingled basins in coupled Lorenz systems.
ABSTRACT We consider a system of two identical linearly coupled Lorenz oscillators presenting synchronization of chaotic motion for a specified range of the coupling strength. We verify the existence of global synchronization and antisynchronization attractors with intermingled basins of attraction such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor and vice versa. We investigated this phenomenon by verifying the fulfillment of the mathematical requirements for intermingled basins and obtained scaling laws that characterize quantitatively the riddling of both basins in this system.
arXiv:1111.5581v2 [nlin.CD] 30 Jan 2012
Intermingled basins in coupled Lorenz systems
Sabrina Camargo1, Ricardo L. Viana2, and Celia Anteneodo1,3
1Department of Physics, PUC-Rio, Rio de Janeiro, Brazil
2Department of Physics, Federal University of Paran´ a, Curitiba, Brazil
3National Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil.
(Dated: January 31, 2012)
We consider a system of two identical linearly coupled Lorenz oscillators, presenting synchro-
nization of chaotic motion for a specified range of the coupling strength. We verify the existence
of global synchronization and antisynchronization attractors with intermingled basins of attraction,
such that the basin of one attractor is riddled with holes belonging to the basin of the other attractor
and vice versa. We investigated this phenomenon by verifying the fulfillment of the mathematical
requirements for intermingled basins, and also obtained scaling laws that characterize quantitatively
the riddling of both basins in this system.
PACS numbers: 0.45.Xt,05.45.Df,05.45.Pq,05.45.-a
The Lorenz system
˙ x = α(y − x),˙ y = βx − y − xz,˙ z = −γz + xy , (1)
for α = 10, β = 28, and γ = 8/3, displays a chaotic
attractor with the familiar butterfly-like shape  . It
is often quoted as a paradigmatic system in nonlinear
dynamics, since it displays many interesting dynamical
properties of chaotic dissipative systems. Moreover its
equations mimetize the dynamical behavior expected to
occur in some physically relevant systems, as convection
rolls in the atmosphere , single-mode lasers , and
segmented disk dynamos . Coupled Lorenz systems
could arise as well in the mathematical modeling of re-
lated physical problems. The simplest case in the latter
category is the coupling of two identical Lorenz systems.
In identical coupled systems, even if chaotic, synchro-
nization of trajectories may occur . This phenomenon
has been studied for more than two decades, motivating
a wealth of analytical, numerical, and even experimen-
tal results . Synchronization of chaos, besides its own
interest as a mathematical problem, finds applications
for instance in secure communications . The chaotic
nature of the dynamics of one of the systems can be ex-
ploited to code messages which could be sent to an iden-
tical system through some form of coupling. If the latter
system is synchronized with the former, the message can
be securely uncoded.
For two completely synchronized systems, either peri-
odic or chaotic, their dynamical variables are equal for all
times. On the other hand, if instead of the difference, it
is the sum of some of their dynamical variables that van-
ishes, the two systems are said to antisynchronize. Due
to phase-space symmetries, coupled Lorenz systems can
exhibit both synchronized and antisynchronized states.
Then, for secure communications purposes, the existence
of another, antisynchronized, state is in principle a source
of troubles since, depending on the initial condition, the
receiver system could be tuned to the antisynchronized
attractor. This situation can still be dramatically worsen
when the riddling phenomenon occurs.
As a matter of fact, multistable dynamical systems
typically have a very complicated structure of basins
of attraction, that may be delimited by fractal bound-
aries . Suppose, for instance, that a dynamical system
has two attractors, with the corresponding basins of at-
traction sharing a common basin boundary in the phase
space. If a ball centered at a given initial condition and
with a radius equal to the uncertainty level intercepts the
basin boundary, we cannot say a priori which attractor
the system will asymptote to . If that boundary is
a curve, even if fractal, the final-state sensitivity prob-
lem can be circumvented by decreasing the radius of the
uncertainty ball (this can be done in experimental or nu-
merical settings, by increasing the precision in determin-
ing the initial condition in phase space). However, such
reduction of uncertain initial conditions is not possible in
the limit case in which the fractal boundary is area-filling,
i.e., the (box-counting) dimension of the basin boundary
gets close to the dimension of the phase space itself .
In that limit case, the fraction of uncertain initial con-
ditions will likely not decrease no matter how much we
decrease the uncertainty balls of each initial condition.
The latter situation occurs for riddled basins .
From the mathematical point of view, riddled basins
are observed in dynamical systems that exhibit an invari-
ant smooth hypersurface with a chaotic attractor lying
on it, another asymptotic final state, out of the invariant
subspace, and negative Lyapunov exponent transverse to
the invariant subspace with positive finite-time fluctua-
tions [10–12]. Under the conditions above, riddling orig-
inates from the loss of transversal stability of unstable
periodic orbits embedded in the chaotic attractor ,
despite the attractor being transversely stable in aver-
age. In this context, attractors must be understood in
the weak sense of Milnor . The transition associated
to the first unstable orbit on the attractor that losses
transversal stability determines the riddling bifurcation
(see for instance Ref.  for an overview). Depending
on the way these orbits loss stability and even on the
dynamics outside the invariant manifold, different bifur-
cation scenarios and different forms of riddled basins can
occur [13, 16–18] (to cite a few examples).
If riddled basins exist in a multistable chaotic system,
their final states are utterly unpredictable, i.e. we can-
not say - with any degree of certainty - which attractor
the system will evolve to for long times . The situa-
tion, in this case, is similar to that for a random process,
for which there can only be determined a probability for
predicting the final state of the system. In fact, some
phenomena formerly attributed to random variations in
initial conditions can be also interpreted as a consequence
of riddling .
The simplest case of riddling is when only one of the co-
existing attractors have a riddled basin. However, when
there is more than one invariant subspace, then more
than one attractor can be riddled. In this case, the basin
structure is called intermingled .
The aim of the present work is precisely to show the
existence of intermingled basins of attraction for the syn-
chronized and antisynchronized states of two coupled
Lorenz oscillators. In previous literature there are al-
ready clues of such phenomenon. Kim and coworkers
, in a work about anti-synchronization of coupled
chaotic oscillators, point to the possibility of a riddled
basin of synchronization in coupled Lorenz systems, but
without going further on that issue. Furthermore, a one-
dimensional reduction of the Lorenz system (to a piece-
wise approximation to the well-known Lorenz map) was
low-dimensional enough for an analytical treatment to
be feasible and show the riddling of the synchroniza-
tion basin . The verification of the transversal sta-
bility conditions through direct methods (i.e., by mak-
ing a linear stability analysis of each invariant subspace)
is quite difficult in two coupled Lorenz systems, since
the phase space is six-dimensional. Then, we investigate
those properties numerically. We also characterize quan-
titatively the riddled basins by means of the scaling laws
giving the probability of making wrong predictions on the
final state of the system, with respect to two quantities
of interest: (i) the phase-space distance to the invariant
subspace; and (ii) the uncertainty radius for each initial
condition . We have verified that, for both quanti-
ties, the probability scales as a power-law, as required
for riddled basins.
The rest of the paper is organized as follows: Section II
describes the coupled system of Lorenz oscillators, as well
as the existence of both synchronized and antisynchro-
nized states. Section III presents a preliminary discus-
sion of the basins of attraction of both the synchronized
and antisynchronized states. The mathematical proper-
ties required for riddled basins and the necessary tools are
the object of Section IV. Section V discusses the quanti-
tative characterization of riddled basins through scaling
laws and the theoretical results supporting them. The
last Section contains our conclusions and final remarks.
II. COUPLED LORENZ SYSTEMS
Many different coupling schemes are possible for two
identical Lorenz systems .
symmetry reasons, a diffusive coupling through the z-
variable, as follows
We have chosen, for
˙ x1= α(y1− x1),
˙ y1= βx1− y1− x1z1,
˙ z1= −γz1+ x1y1+ ε(z2− z1),
˙ x2= α(y2− x2),
˙ y2= βx2− y2− x2z2,
˙ z2= −γz2+ x2y2+ ε(z1− z2),
where we will use the same values for α, β, and γ, as in
the uncoupled case, and ε is the coupling strength.
On considering the dynamical behavior of the coupled
system, it is convenient to perform the changes of vari-
x =(x2− x1)
X =(x2+ x1)
,y =(y2− y1)
, Y =(y2+ y1)
, Z =(z2+ z1)
z =(z2− z1)
after which the coupled system (2) becomes
˙ x = α(y − x),
˙ y = βx − y − (Xz + Zx),
˙ z = −(γ + 2ε)z + Xy + Y x,
˙X = α(Y − X),
˙Y = βX − Y − (XZ + xz),
˙Z = −γZ + XY + xy .
Whenever more convenient to the analysis, we will refer
either to the new or the old variables.
From inspecting Eqs. (4) there follows that the dynam-
ics of the coupled system is invariant with respect to the
transformation (x,y,z) → (−x,−y,−z). Hence the con-
ditions x = y = z = 0 define an invariant subspace Ms:
one initial condition that belongs to this subspace gener-
ates a trajectory in phase space that remains in Msfor
any time. This three-dimensional subspace defines the
complete (or global) synchronization manifold character-
ized by x1= x2, y1= y2, z1= z2.
The dynamics in the invariant subspace Ms, described
by the variables (X,Y,Z), is governed by the equations
of the uncoupled Lorenz system, hence there is a chaotic
attractor As(butterfly-like shape) lying in Ms.
Analogously,due to the symmetry (X,Y,z)
(−X,−Y,−z), the states for which X = Y = z = 0 define
another invariant subspace Ma (anti-synchronization
manifold), associated to the attractor Aa, in which
(x,y,Z) follows the dynamics of the uncoupled system,
i.e. Aais a Lorenz chaotic attractor in Ma.
There are also other symmetries already present in the
uncoupled Lorenz system. Notice in Eqs. (2) that either
(x1,y1) → (−x1,−y1) or (x2,y2) → (−x2,−y2) lead, to
four-dimensional invariant subspaces, while both symme-
tries together lead to a two-dimensional invariant sub-
space with a saddle point at the origin. Finally, included
in this two-dimensional subspace, the lines at z1 = z2
and z1 = −z2 also represent invariant subsets. We did
not find any other relevant attractor other than Asand
Aa, which are attractors of the dynamics in the respec-
tive subspaces Msand Ma, and can become attractors
for the whole phase space depending on their transversal
III.BASINS OF ATTRACTION
In dynamical systems with more than one attractor,
the corresponding basins may have fractal boundaries
and even more complicated structures like the Wada
property . Accordingly, in the coupled Lorenz system
(2), the two coexisting attractors representing synchro-
nized and antisynchronized states are expected to have
such complex basin boundary structure.
Since the phase space of the coupled system is six-
dimensional, the visualization of the basins of attraction
depends on convenient phase space sections or projec-
tions. Figure 1 shows a section of the basin of the anti-
synchronization (synchronization) attractor Aa(As), for
different values of the coupling parameter.
Each initial condition was integrated using a fourth-
order Runge-Kutta scheme with fixed timestep 10−3and
for a time t = 103, after which we determined to what
attractor the corresponding orbit has asymptoted . If
an orbit has asymptoted to an antisynchronized (synchro-
nized) state in Ma(Ms), its initial condition was painted
black (white). Hence the area painted black (white) is a
numerical approximation of a section of the basin of at-
tractor Aa (As). We considered 105initial conditions
with x1 = x2 = y1 = y2 = 1.0 while z1 and z2 were
FIG. 1: Section at x1 = x2 = y1 = y2 = 1.0 of the basins of
synchronization (white pixels) and antisynchronization (black
pixels) attractors of the coupled Lorenz system, for ε = (a)
1.0, (b) 2.0, (c) 2.5 and (d) 2.8.
randomly chosen in the interval [20,24) according to a
uniform probability distribution.
For instance, for a coupling strength ε = 1.0 [Fig.
1(a)], the section of the basin of attractor Aa is a se-
ries of thin filaments stemming from the diagonal. The
filaments are non-uniformly distributed and have a sug-
gestive self-similar appearance. In fact, successive mag-
nifications of [Fig. 1(a)] reveal similar patterns (see Ref.
). Such scenario is also observed for other values of ε,
as illustrated in Fig. 1(b-d), even if some features change
with ε, such as the relative area of each basin, or the def-
inition of the tongues anchored in the diagonal. Let us
note that other cuts also display a tongue structure, as
depicted in Fig. 2 for ε = 2.0.
FIG. 2: Section at z1 = z2 = 22.0 and (a) y1 = y2 = 1.0,
x1,x2 random in [−1,3), (b) x1 = x2 = 1.0, y1,y2 random
in [−1,3) of the basins of synchronization (white pixels) and
antisynchronization (black pixels) attractors of the coupled
Lorenz system, for ε = 2.0.
The structure of the basins of attraction is indeed ex-
pected to be altered by the coupling strength. As an
example, in Fig. 3 we show that, for a given initial con-
dition (x1= y1= z1= 1.0 and x2= y2= z2= 0.5) inte-
grated up to time 102, the trajectories in the subspace of
each oscillator are distinct for ε = 0.5, while for ε = 1.0
trajectories tend to coincide due to synchronization. In
the latter case, the overlapping segments reproduce a cut
of the familiar attractor of the single Lorenz system, since
for synchronized orbits, the evolution proceeds towards
the attractor in Mswhich is defined by x = y = z = 0,
and X = x1= x2, Y = y1= y2, Z = z1= z2follow the
dynamics of an uncoupled system, as described above.
Differently, in the former case (ε = 0.5), the trajecto-
ries of each system depart from those of the uncoupled
Moreover, the observation of synchronized or antisyn-
chronized states depends on the coupling strength. Re-
call that the existence of Ms(i.e. the synchronized state
being a possible solution of the coupled equations) does
not mean necessarily that synchronized states, and in
particular states in its attractor As, can be observed
in numerical simulations. This occurs only if there is
transversal stability, in the sense that any infinitesimal
displacement along directions transversal to Msdecays
exponentially with time.
Let us remark that, due to the symmetry of the
equations with respect to synchronized/antisynchronized
FIG. 3: Trajectories of the coupled Lorenz system for the
same initial condition (x1 = y1 = z1 = 1.0 and x2 = y2 =
z2 = 0.5) up to t = 100 and different coupling values: (a)
ε = 0.5; (b) 1.0. In each case, the time evolution of the
differences of coordinates are also shown (c)-(d) for ε = 0.5
and (e)-(f) for ε = 1.0.
states, comments for attractor Asare also valid for Aa.
In order to visualize the existence of a transversely sta-
ble synchronization manifold, we consider the differences
x1− x2, y1− y2, and z1− z2, which must vanish if a
synchronized attractor is achieved. For ε ? 0.7, z1− z2
vanishes [Fig. 4(c)], while the other two differences may
also vanish (global synchronization) or not (local syn-
chronization) [Fig. 4(a) and (b)]. Similar plots are ob-
FIG. 4: Difference between the coordinates (a) x1− x2; (b)
y1 − y2; (c) z1 − z2 of two coupled Lorenz systems at time
t = 103, as a function of the coupling strength. One hundred
initial conditions were randomly chosen (as in Fig. 1) for each
value of ε (varied in steps of 0.01).
tained for the sums x1+ x2, y1+ y2, indicating that the
basins of the synchronized and antisynchronized states
are complementary to each other.
0 100 200 300 400 500 600 700 800 900 1000
Fraction of initial conditions
0 500 1000
FIG. 5: (a) Fraction of initial conditions (over a total of 104)
yielding trajectories asymptoting synchronized fs or antisyn-
chronized fa states as a function of time, for ε = 2.0. We
also indicated the fraction of trajectories reaching either state
(fs+fa) or none of them (1−fs+fa). (b) Fraction of initial
conditions not reaching these states for different values of the
We did not find any relevant attractor for the coupled
system other than As and Aa. Besides the symmetry
considerations at the end of Sect. II, we performed the
following numerical experiment: we considered the ini-
tial conditions used to plot the sections in Fig. 1 and,
for each time t we computed the fraction of initial con-
ditions that go either to Asor Aa[Fig. 5(a)]. The sum
of these fractions rapidly approaches 100% [Fig. 5(a)],
meaning that the fraction of initial conditions that do
not asymptote to them goes to zero (filled squares in Fig.
5(a)), suggesting the existence of only two attractors for
the coupled system. This conclusion has been observed
to hold for ε ? 0.7 as illustrated in [Fig. 5(b)]. Oth-
erwise, neither synchronized nor antisynchronized states
are approached, as illustrated in Fig. 3(c)-(d) for ε = 0.5.
(Therefore, basin diagrams as those shown in Fig. 1 will
be left blank).
IV.RIDDLED AND INTERMINGLED BASINS
The standard requirements for the existence of a rid-
dled basin are the existence of (i) a smooth invariant sub-
space (of lower dimension than the phase-space) contain-
ing a chaotic attractor, (ii) another asymptotic final state
(not necessarily chaotic) out of the invariant subspace,
(iii) negativity of the Lyapunov exponents transverse to
the invariant subspace with (iv) positive finite-time fluc-
tuations [9–11], which are associated to the transversal
stability properties of unstable periodic orbits (UPOs)
embedded in the attractor. For two symmetrically inter-
mingled basins, the requirements for mutual riddling can
be summarized as follows:
1. There are invariant manifolds Ssand Sacontained
in the phase space H.
2. The dynamics on each manifold Ss,ahas a chaotic
3. Ss,a are transversely stable, meaning that the
largest transversal Lyapunov exponent λ⊥is nega-
4. Although weak stability holds in average (condition
3), UPOs embedded in the chaotic attractor are
In Section III we showed that conditions 1 and 2 are
fulfilled for the coupled Lorenz systems.
two (three-dimensional) manifolds, Ss,a= Ms,a, in the
six-dimensional phase space. They are invariant since
trajectories starting in each subspace will remain there
forever. Because the dynamics in each subspace coincides
with that of the uncoupled map, then, it will evolve to-
wards the respective well known Lorenz attractor As,a
lying in the corresponding manifold.
Moreover, for each invariant subspace, there are out of
three transversal directions. Condition 3 means that the
transverse Lyapunov exponents of typical orbits lying in
the invariant subspaces (Ma and Ms) are all negative.
The point in parameter space where they become posi-
tive defines the blowout bifurcation . To investigate
condition 3, it suffices to consider the largest transversal
exponent, denoted as λ⊥= limt→∞˜λ⊥(x0,t) < 0, where
x0 is an initial condition on the basin of attraction of
either Aaor As.
We computed Lyapunov exponents using two differ-
ent methods. The Lyapunov spectrum was obtained fol-
lowing the algorithm of Wolf et al. , with a Gram-
Schmidt normalization step of 0.1. We integrated Eqs.
(2) using initial conditions given by x1 = y1 = x2 =
y2= 1.0 and z1,z2were randomly chosen in the interval
[20,24) from a uniform probability function, as in Fig. 1.
These initial conditions lead to trajectories that asymp-
tote to either Asor Aa. As a matter of fact, this is not
relevant since both attractors have the same Lyapunov
The second method we used, and which can be applied
to obtain only the largest transversal exponent, is to con-
sider the time evolution of an infinitesimal displacement
along a direction transversal to the synchronized sub-
space Ms, which is given by ,
where δ(t) =
transverse displacement, whose evolution is given by the
variational equations for (x,y,z), setting x = y = z = 0,
?(δx)2+ (δy)2+ (δz)2is the norm of the
˙δx = α(δy − δx),
˙δy = βδx − δy − Xδz − Zδx,
˙δz = −(γ + 2ε)δz + Xδy + Y δx,
˙X = α(Y − X),
˙Y = βX − Y − XZ,
˙Z = −γZ + XY .
Figure 6 shows (in gray symbols) the three largest
(infinite-time) Lyapunov exponents as a function of the
coupling strength ε, obtained by means of the algorithm
by Wolf et al., and the largest transversal exponent given
by (5) is indicated by a thick black line. One of the ex-
ponents is always zero, corresponding to displacements
along the trajectory. The largest exponent is practically
always equal to ∼ 0.9 and corresponds to the chaotic dy-
namics on As (Aa). The third exponent is the largest
transversal exponent which we focus our attention on,
both methods being in good accord in the region of in-
terest (as shown in Fig. 6). For the chaotic attractors in
both synchronization and antisynchronization manifolds,
0.01.0 2.0 3.04.0 5.0
FIG. 6: The three largest Lyapunov exponents of the coupled
Lorenz system as a function of the coupling strength. Gray
symbols correspond to the algorithm by Wolf et al., whereas
the thick black curve is the result of the variational equations
(6). Inset: the largest exponent for a wider range of ε.
the largest transversal exponent vanishes, changing sign,
at ε1 ≈ 0.714 ± 0.005 and ε2 ≈ 3.061 ± 0.005, defin-
ing the critical points of the blowout bifurcation. (There
is also another critical value for large ε, as can be seen
in the inset of Fig. 6, but we will restrict our analysis
to the lower range only). The largest transversal expo-
nent is negative for ε1< ε < ε2, hence, in that interval,
condition 3 for intermingled basins is fulfilled. However,
while the invariant subspaces Ms,a are stable in aver-
age, with negative transversal Lyapunov exponents, there
may be particular unstable periodic orbits embedded in
the chaotic attractors As,athat are also transversely un-
stable, with positive largest transversal Lyapunov expo-
nent  (condition 4). When trajectories come close to
these unstable orbits, they will be repelled from the vicin-
ity of the attractor. This will be reflected in positive val-
ues of the finite-time largest transversal Lyapunov expo-
nent [10–12]. Then we numerically computed the finite-
time largest transversal Lyapunov exponents˜λ⊥(x0,t),
by means of Eq. (5) but at finite t. For large enough t,
one recovers the infinite time exponent λ⊥, which does
not depend on x0, for almost all initial conditions in the
attractors As,a, in contrast to the finite-time ones that
may depend on the initial condition.
We quantify the contributions of the finite-time largest
transversal exponent by obtaining a numerical approx-
imation to the corresponding probability distribution
function P(˜λ⊥(x0,t)). We considered a large number of
points in Ms(with x = y = z = 0, X = Y = 1.0, and Z
randomly chosen in the interval [20,24)), and discarded a
transient. These were the initial conditions used to com-
pute the time-t largest transversal Lyapunov exponents.
Alternatively, we generated a single long chaotic trajec-
tory (after the transient has elapsed) and divided it into
time-t segments, using then the ergodicity of the dynam-
ics to ensure that the conditions are randomly chosen
according to the natural measure of the attractor. The
results were essentially the same.
-1.2 -1.0 -0.8-0.6-0.4
FIG. 7: Probability density functions of the time-30 largest
transversal Lyapunov exponent for different values of ε. Ini-
tial conditions were taken as in Fig. 1. The full lines are
Figure 7 shows probability distribution functions
(when t = 30) for different values of the coupling
strength. In all the considered cases the distribution is
nearly Gaussian and presents positive tails. Then, we
computed the positive fraction of finite-time exponents,
plotted in Fig. 8 as a function of the coupling strength
for different values of the time-t interval used to sample
the finite-time exponents.
0P(˜λ⊥(t))d˜λ⊥(t) > 0. The positive fraction is
0.01.02.0 3.04.0 5.0
FIG. 8: Positive fraction of the largest time-t transversal Lya-
punov exponent as a function of the coupling strength, for
different values of t.
For ε ? ε1, finite-time exponents soon become positive.
This is in agreement with the positivity of the infinite-
time exponent shown in Fig. 6 for this region, and also
with the fact that trajectories do not approach the in-
variant subspaces, but are soon repelled, as already noted
when we tried to plot the basins in Section III, which is
not possible for that parameter range. The positive frac-
tion drops rapidly to 50%, which corresponds to the case
for which the infinite-time exponent vanishes (consistent
with symmetric P(˜λ⊥(t))), and then drops below 50%,
when the infinite-time exponent is negative. For ε ≃ 1.4,
the positive fraction is minimal. For larger values of ε, it
increases, crossing the 50% level again for ε ≃ ε2, where
the infinite-time exponent is again zero at that point (see
Fig. 6). After that, the positive fraction tends to 1 gently
with ε, yielding a positive infinite-time exponent. This
smooth behavior, different from the abrupt one in the
lower limit of the region of negative λ⊥, is consistent with
the observation, for ε > ε2, of a basin structure reminis-
cent of those in Fig. 1, although the filaments from the
diagonal are not neat. Even if trajectories are ultimately
repelled, they can spend long time intervals close to each
The fraction of positive finite-time exponents is non-
null. However, for the range ε1< ε < ε2, that fraction
decreases with t, as expected because the distribution
of finite-time exponents collapses towards a Dirac delta
centered at λ⊥in the long time limit. The decay is expo-
nential, the faster, the closer to the minimum at ε ≃ 1.4.
Hence, the absence of an abrupt decay of the positive
fraction indicates a nonvanishing fraction for finite times.
Then, from this analysis, condition 4 cannot discarded for
any range within the interval ε1< ε < ε2. This suggests
that at least one of UPOs should be transversely unstable
in that interval.
0.0 0.20.4 0.6 0.81.01.21.4
0.00.5 1.01.5 2.02.53.0 3.54.0
0.00.51.0 1.5 2.02.5 3.03.5 4.0
exponent, as a function of ε, for the particular unstable peri-
odic orbits (UPOs) embedded in the Lorenz chaotic attractor
(up to period 5), indicated on the figure by means of the
sequence of symbols A, B denoting the turns around each un-
stable fixed points C+and C−of the Lorenz system. The
curve for typical chaotic trajectories in the attractor is also
shown. (b) Magnification of panel (a). (c) Stability intervals
for each UPO, in order of increasing stability at ε = 0 from
top to bottom: stable (full segment), unstable (dotted). The
vertical lines indicate ε1 and ε2. The symbols delimiting the
stability intervals correspond to µ = 1 (full) and −1 (hollow).
(Color online) (a) Largest transversal Lyapunov
Then we inspected the transversal stability of those
orbits along the lines of periodic orbits threshold theory
. Once localization in phase space and periods of low
period UPOs are available in the literature for the Lorenz
system , we computed Floquet multipliers [28, 30].
Namely, we integrated Eqs. (6), to obtain the matrix Q
such that?δ(τ) = Q?δ(0), with τ the time period of the
orbit and?δ(t) = (δx,δy,δz) the column vector of trans-
verse deviations. The eigenvalue of Q, µ, with maximal
modulus furnishes λ⊥= ln|µ|/τ, for a particular periodic
orbit. Fig. 9 shows the behavior of λ⊥as a function of ε
for particular UPOs, up to period 5. UPOs are labeled
by means of the sequence of symbols A, B denoting the
turns around each unstable fixed point C+and C−of the
Lorenz system. Symmetric orbits obtained by exchang-
ing A↔B or with cyclic symmetry were omitted.
One observes that the lowest period orbit AB (period
2) appears to be the first in destabilizing the vicinity of
ε2, hence defining a riddling bifurcation. Then, between
this point and ε2riddling can occur. This interval cov-
ers most of the range ε1 < ε < ε2, except for a very
small interval in the vicinity of ε1. However, note that
orbits of the type AnB, with n = 1,2,..., have a maximal
transversal Lyapunov exponents that increases with n in
the vicinity of ε1, hence shrinking the remaining small
region of stability around ε ≃ 1. To confirm whether this
region of strong stability (with no transversely unstable
orbits) actually disappears would require the analysis of
higher period orbits, a hard task for this system, since the
number of UPOs increases exponentially with the integer
Near the blowout bifurcation at ε1, the low-period
UPOs (up to period 5) destabilize for coupling strength
either weaker or stronger than the critical value, but
close to it. Let us remark that, differently to the cou-
pled R¨ ossler system studied by Heagy et al. , here
the ordering of the exponents of the lowest period or-
bits in the neighborhood of ε1is inverted with respect to
the uncoupled case as depicted in Fig. 9. This implies
that paradoxically the most stable orbits in the attrac-
tor are those responsible for the transversal destabiliza-
tion in this parameter region. A similar inversion occurs
on some domains of the parameter space of a system of
symmetrically coupled R¨ ossler oscillators[15, 17]. This
characteristic turns difficult the determination of the rid-
dling bifurcation (first destabilized orbit) related to the
blowout at ε1, apparently triggered by higher period or-
Furthermore, our outcomes point to a different nature
of the blowout bifurcations at ε1 and ε2.
UPOs destabilize in its vicinity. Moreover, for all the an-
alyzed orbits, the multiplier µ crosses the circle |µ| = 1
along the real positive semi-axis (associated to a pitch-
fork bifurcation). This is in contrast to the scenario at ε2,
where there are orbits destabilizing far from ε2and with
multiplier µ either +1 or −1. In particular, the first orbit
AB loses stability with µ = 1. The differences are consis-
tent with the picture given by finite-time exponents, for
instance in connection with Fig. 8, where a much abrupt
behavior of the positive fraction was encountered near
The intervals where riddling can occur are delimited on
one side by the blowout bifurcation and on the other by
the riddling bifurcation. In our case there are two of such
intervals and they apparently overlap, such that at least
one UPO has lost transversal stability in the full range
ε1< ε < ε2, although this would have to be confirmed
by the analysis of high period orbits, it is supported by
the analysis of finite-time Lyapunov exponents.
V. SCALING LAWS FOR RIDDLED BASINS
In this Section, we will focus on the determination of
the scaling properties of the basins, which provide a mea-
sure of their structure. Let us focus on the black fil-
aments in Fig. 1(b), which belong to the basin of the
antisynchronization attractor. They are anchored at the
diagonal line, which is a cut of the synchronization man-
ifold Ms, given by x = y = z = 0 and containing a
chaotic attractor As, while the antisynchronization at-
tractor Aa lies elsewhere. In Fig. 10(a) we portrait a
schematic picture of that structure. The filaments of the
basin of Aa are tongues anchored at points of As, and
the complement of the filament set belongs to the basin
of As. If an initial condition starts within any of these
narrow tongues, even if it is very close to As, the resulting
trajectory will asymptote to the other attractor.
The set of basin filaments for Aa is expected to be
self-similar by quite general grounds. Once the riddling
bifurcation occurs for a given periodic orbit, it also oc-
curs for every preimage of this orbit, yielding a dense set
of tongue-like sets anchored at the corresponding preim-
ages on As. The tongue-like shape is a consequence of the
nonlinear terms in the equations describing the transver-
sal dynamics. The characteristic feature of riddling is
that those tongues have widths that tend to zero as we
approach As. Hence the basin of As always contains
pieces of the basin of the other attractor, regardless the
transversal distance to As, so forming a fine structure of
basin filaments (the same applying to Aa).
This fine structure can be quantitatively characterized
by the following numerical experiment [11, 23]: let us
consider the invariant manifold at x = y = z = 0 and,
depart from that manifold, for instance, by increasing z,
up to a distance l = 2|z| ≡ |z1− z2| [as depicted by the
red line segment in Fig. 10(a)]. Then we evaluate the
fraction Vlof points in that segment that belongs to the
basin of As. We obtained a numerical approximation of
this fraction by considering a number of initial conditions
x = y = 0, |z| = l/2, X = Y = 1.0, and Z randomly cho-
sen in the interval [20,24). If the trajectories did not
synchronize (within a small tolerance) up to a time such
that transients have elapsed and stationarity holds (typ-
ically, t = 103), we consider that they asymptote to the
antisynchronization attractor Aaand, accordingly, they
do not belong to the basin of As. If the latter is riddled
with tongues belonging to the basin of Aa, for any dis-
tance l (no matter how small) there is always a nonzero
value of Vl. This fraction tends to zero as l → 0. The
fraction of length belonging to the basin of Aa(fraction
of trajectories that do not synchronize) can be written as
X, Y, Z
x = y = 0
FIG. 10: (a) Schematic figure showing the structure of riddled
basins near the invariant subspace that contains a chaotic at-
tractor. (b) Fraction of trajectories P⋆ that asymptote to the
antisynchronized state as a function of the distance l = 2|z|
to the synchronized state, for different values of the coupling
strength ε. The full lines are least squares fits. (c) P⋆ for
ε = 1.5 and different orientations of the deviation l.
P⋆= 1−Vl, and is expected to scale with l as a power law
P⋆(l) ∼ lη, where η > 0 is a scaling exponent. We inte-
grated several initial conditions at the same distance l to
the synchronization subspace and computed the fraction
P⋆of initial conditions that do not synchronize, repeating
this procedure varying the distance l. The results shown
in Fig. 10(b) confirm the existence of a power law for this
fraction, for many values of the coupling strength.
The results do not vary appreciably when one departs
from the synchronization manifold in other directions
other than z. In Fig. 10(c) we plotted, for the same
coupling strength (ε = 1.5), the fraction P⋆ for initial
conditions with |z| = l/2, x = y = 0 (open squares),
and also for |x| = l/2, y = z = 0 (filled circles) and ran-
dom values of x,y,z such that d ≡
(filled squares), obtaining essentially the same scaling ex-
ponent. This scaling behavior is observed within the in-
terval (ε1,ε2), below ε1, no trajectories synchronize as
seen in the previous sections, above ε2, one observes a
synchronized fraction but it does not change with l. The
dependence of the numerically determined scaling expo-
nents η on the coupling strength is depicted in Fig. 11.
?x2+ y2+ z2= l/2
FIG. 11: Scaling exponent η for the fraction of trajectories
that asymptote to the antisynchronized state obtained by a
numerical experiment (filled circles). For comparison, the the-
oretical values, given by η = |λ⊥|/D are also plotted (open
An analytical expression for the exponent η was de-
rived by Ott and coworkers for a simple model (piecewise
linear non-invertible map) . Their theoretical predic-
tion arises from a diffusion approximation for a biased
random walk that mimics the fluctuations of finite-time
largest transversal Lyapunov exponents λ⊥(t). They ob-
tain the law P⋆∼ lη, with η = |λ⊥|/D where D is the
diffusion coefficient. This diffusion approximation is ex-
pected to be valid near the blowout bifurcation (λ⊥≃ 0)
of an attractor with a riddled basin. The authors conjec-
ture that a similar diffusion approximation, hence a simi-
lar relation involving parameters λ⊥and D, must rule the
scaling relation in a large class of systems. The distribu-
tions shown in Fig. 7 already display a Gaussian charac-
ter, which improves with larger time-t interval, consistent
with the probability distribution function of independent
random innovations, that, by the central limit theorem,
is Gaussian. Additionally, we plot in Fig. 12 the variance
as a function of time, for different values of the coupling
strength. As a matter of fact, the variance decays with
time towards zero following asymptotically a power-law
with exponent −1, as required for a normal diffusion pro-
cesses, so validating the stochastic approach of Ott et al..
Accordingly, the diffusion coefficient D can be estimated
from the numerical curves, following σ2
λ⊥(t)of the probability distribution functions for˜λ⊥(t)
λ⊥(t)∼ 2D/t. The
estimates η = |λ⊥|/D are plotted in Fig. 11 together
with the numerical values. Numerical and estimated val-
ues are in very good agreement in the proximity of the
critical values, as expected . For intermediate values
(0.75 < ε < 2) there is a discrepancy, and the numerical
exponent remains close to one (linear behavior), as also
observed for other systems with intermingled basins .
FIG. 12: Time decay of the variance of the finite-time largest
transversal Lyapunov exponent for different values of the cou-
pling strength ε. The lines are least squares fits of the function
f(t) = 2D/t to the numerical points, for large t, allowing to
estimate the diffusion coefficient D.
Another scaling law typical of riddled basins is related
to the fraction of uncertain initial conditions, with re-
spect to their final-state . We may regard riddled
basins as an extreme case of fractal basins, for which
there is final-state sensitivity and the uncertainty fraction
scales as a power-law with the uncertainty level, whose
exponent gives a measure of the extreme final-state sen-
sitivity due to riddling. Consider again the points at
x = y = 0 and |z| = l/2 drawn in the phase space por-
trait in Fig. 13(a), as described earlier, and choose ran-
domly an initial condition x0on that region. Now choose
randomly another initial condition x′
ability within an interval of length 2ξ and centered at x0
[Fig. 13(a)]. If both points belong to different basins,
they can be referred to as ξ-uncertain [32, 33].
The fraction of ξ-uncertain points, or uncertainty frac-
tion, denoted by ?p?, is the probability of making a mis-
take when attempting to predict which basin the initial
condition belongs to, given a measurement uncertainty
ξ. This probability scales with the uncertainty level as a
power law of the form ?p? ∼ ξφ, where φ ≥ 0 depends on
both x0and l. Numerical results are shown in Fig. 13(b).
The stochastic model of Ott et al. predicts a power-law,
with exponent given by φ = λ2
that agrees with our numerical results close to the crit-
ical points. However, for intermediate values, while the
stochastic model predicts small (though nonzero) values
(φ < 0.28), our numerical results for φ, as illustrated in
Fig. 13(b), yield much smaller values (by a factor greater
than ten). As a matter of fact, for riddled basins, the ex-
0with uniform prob-
⊥/(4Dλ?), a prediction
x = y = 0
X, Y, Z
FIG. 13: (a) Schematic figure showing the numerical determi-
nation of the uncertainty fraction. (b) Fraction of uncertain
initial conditions as a function of the uncertainty level, for
different values of the coupling strength. The solid lines are
least squares fits.
ponent φ should be rigorously zero (i.e. there would be
no way to decrease the uncertainty fraction by decreas-
ing the uncertainty level). Indeed, our results [Fig. 13(b)]
support this scaling law, with numerically obtained ex-
ponents close to zero.
VI.CONCLUSIONS AND FINAL REMARKS
Riddled basins for the synchronization attractor of cou-
pled Lorenz systems have been previously suggested in
the literature but without a detailed characterization. In
this work we offer numerical evidence that, for a spec-
ified range of the coupling parameter (ε1 < ε < ε2),
coupled Lorenz systems exhibit symmetrically riddled
basins of attraction for synchronized and antisynchro-
nized states. Since there are only two symmetric attrac-
tors, their basins are intermingled. We firstly showed
that the mathematical conditions for the existence of rid-
dled basins are fulfilled, with the help of properties of
finite-time largest transversal Lyapunov exponents and
of the largest transversal exponent for particular or-
bits. This is important as furnishes the sources of local
transversal instability of the attractor even if stable in
average. In a second place, we verified the existence of
two scaling laws characterizing quantitatively the degree
of uncertainty related to the riddled basins. These nu-
merical results were compared to an analytical prediction
(the stochastic model ), yielding a good accord where
expected. Beyond the characterization of the structure
of a riddled basin, these scaling laws allow to quantify
the limitations to improve the ability in determining the
final state of the system by increasing the accuracy level.
Let us remark that intermingling, in particular of sym-
metric basins, has also been observed in other systems
with either continuous (e.g., mechanical system  and
coupled R¨ ossler oscillators) or discrete time dynam-
ics (coupled logistic maps ). In the latter case, the
analysis of the lowest period orbit was enough to furnish
the conditions for the occurrence of riddling in certain
parameter region. In fact, as anticipated by the results
presented in Fig. 9, deepening in that point of view may
furnish precise information on the nature of the bifurca-
tions triggering riddling, although this may be a difficult
task for the present system. As other perspectives for fu-
ture work on this system, let us also mention the plausible
occurrence, beyond the blowout bifurcation, of two-state
on-off intermittency  for which there is some evidence
. Finally, it can still be worthy to explore other re-
gions of phase space, as well as other ranges (negative or
large values) of the coupling parameter.
In any case, for applications, multistability is already
a source of troubles. Still worst, the existence of inter-
mingled basins of attraction for the synchronized and an-
tisynchronized chaotic states of this system jeopardizes
the solution of the problem of ensuring a given final state,
since the initial condition determination is always done
within a certain uncertainty level. With riddled basins,
any uncertainty level, however small, lead to complete
indeterminacy of the future state of the system. Hence
in this case we cannot use synchronization of chaos for
any practical purpose, since we will always be haunted
by the existence of the another, antisynchronized state,
with a basin intermingled with the basin of the synchro-
nized state. Of course, the same difficulties concern the
predictability of natural phenomena modeled by coupled
Lorenz systems. Therefore, the importance of detecting
the regimes where riddling can occur in a dynamical sys-
This work was made possible with help of CNPq,
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