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Basin of attractionYou +1'd this

Figure 1 : (a) Double w ell potential

an d (b) the resultin g basins of attra ction.

Edward Ott, University of Maryland, MD, USA

Roughly speaking, an attractor of a dynamical system is a subset of the state space to which orbits

originating from typical initial conditions tend as time increases. It is very common for dynamical sy stems to

have mor e than one attractor. For each such attractor, its basin of attraction is the set of initial conditions

leading to long-time behav ior that approac hes that attractor . Thus the qualitative behavior of the long-time

motion of a giv en system can be fundamentally different depending on which basin of attraction the initial

condition lies in (e.g., attractors can corre spond toperiodic , quasiperiodic or chaotic behav iors of different

ty pes). Regarding a basin of attraction as a region in the state space, it has been found that the basic

topological structure o f such regions can vary greatly from sy stem to sy stem. In what follows we give

ex amples and discuss seve ral qualitatively different kinds of basins of attraction and their practic al

implications.

Example

A simple ex ample is that o f a point particle mov ing in a two-

well po tential with friction, as in Figure 1 (a). Due to the

friction, all initial conditions, ex cept those at

or on its stable manifold eventuallycome to rest at either

or which are the two attractors of the

system. A po int initially placed on the unstable equilibrium

point, will stay there forever; and this state has a one-

dimensional stable manifold. Figure 1(b) shows the basins of

attraction of the two stable equilibrium point,

where the c ro sshatched region is the basin for the attractor

at and the blank region is the basin for the attracto r

at The boundary separating these two basins is

the stable manifold o f the unstable equilibrium

Fractal basin boundaries

In the abov e ex ample, the basin boundary was a smooth curv e. Howev er, other po ssibilities exist. An

example of this occ urs for the map

For almost any initial condition (ex cept for those precisely o n the boundary between the basins of

attraction), is either or which we may regard as the two attracto rs of the

system. Figure 2 shows the basin structure for this map, with the basin for the attracto r black and

the basin of the attracto r blank. In contrast to the prev ious ex ample, the basin boundary is no

longer a smooth curv e. In fact, it is a fractal cur ve with a box-counting dimension 1 .62.... We emphasize that,although fractal, this basin boundary is still a simple cur ve (it can be written as a co ntinuous parametric

functional relationship for such that if

)

V(

x ) ,

x= d x / d t = 0

x=

x

0

x= ,

x

0

x= 0 ,

x= ,

x

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x=

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x= .

x

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x= 0 .

= ( 3 ) m o d 1 , x

n+ 1

x

n

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n+ 1

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y=

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s) ,

y=

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s> 0 (

x( ) ,

y( ) ) (

x( ) ,

y( ) )

s

1

s

1

s

2

s

2

.s

1

s

2
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Figur e 2: A case where th e basin boun dary is a

f ra cta l curv e .

Anothe r example o f a sy stem with a fractal basin boundary is the forced damped pendulum equatio n,

For these parameters, there are two attracto rs which are both periodic orbits (Grebo gi, Ott and Yo rke,

1987 ). Figure 3 shows the basins of attraction of these two attracto rs with initial values plotted horizontally

and initial values of plotted vertically . The figure was made by initializing many initial conditions on a

fine rec tangular grid. Each initial condition was then integrated forward to see which attrac tor its orbit

approac hed. If the orbit approached a particular one of the two attractors, a black dot was plotted on thegrid. If it approached the other attracto r, no dot was plotted. The dots are dense enough that they fill in a

solid black region exc ept near the basin boundary . The speckled appearance of much of this figure is a

consequence o f the intricate, finescaled structure of the basin boundary . In this case the basin boundary is

again a fractal set (its box-counting dimension is about 1 .8), but its topology is more complicated than that of

the basin boundary of Figure 2 in that the Figure 3 basin boundary is not a simple curv e. In both o f the above

ex amples in which fractal basin boundaries occur, the fractality is a result ofchaotic motion (see transient

chaos) of orbits on the boundary , and this is generally the case for fractal basin boundaries (McDonald et al.,

1985).

Basin Boundary Metamorphoses

We hav e seen so far that there c an be basin

bo undaries of qualitativ ely different types. A s in the

case of attractors, bifurcations can occur in which

basin boundaries undergo qualitativ e c hanges as a

system parameter passes through a c ritical

bifurcation v alue. For example, for a sy ste m

parameter the basin boundary might be a

simple smooth curv e, while for it might befractal. Such basin boundary bifurcations have been

called metamorphoses (Grebogi, et al., 1987 ).

The Uncertainty Exponent

Fractal basin boundaries, like those illustrated

abov e, are ex tremely c ommon and have po tentially

important practical consequences. In particular, they may make it more difficult to identify the attrac tor

corresponding to a given initial condition, if that initial condition has some uncertainty. This aspect isalready implied by the speckled appearance of Figure 3. A quantitativ e measure of this is provided by the

uncertainty exponent(McDonald et al., 1985). For definiteness, suppose we randomly c hoose an initial

condition with uniform pro bability density in the area of initial condition space co rresponding to the plot in

Figure 3. Then, with probability one, that initial condition will lie in one of the basins of the two attrac tors

[the basin boundary has zero Lebesgue measure (i.e., 'zero area') and so there is zero pro bability that a

random initial condition is on the boundary ]. Now assume that we are also told that the initial condition has

some given uncertainty, and, for the sake of illustration, assume that this uncertainty can be represented

by say ing that the real initial c onditio n lies within a c irc le o f radius centered at the c oo rdinates

that were randomly chosen. We ask what is the probability that the could lie in a basin that is

different from that of the true initial condition, i.e., what is the probability, that the uncertainty couldcause us to make a mistake in a determination of the attractor that the orbit goes to. Geometr ically, this is the

same as asking what fraction of the area of Figure 3 is within a distance of the basin boundary . This fraction

scales as

/

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Figure 3: Basins of attr action for a forced dam ped

pendulum (pictur e m ade by H.E. Nusse) .

where is the uncertainty exponent(McDonald et al., 1985) and is given by where is the

dimension of the initial condition space ( for Figure 3) and is the box-counting dimension of the

basin boundary . For the ex ample o f Figure 3, sinc e we hav e For small it bec omes

very difficult to impro v e predic tiv e c apacity (i.e ., to predict the attractor from the initial condition) by

reducing the uncertainty. For example, if to reduce by a factor of 10 , the uncertainty would

have to be reduc ed by a factor of Thus, fractal basin boundaries (analogous to the butterfly effect of

chaotic attractors) pose a barrier to prediction, and this barrier is related to the presence of chaos.

Riddled Basins of Attraction

We now disc uss a ty pe of basin to pology that may oc cur in certain spec ial systems; namely , sy stems that,

through a symmetry or some other constraint, have a smooth invariant manifold. That is, there ex ists a

smooth surface or hypersurface in the phase space, such that any initial condition in the surface generates an

orbit that remains in the surface. These systems can have a particularly bizarre type of basin structure c alled

a riddled basin of attraction (Alex ander et al., 1992; Ott et al., 1994). In order to discuss what this means, we

first hav e to c learly state what we mean by an "attracto r". For the pur poses of this discussion, we use the

definition of Milnor (1 985): a set in state spac e is an attrac tor if it is the limit set of orbits o riginating from a

set of initial conditions of positiv e Lebesgue measure. That is, if we randomly choo se an initial condition with

uniform probability density in a suitable sphere o f initial condition space, there is a non-zero probability that

the orb it from the c hosen initial condition goes to the attrac tor. This definition differs from another co mmon

definition of an attractor which re quires that there ex ists some neighborhood o f an attractor such that all

initial conditions in this neighborhood generate orbits that limit on the attractor. A s we shall see, an

"attracto r" with a riddled basin conforms with the first definition, but not the sec ond definition. The failure to

satisfy the seco nd definition is because there are points arbitrarily c lose to an attractor with a riddled basin,

such that these points generate orbits that go to another attractor (hence the neighborhood mentioned

abov e does not exist.)

We are no w ready to say what we mean by a riddled basin. Suppose o ur sy stem has two attractors which we

denote and with basins and We say that the basin is riddled, if, for every point in an -

(

) ,

=D

D

0

D

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D

0

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0

0 . 2 .

= 0 . 2 ,

(

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.1 0

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.C

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http://www.scholarpedia.org/w/index.php?title=Probability_density&action=edit&redlink=1http://www.scholarpedia.org/w/index.php?title=Lebesgue_measure&action=edit&redlink=1http://www.scholarpedia.org/article/Attractorhttp://www.scholarpedia.org/article/Invariant_manifoldhttp://www.scholarpedia.org/article/Butterfly_effecthttp://www.scholarpedia.org/w/index.php?title=Basin_of_attraction&printable=yes#fig:Ott_fig3.gifhttp://www.scholarpedia.org/article/Box-counting_dimensionhttp://www.scholarpedia.org/w/index.php?title=Basin_of_attraction&printable=yes#fig:Ott_fig3.gifhttp://www.scholarpedia.org/article/File:Ott_fig3.gif


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Figure 4 : Schem atic i l lustrat ion of a situation with a riddled basin of

attract ion.

radius ball, centered at contains a positive Lebesgue measure of points in for any This

circumstance has the following surprising implication. Say we initialize a state at and find that the resulting

orbit goes to Now say that we

attempt to repeat this experiment. If

there is any error in our resetting of

the initial condition, we cannot be

sure that the orbit will go to

(rather than ), and this is the caseno matter how small our error is.

Put another way, ev en though the

basin has positiv e Lebesgue

measure (non-zero v olume), the set

and its boundary set are the same.

Thus the existence of riddled basins

calls into question the repeatability

of expe riments in such situations.

Figure 4 illustrates the situation we

have been discussing. As shown in

Figure 4, the attracto r with a riddled basin lies on a smooth invariant surface (or manifold) and this is

general for attractors with riddled basins. Ty pical syste ms do not admit smooth invariant manifolds, and this

is why riddled basins (fortunately?) do not occ ur in generic cases. Examples, where a dy namical system has a

smooth invariant surface are a system with reflection symmetry of some coordinate about in which

case would be an invariant manifold, and a predator-prey model in population dy namics, in which

case one of the populations being zero (ex tinction) is an invariant manifold of the model.

References

Alex ander, J., Y orke, J.A., Y ou, Z., and Kan, I . Riddled Basins (1 992) Int. J. Bif. Chao s 2:7 95 .

Grebogi, C., Ott, E. and Yo rke, J.A. (1987 ) Basin Boundary Metamorphoses: Changes in Ac cessible

Boundary Orbits, Physica D 24:243.

McDonald, S.W., Grebogi, C., Ott, E., and Yorke, J.A . (1985 ) Fractal Basin Boundaries, Physica D 17 :125 .

Milnor, J. (1985) On the Concept of an Attractor, Comm. Math. Phys. 99:17 7 .

Ott, E., Chapter 5 in Chaos in Dy namical Sy stems, Cambridge Univ ersity Press, seco nd edition 200 3.

Ott, E., Sommerer, J.C., Alex ander, J.C., Kan, I., and Y orke, J.A . (1994) The Transition to Chaotic

Attractors with Riddled Basins, Phy sic a D 7 6:384.

Internal references

John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.

Peter Ashwin (2006) Bubbling transition. Scholarpedia, 1(8):17 25.

Edward Ott (2006) Controlling chaos. Scholarpedia, 1(8):1699.

Edward Ott (2006) Crises. Scholarpedia, 1(10 ):17 00.

Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (200 6) Periodic orbit. Scholarpedia, 1 (7 ):13 58.

Frank Hoppensteadt (2006) Predator-prey model. Scholarpedia, 1(10):1563.

Philip Holmes and Eric T. Shea-Brown (20 06) Stability. Scholarpedia, 1 (10):1838.

External Links

Author's webpage (http:// www.ee.umd.edu/faculty/ott.html)

(p

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p

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See Also

Attractor Dimensio n, Bubbling Transition, Chaos, Crises, Controlling Chaos, Dynamical Systems, Inv ariant

Manifolds, Periodic Orbit, Stability, Transient Chaos, Unstable Periodic Orbits

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