Full text of DOI:10.1063/1.166495

Stalactite basin structure of dynamical systems with transient chaos in an invariant
manifold
Vasily Dronov and Edward Ott
Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 10, 291 (2000); doi: 10.1063/1.166495
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CHAOS
VOLUME 10, NUMBER 2
JUNE 2000
Stalactite basin structure of dynamical systems with transient chaos
in an invariant manifold
Vasily Dronova) and Edward Ott
Institute for Plasma Research, University of Maryland, College Park, Maryland 20742
͑Received 1 September 1999; accepted for publication 22 February 2000͒
Dynamical systems with invariant manifolds occur in a variety of situations ͑e.g., identical coupled
oscillators, and systems with a symmetry͒. We consider the case where there is both a nonchaotic
attractor ͑e.g., a periodic orbit͒ and a nonattracting chaotic set ͑or chaotic repeller͒ in the invariant
manifold. We consider the character of the basins for the attracting nonchaotic set in the invariant
manifold and another attractor not in the invariant manifold. It is found that the boundary separating
these basins has an interesting structure: The basin of the attractor not in the invariant manifold is
characterized by thin cusp shaped regions ͑‘‘stalactites’’͒ extending down to touch the nonattracting
chaotic set in the invariant manifold. We also develop theoretical scalings applicable to these
systems, and compare with numerical experiments. © 2000 American Institute of Physics.
͓S1054-1500͑00͒00502-4͔
Dynamical systems that exhibit chaotic dynamics on an
invariant manifold embedded in their state space are of
interest in a variety of situations, and examples show that
these situations may yield novel dynamical behaviors.
One such situation is the case where the chaotic set in the
invariant manifold is attracting on average but has
within it invariant subsets that are repelling transverse to
the invariant manifold. In appropriate circumstances this
leads to so-called ‘‘riddling’’ of the basin of attraction. In
this paper we consider a related but different situation
where there is both a nonattracting chaotic set and an
attractor which is ‘‘absolutely attracting’’ in the invari-
ant manifold. Here, by ‘‘absolutely attracting’’ we mean
that the attractor is attracting for points in the invariant
manifold and contains no invariant subsets that are non-
attracting transversely „e.g., the absolutely attracting at-
tractor might be a stable periodic orbit…. Thus in the full
phase space there is an open neighborhood of the abso-
lutely attracting attractor that is part of its basin of at-
traction. It is found that in such a case the basin bound-
ary of the attractor in the invariant manifold may be
characterized by a thin stalactitelike structure emanating
from the nonattracting chaotic set in the invariant mani-
fold. We employ a Fokker–Planck type model which
shows that, near the invariant manifold, the state space
measure „volume… not in the basin of the attractor on the
invariant manifold scales as a power law of the displace-
ment from the invariant manifold. The solution to the
Fokker–Planck model for the power law exponent is in a
good agreement with numerical tests.
phase space have been studied, and some important dynami-
cal consequences of this have been revealed. An important
class of systems that have invariant manifolds are those that
possess an appropriate symmetry. In this case any initial
state that has the same symmetry as the system evolves to
other states that also respect the symmetry of the system. The
set of such symmetric initial states then forms a manifold
that is invariant under the dynamics of the system. An ex-
ample of an invariant manifold not accompanied by a sym-
metry of the dynamical system is the case of one-way cou-
pling of two identical oscillators, for which the states where
the oscillators are synchronized form an invariant manifold.
An invariant manifold ͑whether induced by symmetry or
not͒ can also have the property that the dynamics restricted
to this manifold is chaotic, i.e., initial states in the manifold
can be attracted to some chaotic set A in the invariant mani-
fold. Thus A is an attractor for initial conditions in the in-
variant manifold. Note, however, that A may or may not be
an attractor for the full system. Here, by an attractor for the
full system we mean that the set A attracts a set of initial
conditions of positive Lebesgue measure in the full phase
space.1 In what follows, if we say, without qualification, that
A is an attractor, then we mean that it is an attractor for the
full system.
Considering the case where A is an attractor, there are
typical cases where there is a small set of points in any
neighborhood of the invariant manifold that move away from
A as the dynamical system evolves. If the global dynamics of
the system is such that these repelled orbits are attracted to a
set other than A, then the basin of attraction of A is riddled.2,3
That is, if r is any point in the basin of A, then, for every ␯,
no matter how small, there is a displacement ␦,
that the point rϩ␦ is in the basin of another attractor, and the
set of such points, rϩ␦, Ͻ␯, has nonzero phase space
volume ͑i.e., positive Lebesgue measure͒. This presents a
basic obstruction to determinism in such systems: If one does
an experiment by preparing an initial condition and observes
that the resulting orbit goes to A, then no matter how great
͉␦͉Ͻ␯, such
I. INTRODUCTION
͉␦͉
Recently, physically important examples of dynamical
systems that have invariant manifolds embedded in their
a͒
Electronic mail: dronov@glue.umd.edu
1054-1500/2000/10(2)/291/8/$17.00
291
© 2000 American Institute of Physics
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292
Chaos, Vol. 10, No. 2, 2000
V. Dronov and E. Ott
in the infinite number of cusps of the yÞ0 attractor emanat-
ing from the invariant manifold yϭ0. To characterize this
situation we shall be interested in the scaling of the size of
the yÞ0 attractor’s basin in the region near the invariant
manifold (yϭ0). In particular, we will consider a horizontal
line yϭy1 ͑see Fig. 1͒ and ask what is the Lebesgue measure
of x values in this line that are attracted to the yÞ0 attractor.
We find that for y1 small this measure, denoted Pϱ(y1),
exhibits a power law scaling, Pϱ(y)ϳy␣. To arrive at this
result, following Ref. 7, we modify the diffusion model of
Ref. 4 previously used for the riddled case to account for the
new situation pictured in Fig. 1, and we use it to derive
results for the scaling exponent ␣. We claim that these re-
sults are universal in the sense that they are valid for the
class of dynamical systems of the type considered previ-
ously.
In order to check our scaling relations numerically we
studied three dynamical systems. The first two are repre-
sented by two-dimensional maps. In the first two-
dimensional map case the dynamics in the invariant manifold
is described by the well-known logistic map, and we will be
interested in parameter values in the vicinity of a type-one
intermittency transition.10 In the second two-dimensional
map case the logistic map is replaced with a map11 exhibiting
type-three intermittency.10 The third example is a system of
ordinary differential equations ͑a flow͒ which is a modifica-
tion of a previously studied system3 that describes the motion
of a particle in a two-dimensional potential well. As before,
we are interested in parameter values near an intermittency
transition. Our predicted scaling relations were tested for all
three numerical models and reasonably good agreement with
the theory was observed.
FIG. 1. Schematic of the situation where there is a chaotic repeller ͑vertical
tic marks͒ and a periodic orbit ͑closed circle͒ in the invariant manifold (y
ϭ0).
one’s precision in preparing the initial condition, one cannot
be sure that an attempted repeat of the experiment will result
in the same outcome.
The unusual properties of riddled basins have received
much attention. Recent work4–7 has investigated the transi-
tion to chaotic attractors with riddled basins and the effect of
noise and asymmetry on the dynamics of systems with
riddled basins. To treat the transition to chaotic attractors
with riddled basins, a simple analyzable diffusion model4
was proposed and scaling relations consistent with numerical
simulations were obtained.5
The main question addressed in this paper is what hap-
pens if, instead of a chaotic attractor embedded in the invari-
ant manifold, there is a chaotic repeller for the initial condi-
tions in the invariant manifold.8 By a chaotic repeller we
mean an invariant set on which the dynamics is chaotic, but
which does not attract a positive Lebesgue measure set of
initial conditions in the invariant manifold. Typically, such
nonattracting chaotic sets manifest themselves as chaotic
transients.9 For the case we consider, an initial condition in
the invariant manifold can experience a chaotic transient,
after which it is attracted to a periodic orbit in the invariant
manifold.
As we shall show, when a chaotic repeller is in the mani-
fold, the basin of the periodic orbit attractor in the invariant
manifold is no longer riddled, although it still has unusual
properties. This new situation is illustrated schematically in
Fig. 1, where we use a two-dimensional x– y representation.
In Fig. 1, yϭ0 represents the invariant manifold. The closed
circle represents the nonchaotic attractor in the yϭ0 invari-
ant manifold, and the vertical tic marks represent the chaotic
repeller in yϭ0. From every point in the chaotic repeller
there emanates a cusp-shaped region ͑a stalactite͒ of the ba-
sin of the attractor in yÞ0. Only one of these cusp-shaped
regions is shown in Fig. 1, but we emphasize that such re-
gions exist for all points in the chaotic repeller. Note that
since the attractor ͑the closed circle in Fig. 1͒ attracts a
neighborhood of itself, its basin is not riddled. Nevertheless,
there is a remnant of the previously studied riddled behavior
In a recent paper, Lai and Grebogi12 consider the same
situation that we consider here. A main claim in that paper is
that the basin is of a mixed type with the property that, in the
vicinity of the saddle, the basin is riddled, while in the region
of the attracting periodic orbit in the invariant manifold, the
basin is solid ͑i.e., it consists of open volumes and is not
riddled͒. Such a mixed basin cannot occur, and the basin
cannot be riddled anywhere. In general, if a basin is open in
any neighborhood N of the attractor, it must be open every-
where. A simple argument showing this is as follows. Say p
is a point in the basin. Evolving p forward, it must eventually
approach the attractor. Thus, at some finite time, the orbit
from p must eventually enter N, say at point p . The point
Ј
p in N necessarily has an open neighborhood in the basin.
Ј
Since p iterates to p in a finite number of iterates, p must
Ј
also have an open neighborhood in the basin. Hence, in con-
tradiction to the claim of Ref. 12, the basin cannot be riddled
anywhere. Lai and Grebogi12 also attempt to obtain the scal-
ing Pϱ(y)ϳy␣. However, they use a crude model for the
chaotic transient; in particular, in their model all points in the
chaotic transient phase abruptly leave the transient at a fixed
time equal to the average transient lifetime. In fact, there is a
continuous long-time exponential decay of orbits in the cha-
otic transient, and it is necessary to include this in the model
to obtain the correct scaling and the correct exponent ␣.
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Chaos, Vol. 10, No. 2, 2000
Stalactite basin structure
293
Let h denote the y-Lyapunov exponent for a typical
͗
͘
Ќ
orbit on the yϭ0 chaotic invariant set. Here, by ‘‘typical’’
we mean with respect to the natural transient measure for the
x map. In this case h is calculated by sprinkling N initial
͗
͘
Ќ
0
conditions in yϭ0, iterating them for n iterates, discarding
those that are near the nonchaotic orbit ͑say, further than
some appropriate distance l͒, calculating the hЌ exponents
over the time interval 0 to n for each orbit not near the
nonchaotic orbit at time n, and averaging these values. In the
limit N →ϱ, n→ϱ, this average is h , the y-Lyapunov
͗
͘
0
Ќ
exponent, calculated with respect to the natural transient
measure for the x map.
We show in the following that the basin of ͉y͉Ͼyc can
be in the form of an infinite set of cusp-shaped regions ema-
nating from the chaotic transient invariant set in the plane
yϭ0. We call these cusp-shaped regions stalactites. We
show for our examples that the Lebesgue measure in x of
these tongues in a yϭy1 cross section scales as y␣1 , where
␣Ͼ1. We also give a theory for determining ␣ and obtain
good agreement of the theory with numerical experiments.
FIG. 2. Period-three window in the bifurcation diagram of the logistic map,
x
nϩ1ϭrxn(1Ϫxn).
II. THE MODEL
Consider a smooth two-dimensional map of the form
ynϩ1ϭN x ,y ,⑀͒,
We note that stalactite boundaries can occur for both h
͗
͘
Ќ
Ͼ0 and h Ͻ0. For stalactites in the case h Ͻ0, how-
͗
͘
͗
͘
Ќ
Ќ
͑1͒
͑2͒
͑
n
n
ever, it is also required that the chaotic saddle ͑although
transversely attracting͒ have embedded periodic orbits with
positive transverse Lyapunov exponents.12
xnϩ1ϭM x ,r͒,
͑
n
where ⑀ and r are parameters. Suppose that N(x,0,⑀)ϭ0 so
that there is an invariant manifold at yϭ0. We also assume
that N(•) is such that if
͉yn͉Ͼyc then all subsequent y iter-
ates are also in ͉y͉Ͼyc ; in particular, the orbit never comes
back close to the yϭ0 plane. Thus there is another attractor
III. DIFFUSION APPROXIMATION
͑or attractors͒ in
ycϭ1.
͉y͉Ͼyc . Without loss of generality we let
Consider the map ͑1͒–͑2͒ and assume that there is a
Following Ref. 7 we refer to ⑀ as a ‘‘normal’’ parameter
since it affects the dynamics normal to the yϭ0 invariant
manifold, but has no influence on the dynamics in the yϭ0
invariant manifold ͓the dynamics in yϭ0 is governed by Eq.
͑2͔͒. Similarly,7 we refer to r as a ‘‘non-normal parameter.’’
Suppose that for some value of r, M(x,r) has a chaotic
attractor in x and ⑀ is such that riddling occurs. In this case
yϭ0 is an attractor and its Lyapunov exponent, which we
denote hЌ ͓calculated by taking a differential y variation of
Eq. ͑1͔͒, is negative, hЌϽ0. In the riddled case, in any neigh-
borhood of the yϭ0 attractor there are initial conditions that
stay in the neighborhood and go to the attractor, as well as
other initial conditions that leave the neighborhood of y
fixed point attractor xϭx for almost any initial condition in
yϭ0. Also assume that there is a chaotic transient set in y
ϭ0. The transverse tangent map ͑evolving differential y dis-
placements from the invariant set yϭ0͒ is
*
␦ynϩ1ϭNy 0, x ͒␦y ,
͑
n
n
where Ny(0, x) denotes ץN/ץy evaluated at yϭ0. For an
initial condition x0 precisely on the chaotic transient set, the
orbit xn is typically chaotic. For x1 on the nonchaotic attrac-
tor in yϭ0 ͑i.e., xϭx ͒, we have
͉
Ny(0, xn)
. Then the change in ␦z is given by ␦znϩ1
ϭ␦znϩ⌬n , where ⌬nϭϪln Ny(0, xn) . During the chaotic
transient xn is chaotic, and hence ⌬n varies in a random
manner. After the chaotic transient ⌬nХln Ny(0, x ) Ͻ0.
We follow Ref. 7 and model this situation as follows: ⌬n is
taken to be of constant magnitude ϭ⌬ when the orbit is
͉Ͻ1. Let ␦zn
*
ϭϪln ͉␦yn ͉
͉
͉
ϭ0, possibly moving to the ͉y͉Ͼyc attractor. What will hap-
pen if we change r in such a way that, for a typical initial
condition x0 , M(•) generates a chaotic transient instead?
͓For example, if M(•) is the logistic map we can imagine a
change of r such that we enter a periodic window, e.g., the
period-three window on the bifurcation diagram of the logis-
tic map; see Fig. 2.͔ This means that the original chaotic
attractor in the invariant plane yϭ0 does not exist any more
and typical initial conditions in yϭ0 eventually go to a non-
chaotic attractor ͑e.g., a periodic orbit or a fixed point͒ as n
→ϱ. There will still exist, however, an infinite nonattracting
set of initial conditions xinv ͑representing the ghost of the
͉
͉
*
͉
⌬ ͉
n
on the chaotic transient and fluctuations in ⌬n are modeled
by random assignment of its sign ⌬nϭϮ⌬, and we assume
that the linearized dynamics is a good approximation to the
actual dynamics for finite y in a neighborhood of yϭ0. Thus
on any given
¯
yϵln 1/
͉y͉ an iterate is represented by the step
¯
y→yϮ⌬, with
¯
¯yϭϩϱ(␦yϭ0) corresponding to yϭ0. We
consider a Markov chain model described by a semi-infinite
chain of states Si , iϭ0,1,2,... ͑see Fig. 3͒. We call the chain
Si the ‘‘chaotic chain.’’ The model has three parameters ␤ϩ ,
␤Ϫ , and ␩, which represent the probabilities of the follow-
ing transitions:
͕
͖
0
former chaotic attractor͒ for which infinitely long chaotic
orbits can be generated, and this set has zero Lebesgue mea-
sure in x.
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294
Chaos, Vol. 10, No. 2, 2000
V. Dronov and E. Ott
For the purposes of obtaining a solution for p(i), we
will apply the diffusion approximation, valid for (␤ϩϪ␤Ϫ)
small,4 to the Markov model. Then the basic parameters of
the diffusion model are the following:
͑1͒ the average drift along the Si chain per iterate,
v
v
ϭ ␦
͗
¯
y , where ␦
¯y is the increment in
¯y in one time step;
͘
by definition of the transverse Lyapunov exponent
ϭϪ h
;
͘
͗
Ќ
͑2͒ the diffusion per iterate, Dϭ 12 (␦
¯yϪ ¯
y )2 , where ͗¯͘
FIG. 3. Markov chain.
͗
͗ ͘ ͘
denotes an average over the initial random values of x1 ;
͑3͒ and an average lifetime of a typical chaotic transient ␶.
␤ϩ :Siϩ1→Si and S1→X,
Let P(
y ͑given that x1 is randomly chosen on the horizontal line
segment yϭy1͒. Considering n to be approximated by a con-
tinuous variable ͑valid for nӷ1͒, P(¯y,¯y1 ,n) obeys the fol-
lowing drift-diffusion equation:
¯y,¯y1 ,n) be the probability distribution function
for
¯
␤Ϫ :Si→Siϩ1
,
␩:Si→Q.
The transition Siϩ1→Si(Si→Siϩ1) represents a step in y
away from ͑toward͒ the invariant surface yϭ0, and in y it is
represented by a step yϪ⌬(y→yϩ⌬). X represents the
y→
condition Ͼyc for which the orbit goes to an attractor ͑or
ץ2P
ץ¯
y2
¯
ץP
ץn
ץP
P
¯
¯
¯ ¯
ϩ
ϭϪ ϩD
.
͑3͒
v
ץ
¯y
␶
͉y͉
attractors͒ not in yϭ0. After the transition Si→Q(S1→X)
Since we imagine initial conditions all to start at yϭy1 , we
have
the orbit is assumed to move uniformly in x ͑in y͒ toward the
periodic attractor in yϭ0 ͑the attractor in ͉y͉Ͼyc͒. Once the
P
¯
y,
¯
y ,0͒ϭ␦
¯yϪ
͑
¯
y1͒,
¯
y1Ͼ0.
͑4͒
͑
1
chaotic chain is exited by transmission to Q or X, the chaotic
transient is considered to be ended. We identify the trans-
verse Lyapunov exponent with the random walk parameters
Since any orbit which crosses yϭycϵ1 (
yϾ1 attractor, we have
¯
yϭ0) is lost to the
via h ϭ(␤ Ϫ␤ )⌬. The question now becomes what is
͗
͘
Ќ
ϩ
Ϫ
P 0, ¯y1 , n͒ϭ0.
͑
͑5͒
the probability of reaching the state X if starting at the state
Si . Our purpose in introducing this Markov model is to ob-
tain a result for the scaling of the size of the yÞ0 attractor
basin in the region near yϭ0. In particular, consider the
horizontal line yϭy1 . Different initial conditions xϭx1 on
this line produce orbits, some of which go to the yϭ0 at-
tractor, and some of which go to the yÞ0 attractor. We ask
what fraction of the x-Lebesgue measure on yϭy1 goes to
the yÞ0 attractor. Denote this fraction Pϱ(y1). The above-
mentioned Markov model is relevant to this question in the
following way. If we choose a point x1 at random on the line
yϭy1 , then the probability that x1 is in the basin of the y
Þ0 attractor is Pϱ(y1). Furthermore, as argued in Ref. 4,
while the orbit is chaotic, the orbit generated for random x1
can be regarded as random and as a stochastic process cor-
responding to a random walk on the Si chain. However, the
orbits following the chaotic transient can leave the Si chain
either by repulsion into the x region where the orbit moves
As discussed in Ref. 4, for this diffusion approximation to be
valid we require two conditions:
͑1͒ Many steps must be taken to reach yϭ1 ͑corresponding
to
͑2͒ The drift on each iterate must be small, which means that
Ӷ1.
¯yϭ0͒, or ¯y1ӷ1.
͉
h ͉
͘
Ќ
͗
In order to solve Eq. ͑3͒ we will introduce the Laplace
transform of P(
variable n,
¯y,¯y1 ,n) with respect to the continuous time
ϱ
¯
P
¯y,
¯
y ,s͒ϭ
e
ϪsnP dn.
͑6͒
͑7͒
͑
͵
1
0
Equations ͑3͒–͑5͒ yield
2
2
¯ ¯ ¯
¯y Ϫ dP/d¯yϪ sϩ1/␶͒PϭϪ␦ ¯yϪ¯y1͒
v ͑ ͑
Dd P/d
¯
with the boundary condition P(0,
¯y1 ,s)ϭ0. Solving Eq. ͑7͒
toward the nonchaotic orbit (xϭx on yϭ0; i.e., Si→Q) or
*
is straightforward ͑see the Appendix͒.
by repulsion in y into the region
͉
y
͉
Ͼyc ͑i.e., S1→X͒. Thus
In Eq. ͑3͒, the term P/␶ is the rate at which orbits leave
the chaotic transient to move toward the attractor in yϭ0.
Thus the probability of going to the attractor in the invariant
manifold is
we can identify Pϱ(y1) with a random walker’s probability
p(i) of reaching X starting at iϳ⌬Ϫ1 ln
the large i scaling of p(i),
͉yc /y1͉. In particular,
p i͒ϳeϪKi
,
͑
ϱ
ϱ
1
P
yϭ0ϭ
P
¯y,¯y1 ,n͒d
͑
¯y dn
͑8͒
and the small y1 scaling of Pϱ(y1),
͵ ͵
␶
0
0
␣
Pϱ y ͒ϳ
͉
y1
͉
,
͑
i
and, therefore, the probability of getting attracted to the
are connected. Thus we now attempt to estimate K from the
Markov model. Once this is accomplished, we will be in a
position to use this to conjecture a general result for ␣, which
we will compare with numerical experiments in Sec. IV.
yϾ1 attractor is
ϱ
ϱ
1
Pϱϭ1ϪPyϭ0ϭ1Ϫ
P
¯y,
¯y1 ,n͒d
¯y dn.
͑9͒
͑
͵ ͵
␶
0
0
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Chaos, Vol. 10, No. 2, 2000
Stalactite basin structure
295
Performing the integration we finally get
␣
2ϩ4D/␶Ϫ h /2D.
͑10͒
ͱ h
͗
P ϭy ,␣ϭ
͓
͔
͘
͘
͗
ϱ
Ќ
Ќ
1
2
The special case of small D(D/ h ␶Ӷ1) yields
͗
͘
Ќ
Ϫ1
h
Ќ
␶͒
for
h Ͼ0
͘
Ќ
͑
͗
͘
͗
␣Х
.
͑11͒
ͭ
Ϫ h /D for
h Ͻ0
͘
Ќ
͗
͘
͗
Ќ
Note that, as expected, the result for
agrees with the result for the exponent in the case of a
riddled basin attractor, Ref. 4. In the examples that follow
h
Ќ
Ͻ0 and ␶→ϱ
͗
͘
h
Ќ
Ͼ0.
͘
͗
IV. NUMERICAL EXPERIMENTS
The universality of the phenomena addressed in this pa-
per implies that very general results may be extracted from
simple models that incorporate the essential features respon-
sible for this phenomena. In this spirit we introduce three
illustrative examples.
Example 1: The first example is the following two-
dimensional map:
FIG. 4. ͑a͒, ͑b͒ Basins of attraction for Eqs. ͑12͒ and ͑13͒ for ⑀ϭ0.926 and
0.928, respectively. To obtain ͑c͒ we uniformly sprinkle Lϭ1.5ϫ106 initial
conditions in y0ϭ0, 0Ͻx0Ͻ1. We then iterate them 200 iterates and find
that there are lϭ950 orbits which still stay at a distance larger than 0.1 from
x
y
nϩ1ϭM1 x ,r͒ϭrx 1Ϫx ͒,
͑12͒
͑
͑
n
n
n
the period-three attractor. Then we plot 1/2 z2
versus n, where
¯
nϩ1ϭN1 x ,y ,⑀,⑀ ͒
͑
Ј
͗
͘
͗
͘
IC
n
n
IC
n
denotes an average over the lϭ950 orbits still not near the period-three
attractor. The value of D is then estimated as the slope of the straight line fit
to the numerical data; ͑d͒ the fraction of initial conditions going to yϭϱ as
a function of y. ␣exp is obtained as the slope of the straight line fit to the data
in a log–log scale. ͑c͒, ͑d͒ Correspond to the situation where ⑀ϭ0.926.
ϭ 1ϩ⑀Ϫ⑀ 1Ϫcos 2␲x ͒ y ϩy3.
͑13͒
͓
͑
͔
͖
n
͕
Ј
n
n
This system has N (x,0,⑀,⑀ )ϭ0. Therefore, yϭ0 is an in-
Ј
1
variant manifold for the map. The dynamics on this invariant
manifold are generated by the logistic map and are indepen-
dent of the parameters ⑀ and ⑀Ј. Therefore, the parameters ⑀
and ⑀Ј are normal parameters for the system ͑12͒ and ͑13͒.
The value of ␶ was chosen to be 3.837, which corresponds to
the case of a period-three attractor in the period-three win-
dow of the logistic map ͑see Fig. 2͒. This implies that typical
initial conditions in 0Ͻx0Ͻ1 and y0ϭ0 may generate cha-
otic transients. The average lifetime ␶ of such a transient
To obtain the values of h for the chaotic transient set
͗
͘
Ќ
͑the second column of Table I͒ we first uniformly sprinkle
many initial conditions in y0ϭ0, 0Ͻx0Ͻ1. We then iterate
these initial conditions M iterates. At time Mӷ1 most of the
sprinkled orbits will be close to the attractor ͑say within a
distance of 0.1͒. However, if the number of sprinkled orbits
is large, there will still be orbits that are not near the attrac-
tor. We make the number of initial conditions sprinkled large
enough so that there are still many initial conditions not near
depends on r and in our case is numerically determined to be
␶ϭ21.41. Note that if y Ͼ 1(2⑀ Ϫ⑀)
then ynϩ1Ͼyn
1/2
͓
͔
Ј
n
3
and the orbit evolves toward yϭϰ, which we regard as the
attractor not in yϭ0. A numerical approximation to the ba-
sins of attraction is shown in Figs. 4͑a͒ and 4͑b͒. To generate
this figure we use a 5000ϫ5000 grid of initial conditions,
the attractor at time M. We then approximate h by
͗
͘
Ќ
M
1
h
Х
͘
͉
ln N1y x ,0͒
͉
,
͑
͗
ͳ
͚
n
ʹ
Ќ
M nϭ1
and we iterate each initial condition until it either reaches y
IC
1/2
Ͼ
1(2⑀ Ϫ⑀)
͑in which case we plot a closed circle at the
͓
͔
Ј
where N1y(x,y)ϭץN1 /ץy and
¯
denotes an average
3
͗
͘
IC
location of the initial condition͒ or else reaches a very small
value of y,yϽ2ϫ10Ϫ5. In the latter case we presume that,
with high probability, the orbit will go to the yϭ0 attractor.
͑This is justified by the scaling Pϱϳy␣͒. We see in Figs. 4͑a͒
and 4͑b͒ that there are many thin downward pointing black
regions. From numerical examination of some of them by
magnification of the horizontal scale we find that these thin
black regions apparently extend down and touch the invari-
ant set. They are the stalactites referred to in Sec. I. The
reason the stalactites appear in Figs. 4͑a͒ and 4͑b͒ to termi-
nate before reaching yϭ0 is that they become so thin as y is
decreased that they eventually fail to be resolved by our grid
over those initial conditions whose orbits are not near the
attractor in yϭ0 at time M. The value of D ͑the third column
of Table I͒ was obtained by noting that the quantity
n
z ϭ
n
ln͉
N1y x ,0͒
͉
Ϫ h
͒
͘
Ќ
͑
͑
͗
͚
m
mϭ1
TABLE I. Data for example 1.
⑀
h
D
͗
͘
␣
␣
exp
Ќ
th
0.920
0.923
0.926
0.928
0.0164
0.0200
0.0233
0.0256
0.000 920
0.000 917
0.000 885
0.000 864
2.49
2.12
1.87
1.72
2.14
2.16
1.93
1.74
of initial conditions. In our simulations we took ⑀ ϭ0.8 and
Ј
varied the value of ⑀. For each value of ⑀ we calculated h
,
͘
͗
Ќ
D, ␣th , and ␣exp . The data are presented in Table I.
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296
Chaos, Vol. 10, No. 2, 2000
V. Dronov and E. Ott
TABLE II. Data for example 2.
⑀
h
D
͗
͘
␣
␣
exp
Ќ
th
0.150
0.155
0.005
0.010
0.011
0.011
1.59
1.40
1.62
1.42
This system has the same general properties as in the map of
the previous example. However, M2(xn ,r), the map in the
invariant manifold, is an example11 exhibiting type-iii
intermittency.10 The bifurcation diagram for M2(•) is shown
in Fig. 5. This map has a period-one orbit at xϭ0. As the
parameter r increases through rcϭ1, this period-one orbit
becomes unstable, experiencing an inverse period doubling
bifurcation. The map has a chaotic attractor for rϾ1. The
parameter r was taken to be equal to 0.95; M2(•) is contract-
ing at the fixed point xϭ0 and repelling on a transient. For
this value of r, we obtain ␶ϭ28.0. The basin of attraction is
FIG. 5. Bifurcation diagram for Eq. ͑14͒.
undergoes unbiased diffusion while the orbit is in the chaotic
1
2
transient. Thus we plot z2
vs n, and estimate D as the
͗
͘
IC
n
shown in Fig. 6. Table II gives the values of h , D, ␣ ,
͗
͘
Ќ
th
slope of a straight line fit to the data ͓see Fig. 4͑c͒ for an
example of such a plot͔. The fourth column of Table I. ␣th ,
and ␣exp for ⑀ ϭ0.1, and two different values of the other
Ј
normal parameter ⑀.
is then given by inserting h and D, as found above, into
͗
͘
Ќ
Example 3: The third system we studied is a modifica-
tion of a flow that describes the motion of a particle in a
two-dimensional ͑x,y͒ double potential well3 V(r)ϭ(1
Eq. ͑10͒. The value of the experimental exponent ␣exp ͑the
fifth column of Table I͒ was obtained by estimating the frac-
tion of ‘‘black’’ dots in the basin of attraction ͑those that
lead to y→ϱ͒ as a function of y, and plotting this fraction as
a function of y using a log–log scale. We find that, for suf-
ficiently small y, such plots are, in accord with the predicted
y␣ dependence, well-fit by a straight line ͓see Fig. 4͑d͒, for
example͔. The slope of such a fit gives ␣exp . One can see
from Table I that ␣exp agrees reasonably well with its pre-
dicted value ␣th .
Ϫx2)2ϩ(xϩx)y2. The particle is a subject to friction and
¯
time harmonic forcing,
d2r/dt2ϭϪ␥dr/dtϪٌV r͒ϩf sin ␻t͒xˆ.
͑16͒
͑
͑
0
Here xˆ is the unit vector in the x direction and ␥, f0 , ␻, and
x are parameters. For appropriate (␥, f0 ,␻,x) this system is
¯
¯
known to exhibit a riddled basin.3 The phase space of this
problem is five dimensional with coordinates x, dx/dt, y,
dy/dt, and ␪ϭ(␻t) mod 2␲. From the symmetry of the po-
tential the problem is invariant with respect to y→Ϫy, and,
therefore, yϭdy/dtϭ0 specifies a three-dimensional invari-
ant hyperplane in the full five-dimensional phase space. The
dynamics in this invariant hyperplane is obtained by setting y
and dy/dt equal to zero in Eq. ͑16͒. This yields
Example 2: We consider the two-dimensional map,
x
nϩ1ϭM2 x ,r͒
͑
n
ϭxn r 12.3Ϫ7r͒x4Ϫr 11.3Ϫ7r͒x2ϩx2Ϫr ,
͓ ͑
͑
͔
n
n
n
͑14͒
͑15͒
y
nϩ1ϭN2 x ,y ,⑀,⑀ ͒
͑
Ј
n
n
ϭ 1ϩ⑀Ϫ⑀ 1ϩcos ␲x ͒ y ϩy3.
͓
͑
͔
͖
n
͕
Ј
n
n
d2x/dt2ϩ␥ dx/dtϪ4x 1Ϫx2͒ϭf sin ␻t͒.
͑17͒
͑
͑
0
FIG. 6. Basin of attraction for Eqs. ͑14͒ and ͑15͒ obtained on a 2000
ϫ2000 grid of initial conditions. rϭ0.95, ⑀ϭ0.135, ⑀ ϭ0.1.
FIG. 7. Bifurcation diagram for Eq. ͑17͒.
Ј
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Chaos, Vol. 10, No. 2, 2000
Stalactite basin structure
297
that the basin boundary separating the basins of the attractor
in the invariant manifold and another attractor not in the
invariant manifold is characterized by stalactitelike structure,
thin cusp-shaped regions extending down to touch the invari-
ant manifold at the location of points in the nonattracting
chaotic set. Using a random walk model, we have obtained a
scaling relation for the variation of the measure of the basin
of the attractor not in the invariant manifold. In particular,
we show that this basin’s measure in a surface at a small
distance y from the invariant manifold scales as a power law
y␣, and we have obtained a theoretical prediction for the
scaling exponent ␣. This prediction has been tested with
good results by comparison with various numerical experi-
ments.
FIG. 8. Numerical basin for Eq. ͑16͒ obtained on 1000ϫ1000 initial con-
ditions. Following Ref. 4 we iterate each initial condition forward in time
until the trajectory either gets very close to the invariant plane yϭ0 (
Ͻ10Ϫ8 Ͻ10Ϫ9, and y yϽ0͒, or else is definitely in the repelling re-
gion (
conditions eventually attracted to ϱ.
͉y͉
,
y
͉
͉
vy
Ͼ20,
v
ACKNOWLEDGMENT
͉
͉
͉ ͉Ͼ100, and y yϾ0͒. Closed circles correspond to initial
vy v
The work was supported by the Office of Naval Re-
search ͑Physics͒.
´
We analyzed Eq. ͑16͒ in terms of a stroboscopic Poincare
section corresponding to the period of the forcing, i.e., we
consider the times ␻t mod 2␲ϭ0. This lead to a four-
dimensional discrete-time mapping.
In previous work using model ͑16͒, the parameters ␥, f0 ,
and ␻ were chosen so that Eq. ͑17͒ had a chaotic attractor. In
APPENDIX: DERIVATION OF EQ. „10…
The solution to Eq. ͑7͒ subject to the boundary condition
¯
P(0,
¯y1 ,s)ϭ0 is
this case, appropriate choice of the parameter ¯x yields basin
A exp ␬
¯
y Ϫexp ␬
¯
͔
b
y ͒, 0Ͻ
¯yϽ
¯
y1
͑
͓
͔
͓
riddling for the yϭ0 attractor. Here we consider a modifica-
tion of Eq. ͑16͒ such that the dynamics in yϭ0 is such that
the attractor is periodic, but there is also a chaotic transient.
a
¯
P
¯y,
¯
y ,s͒ϭ
͑
ͭ
1
B exp Ϫ␬ ¯y ¯yϾ¯y1,
͔
a
͓
͑A1͒
In particular, we replace
¯x by a time periodic quantity
¯x
where
ϭ¯
x0ϩx1 sin(␻tϪ␾) and set the parameters at
¯
¯x0ϭ4.4, ¯x1
ϭ2.9, ␾ϭ0.15, ␥ϭ0.05, and ␻ϭ3.5. Varying the remaining
parameter f0 we obtain a bifurcation diagram for Eq. ͑17͒
͑see Fig. 7͒. As f0 is decreased there is a type-iii intermit-
tency transition to chaos ͓similar to the map ͑15͔͒. For f0
ϭ4.0 the average lifetime of a chaotic transient is ␶ϭ12.0.
1
ͱ h
͗
␬aϭ
2ϩ4D sϩ1/␶͒ϩ h ͒,
͑
͗ ͘
Ќ
͑
͑
͘
͘
Ќ
2D
1
ͱ h
͗
␬bϭ
2ϩ4D sϩ1/␶͒Ϫ h ͒,
͑
͗ ͘
Ќ
Ќ
In order to estimate h we integrated the equations for the
͗
͘
2D
Ќ
tangent vector representing the infinitesimal variations from
the invariant plane yϭ ϭ0:d␦y/dtϭϪ␥␦ yϪ2(x
vy
v
exp Ϫ␬b¯y1
͔
͓
AϭϪ
BϭϪ
,
ϩ¯x)␦y, d␦y/dtϭ␦ , where ␦y and ␦ are infinitesimal
vy vy
␬ ϩ␬ ͒D
͑
a
b
variations ͑Fig. 8͒. The results are presented in Table III and
we again find reasonably good agreement between our esti-
exp Ϫ␬
¯
by1 Ϫexp ␬
a¯y1
͔
͓
͔
͓
.
mates of ␣th and ␣exp
.
␬ ϩ␬ ͒D
a
͑
b
V. CONCLUSION
Thus
In this paper we have considered systems with an invari-
ant manifold in which are located both a nonattracting cha-
otic set and a nonchaotic attractor ͑e.g., a periodic orbit͒.
Thus, in this situation, typical initial conditions in the invari-
ant manifold may yield orbits that experience chaotic tran-
sients before approaching the chaotic attractor. It is found
ϱ
¯
ϱ
y
1
¯
¯
¯
P d¯y
P
¯y,
¯y1 ,s͒d
¯yϭ
P d
¯
yϩ
͑
͵
͵ ͵
¯
0
0
y
1
ϭA 1Ϫexp Ϫ␬a¯y1 ͒/
͓͑ ͔
͓
␬ Ϫ exp ␬b¯y1 Ϫ1͒/␬
͓ ͔ ͔
b
͑
a
ϩB exp Ϫ␬a¯y1 )/␬
͔
a
͓
TABLE III. Data for example 3.
1Ϫexp Ϫ␬
b¯y1
͓
͔
ϭ
.
h
D
͗
͘
␣
␣
exp
Ќ
th
sϩ1/␶
0.030
0.0285
1.26
1.29
Finally,
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298
Chaos, Vol. 10, No. 2, 2000
V. Dronov and E. Ott
3 J. C. Sommerer and E. Ott, Nature ͑London͒ 365, 136 ͑1993͒.
4 E. Ott, J. C. Alexander, I. Kan, J. C. Sommerer, and J. A. Yorke, Physica
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intermittency. For example, see T. Yamada and H. Fujisaka, Prog. Theor.
Phys. 76, 582 ͑1986͒; H. Hata and S. Miyazaki, Phys. Rev. E 55, 5311
͑1997͒.
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9 S. W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, Physica D 17, 125
͑1985͒.
ϱ
ϱ
1
P
yϭ0ϭ
P ¯y,¯y1 ,n͒d¯y dn
͑
͵ ͵
␶
0
0
ϱ
1
¯
P ¯y,¯y1,0͒d¯y
͑
ϭ
͵
␶
0
ϭ1Ϫexp Ϫ␣
¯
y1 ϭ1Ϫy␣ ,
͓
͔
1
where ␣ is ␬ with s set equal to zero. Thus
b
Pϱϭ1ϪPyϭ0ϭy␣1 ,
which is Eq. ͑10͒.
10 Y. Pomeau and P. Manneville, Commun. Math. Phys. 74, 189 ͑1985͒.
11 W. Yang, M. Ding, A. J. Mandell, and E. Ott, Phys. Rev. E 51, 102
͑1995͒.
1 J. Milnor, Commun. Math. Phys. 99, 177 ͑1985͒.
2 J. C. Alexander, I. Kan, J. A. Yorke, and Z. You, Int. J. Bifurcation Chaos
Appl. Sci. Eng. 2, 795 ͑1992͒.
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