the Edge of Stability

[Mike Oliker]

[I sent this twice: with and without the attached sample of fractal artwork.
Sorry if it gets printed twice.]

 

After Jonathan Wolfe's presentation to the Albuquerque Complexity Group
(Friam-south?) I got to thinking about the Edge of Chaos. (See Jonathan's
work at the Fractal Foundation: http://www.fractalfoundation.org/  or see
the attachment.  Free Mandelbrot software available for your own creations
there as well).

 

I've heard of the edge of chaos as a hypothesis: that natural systems are
drawn to the edge of chaos, that it is a point of exceptional fecundity of
behavior, and I've always seen the Mandelbrot Set as a great visualization
of it.

 

Looking at the Mandelbrot set as Jonathan showed us, from an artist's
perspective, how to navigate and sculpt the resulting artwork, I realized
that it's boundary, where all this infinity of beauty arises, is actually
the Edge of Stability.  Chaos (aperiodic orbits) exists within the stable
region in tiny pockets within each "buddha replica" (small echoes of the
whole set).  So the edge of the Mandelbrot set is a dense web of
periodicities and behaviors combining chaos and regularity in every part,
all teetering on the edge of instability.  It's not the chaos, it's the
instability which kills ya'.  In fact, for some purposes, it's chaos that is
the most stable.

 

The Edge of Stability seems like a better model for evolution -- a small DNA
perturbation jumps one out a distance, often into instability (and death)
but occasionally onto the filaments of stability which arch out from the
edge of the large stable regions.  It also highlights why, if one is living
on the edge of stability, one would like a map as to where the stable
regions are.

 

That may seem like a small distinction, but it is very clarifying for me.
Does it do anything for you?

-Mike Oliker

(PS: ACG meets 2nd and 4th Tuesdays of each month, June 14 is our next
meeting, email me for details or to get on our mailing list)

 

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[Jochen Fromm]

Message"Edge of chaos" is a good catchword. Is the buzzword well-defined or
ill-defined?
Do we mean (a) The region with highest complexity between order (no change
or
periodic change, rigid or fixed structures, too static) and chaos (constant,
aperiodic change, no rigid or fixed structures, too noisy)? Or simply
(b) a critical point with high correlation length and strong fluctuations?
http://cscs.umich.edu/~crshalizi/notebooks/edge-of-chaos.html

Wikipedia and many other websites say the term was coined by
Christopher Langton in 1990, and refers originally to an area
in the range of lambda, which was varied while examining the
behavior of a cellular automata (CA). As lambda varied, the behavior
of the CA went through a phase transition of behaviors.
http://en.wikipedia.org/wiki/Edge_of_chaos

According to (*), for a given CA rule table, lambda is computed as follows.
For a CA, one state q is chosen arbitrarily to be quiescent. The lambda
value of a given CA rule is then the fraction of non-quiescent output
states in the rule table.

For a two-state CA, lambda=1/2 is a critical point or "phase transition"
between two phases, one phase near lambda=0 where are cells have value 0,
and one phase near lambda=1 where are cells have value 1. This definition
of the "edge of chaos" would be identical or similar to a critical point.

If you consider consider only lambda in the range 0 <= lambda <= 1/2,
because every CA with lambda > 1/2 corresponds to a CA with
lambda < 1/2 after interchanging the roles of black and white cells,
you can observe another "edge of chaos". As lambda increases, CA
roughly go through the four basic Wolfram classes in this order:
I (fixed), II (periodic), IV (complex), III (chaotic).  Class IV can be
seen as the edge of chaos at the boundary between class II (order)
and III (chaos).
http://classes.yale.edu/Fractals/CA/CAPatterns/Langton/Langton.html

If the four Wolfram classes are ill-defined, as Eppstein argues (see
http://www.ics.uci.edu/~eppstein/ca/wolfram.html ), is this definition still
valid? Which definition of "edge of chaos" is right and useful? Both?
In which case do systems "evolve to" the edge of chaos?

-J.


(*) Revisiting the Edge of Chaos
http://www.cs.pdx.edu/~mm/RevEdge.html

[Russell Standish]

I would echo Jochen's post somewhat. Edge of chaos is literally true
in Langton's CAs, which do have a chaotic phase. However, in more
general evolutionary systems, there appears to be a surface of
criticality, on one side of which lies stability (points or limit
cycles), and on the other instability, or completely random
behaviour. There is no evidence of deterministic chaos in this
unstable regime, however, so "edge of chaos" does not seem to be the
correct term (although it is often used). My own preferred term is
"self-organised criticality", although for some people this has its
own baggage.

Cheers

On Wed, Jun 15, 2005 at 08:18:30AM +0200, Jochen Fromm wrote:
> Message"Edge of chaos" is a good catchword. Is the buzzword well-defined or
> ill-defined?
> Do we mean (a) The region with highest complexity between order (no change
> or
> periodic change, rigid or fixed structures, too static) and chaos (constant,
> aperiodic change, no rigid or fixed structures, too noisy)? Or simply
> (b) a critical point with high correlation length and strong fluctuations?
> http://cscs.umich.edu/~crshalizi/notebooks/edge-of-chaos.html
>

--
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is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.

----------------------------------------------------------------------------
A/Prof Russell Standish                  Phone [phone] (mobile)
Mathematics                               0425 253119 (")
UNSW SYDNEY 2052                 [hidden email]            
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[Jochen Fromm-2]

Interesting. Can you define "Edge of Chaos" in terms of chaos theory? I
think I have seen a definition of edge of chaos somewhere as the point right
before the onset of complete chaos in bifurcation diagrams.

-J.

-----Urspr?ngliche Nachricht-----
Von: [hidden email] [mailto:[hidden email]] Im Auftrag
von Russell Standish
Gesendet: Mittwoch, 15. Juni 2005 12:05
An: The Friday Morning Applied Complexity Coffee Group
Betreff: Re: [FRIAM] the Edge of Chaos

I would echo Jochen's post somewhat. Edge of chaos is literally true in
Langton's CAs, which do have a chaotic phase. However, in more general
evolutionary systems, there appears to be a surface of criticality, on one
side of which lies stability (points or limit cycles), and on the other
instability, or completely random behaviour. There is no evidence of
deterministic chaos in this unstable regime, however, so "edge of chaos"
does not seem to be the correct term (although it is often used). My own
preferred term is "self-organised criticality", although for some people
this has its own baggage.

[Russell Standish]

On Wed, Jun 15, 2005 at 12:30:14PM +0200, Jochen Fromm wrote:
>
> Interesting. Can you define "Edge of Chaos" in terms of chaos theory? I
> think I have seen a definition of edge of chaos somewhere as the point right
> before the onset of complete chaos in bifurcation diagrams.
>
> -J.
>

Could be. In Langton's work (and also follow on work by Wuensche)
there is a parameter that moves the CA from point attractors, through
limit cycles to full blown chaos. In period doubling of say the
logistic map, the same happens as the parameter \lambda is varied. I
haven't heard anyone using the term to describe period doubling
before, though.

Cheers

--
*PS: A number of people ask me about the attachment to my email, which
is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.

----------------------------------------------------------------------------
A/Prof Russell Standish                  Phone [phone] (mobile)
Mathematics                               0425 253119 (")
UNSW SYDNEY 2052                 [hidden email]            
Australia                                http://parallel.hpc.unsw.edu.au/rks
            International prefix  +612, Interstate prefix 02
----------------------------------------------------------------------------

[Jochen Fromm-2]

 
This is a good analogy, but only a rough one. In the logistic map, (see
http://mathworld.wolfram.com/LogisticMap.html) the edge of chaos would then
be equal
to the accumulation point r=3.569945672, where periodicity and
period-doubling give way to chaos. In Langton's work, the parameter where
the CA behavior moves periodic
behavior to full blown chaos is not well defined.

The concept of "evolving to the edge of chaos" seems to be most suitable for
self-organized criticality in the sense of Per Bak. A sand-pile for example
evolves automatically to a state with a critical slope, where avalanches of
all sizes are possible.

The evolution "at the edge of chaos" seems to produce naturally the most
complex structures in complex _adaptive_ systems, because the complexity in
the environment reaches its peak here.

Perhaps you can define "edge of chaos" in general systems as the critical
region between order and chaos with unpredictable fluctuations due to a
delicate balance
between two opposite complementary forces, for example one expanding and one
contracting force. If the contracting or damping force prevails, then you
get stability, if the other expanding or amplifying force dominates, you get
instability or completely random behaviour.

In phase transitions between liquid and gaseous forms for example, the
expanding force would be thermal movement, and the contracting force would
be the attractive intramolecular force between the atoms.

-J.


-----Urspr?ngliche Nachricht-----
Von: [hidden email] [mailto:[hidden email]] Im Auftrag
von Russell Standish
Gesendet: Mittwoch, 15. Juni 2005 14:09
An: The Friday Morning Applied Complexity Coffee Group
Betreff: Re: [FRIAM] the Edge of Chaos

In Langton's work (and also follow on work by Wuensche) there is a parameter
that moves the CA from point attractors, through limit cycles to full blown
chaos. In period doubling of say the logistic map, the same happens as the
parameter \lambda is varied. I haven't heard anyone using the term to
describe period doubling before, though.

[Bruce Sawhill]

"edge of chaos" is defined in terms of correlation length in a system.  
It increases with lambda up to a point, then decreases.  The associated
complexity can be measured by the number of states in a finite state
machine required to produce the output of the system (Crutchfield).  
This gets harder to do for systems more complex than CAs, though.

Bruce


On Jun 15, 2005, at 6:30 AM, Jochen Fromm wrote:

>
> Interesting. Can you define "Edge of Chaos" in terms of chaos theory? I
> think I have seen a definition of edge of chaos somewhere as the point
> right
> before the onset of complete chaos in bifurcation diagrams.
>
> -J.
>
> -----Urspr?ngliche Nachricht-----
> Von: [hidden email] [mailto:[hidden email]] Im
> Auftrag
> von Russell Standish
> Gesendet: Mittwoch, 15. Juni 2005 12:05
> An: The Friday Morning Applied Complexity Coffee Group
> Betreff: Re: [FRIAM] the Edge of Chaos
>
> I would echo Jochen's post somewhat. Edge of chaos is literally true in
> Langton's CAs, which do have a chaotic phase. However, in more general
> evolutionary systems, there appears to be a surface of criticality, on
> one
> side of which lies stability (points or limit cycles), and on the other
> instability, or completely random behaviour. There is no evidence of
> deterministic chaos in this unstable regime, however, so "edge of
> chaos"
> does not seem to be the correct term (although it is often used). My
> own
> preferred term is "self-organised criticality", although for some
> people
> this has its own baggage.
>
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9:30a-11:30 at ad hoc locations
> Lecture schedule, archives, unsubscribe, etc.:
> http://www.friam.org
>

[Jochen Fromm-2]

"Edge of chaos" is a nice metaphorical phrase with some poetic
appeal. Like system and self-organization it is an ambiguous and
general word which many meanings. I think it is important
to come to a clear and precise definition if you want to use
and apply the concept in a scientific context.

Bruce, you said on the NECSI Mailing list in 1999:
"'Self-organized criticality' and 'edge of chaos' are very
different terms. Self organized criticality refers to a very
specific class of open (material or energy flows through) systems
like sandpile models, which 'self-organize' around a characteristic
configuration (a sandpile fluctuates around the angle of repose).
'Edge of chaos' is more descriptive and less precise, and
does not refer to any particular set of models. I
http://necsi.org:8100/Lists/complex-science/Message/333.html

Whereas Per Bak said on the same Mailing list:
"Indeed, the "edge of chaos" is the same as the
"selforganized critical state". Actually, in Packards
original article on the edge of chaos from 1988, he does
properly refer to the work on SOC from 1987"
http://necsi.org:8100/Lists/complex-science/Message/340.html

This is confusing, one says they are different terms,
the other says they are equal terms which say the same.
Tam?s Vicsek  says in his Nature article about complexity (*):
"If a concept is not well defined, it can be abused. This is
particularly true of complexity, an inherently interdisciplinary
concept that has penetrated a range of intellectual fields from
physics to linguistics, but with no underlying, unified theory."

Vicsek himself offers no precise definition. He goes on and uses
the term "edge of chaos" in a very vague and general way:
"Such systems [he mentioned before turbulent flows and the brain]
exist on the edge of chaos ? they may exhibit almost
regular behaviour, but also can change dramatically and
stochastically in time and/or space as a result of small
changes in conditions."

His use of "edge of chaos" sounds more like "self-organized
criticality": a system at a critical state or phase transition
shows fluctuations and correlations on all scales.
The point is that you do not always have a sharp edge between
order and chaos in phase transitions or critical states.
It is unclear if there is a "fine line" between order and
chaos in general.

Perhaps you can distinguishes between two forms of
"Edge of Chaos"

(1) Phase Transition or Critical Point with strong
    fluctuations and scale-free correlations. Systems
    can evolve to this point, they are "evolving to the
    edge of chaos" (see self-organized criticality)
    This point can often be measured precisely
    (for example lambda=1/2 for Langton's parameter,
    or a critical temperature like 100 Degrees).

(2) General Point of highest complexity between order and
    chaos. The evolution at this point, "at the edge of chaos"
    produces naturally the most complex structures
    in _adaptive_ systems, because the complexity in
    the environment reaches its peak here.
    This point can often not be measured precisely
    (for example lambda somewhere between 0 and 1/2,
     except the transition or accumulation point on
     the period doubling route to chaos.


What both have in common is the Power-law behavior:
critical exponents and scale-free correlations in (1)
and scale-free distributions of events in (2), and
a delicate balance between two opposite complementary forces,
for example one organizing and regulatory force, and
one random or chaotic force, or one expanding and one
contracting force, or one attracting and one repulsive force.

Remeber we get complex small-world networks if we add
randomness to order in regular networks, and complex
scale-free networks if we add order to randomness
(through preferential attachment) in random networks.

-J.

(*) The bigger picture
Tam?s Vicsek
Nature Vol. 418 (2002) p.131
http://angel.elte.hu/~vicsek/images/complex.pdf