A
In response to Genes comment:
>If I start with an inital state for a stable system it should
>unfold in an orderly fashion to a predictable future
>state...unless the system is operating in the realm of chaos...
Im not sure if Im understanding Genes comment correctly, but if I am, Im
taking it to mean that he is differentiating between a stable system and a
chaotic system.
A model of resistance to change that I developed showed some very interesting
behavior in that it went from stable (fixed) to stable (periodic) to chaotic,
depending upon the amount of "stress" the system was put under.
My interpretation is that any stable system can become chaotic if subjected
to certain stresses.
I point this out so that we understand that assuming a system that is
currently stable differs from one that is currently chaotic may not always be
correct. The key lesson -- the same system will demonstrate radically
different behavior when stressed.
Peter von Stackelberg
Applied Futures, Inc.
1733 Woodstead Court, Suite 101
The Woodlands, TX
77380
Stable vs. Chaotic Systems
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PeterVS1@aol.com
- Junior Member
- Posts: 4
- Joined: Fri Mar 29, 2002 3:39 am
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jsterman@MIT.EDU (John Sterman)
- Senior Member
- Posts: 54
- Joined: Fri Mar 29, 2002 3:39 am
Stable vs. Chaotic Systems
Regarding the discussion of conditions in which chaos can arise:
No linear system, no matter what order (how many levels), can exhibit
chaos.
In continuous time systems, a third order (or greater) nonlinear system
is necessary (but not sufficient) for chaos to arise. Note that while
this is a necessary condition, other conditions must be met for chaos to
arise. Stated heuristically, not formally, these additional conditions
are roughly that the system must be globally stable (that is, its
trajectories must remain bounded within a finite region of phase space)
and it must have at least one local instability within that region. The
local instability can arise from positive feedbacks or from delays in a
negative feedback loop. Typically, any system that produces limit
cycles can be made to go chaotic by adjusting a suitable system
parameter.
In discrete time systems, like the logistic map that Peter Fleissner
described, there are two cases:
If the discrete map is nonlinear but invertible, then chaos requires a
system of order >= 2.
If the discrete map is noninvertible, then chaos can arise even in the
first order case. The logistic map is noninvertible, so it can exhibit
chaos even though it is first order (but discrete time).
One way to think of the relationship between continuous and discrete
time systems is that any discrete time system can be converted to an
equivalent continuous time system by introducing an infinite order
(pipeline) delay with length one time period in the net flow into each
state equation. Since the continuous time system corresponding to the
discrete map then has infinite order, it can exhibit chaos if the other
conditions are met.
John Sterman
jsterman@mit.edu
No linear system, no matter what order (how many levels), can exhibit
chaos.
In continuous time systems, a third order (or greater) nonlinear system
is necessary (but not sufficient) for chaos to arise. Note that while
this is a necessary condition, other conditions must be met for chaos to
arise. Stated heuristically, not formally, these additional conditions
are roughly that the system must be globally stable (that is, its
trajectories must remain bounded within a finite region of phase space)
and it must have at least one local instability within that region. The
local instability can arise from positive feedbacks or from delays in a
negative feedback loop. Typically, any system that produces limit
cycles can be made to go chaotic by adjusting a suitable system
parameter.
In discrete time systems, like the logistic map that Peter Fleissner
described, there are two cases:
If the discrete map is nonlinear but invertible, then chaos requires a
system of order >= 2.
If the discrete map is noninvertible, then chaos can arise even in the
first order case. The logistic map is noninvertible, so it can exhibit
chaos even though it is first order (but discrete time).
One way to think of the relationship between continuous and discrete
time systems is that any discrete time system can be converted to an
equivalent continuous time system by introducing an infinite order
(pipeline) delay with length one time period in the net flow into each
state equation. Since the continuous time system corresponding to the
discrete map then has infinite order, it can exhibit chaos if the other
conditions are met.
John Sterman
jsterman@mit.edu
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George Richardson
- Junior Member
- Posts: 3
- Joined: Fri Mar 29, 2002 3:39 am
Stable vs. Chaotic Systems
"...A model of resistance to change that I developed showed some very
interesting behavior in that it went from stable (fixed) to stable (periodic)
to chaotic, depending upon the amount of "stress" the system was put under.
My interpretation is that any stable system can become chaotic if subjected
to certain stresses..."
Well, a first-order system showing exponential growth wouldnt turn chaotic
unless one "stressed" it with more levels, giving it the chance to oscillate.
So I think the conclusion is a bit too bold, even though it is undoubtedly
insightful in the instance in which in arose.
...GPR
-----------------------------------------------------------------------------
George P. Richardson G.P.Richardson@Albany.edu
Rockefeller College of Public Affairs and Policy GR383@albnyvms.bitnet
University at Albany - State University of New York Phone: 518-442-3859
Albany, NY 12222 Fax: 518-442-3398
-----------------------------------------------------------------------------
interesting behavior in that it went from stable (fixed) to stable (periodic)
to chaotic, depending upon the amount of "stress" the system was put under.
My interpretation is that any stable system can become chaotic if subjected
to certain stresses..."
Well, a first-order system showing exponential growth wouldnt turn chaotic
unless one "stressed" it with more levels, giving it the chance to oscillate.
So I think the conclusion is a bit too bold, even though it is undoubtedly
insightful in the instance in which in arose.
...GPR
-----------------------------------------------------------------------------
George P. Richardson G.P.Richardson@Albany.edu
Rockefeller College of Public Affairs and Policy GR383@albnyvms.bitnet
University at Albany - State University of New York Phone: 518-442-3859
Albany, NY 12222 Fax: 518-442-3398
-----------------------------------------------------------------------------
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peter@iguwnext.tuwien.ac.at (Pet
- Newbie
- Posts: 1
- Joined: Fri Mar 29, 2002 3:39 am
Stable vs. Chaotic Systems
(Moderators note - I have added in some commentary at the end - Bob Eberlein)
I agree to George P. Richardsons comment on the impossibility to produce
chaos by first order LINEAR systems, but I disagree to this statement in
general. Here is a counter-example:
Try the Verhulst-equation (it is a discrete version of the differential
equation
dx/dt = a.x.(b-x)
generating the logistic function):
x(t) = a . x(t-1) . (1 - x(t-1))
It is able to produce nice chaos for some values of the parameter a.
Chaos comes up if the difference equation (or the system of difference
equations) is bounded and thus nonlinear, and some particular solution is
unstable (meaning that the initial error compared with the original
starting point will be amplified, or, what means the same, the
Liapunov-multiplier is greater than 1).
The property of the system to be unstable or chaotic w. r. t. a certain
trajectory (or particular solution) cannot change if the equation is not
changed, or, what is the same, it cannot change if the parameter values are
not changed.
Thus, in my opinion, exerting "stress" to any equation must mean that there
is some change in the values of the parameters or in the structure of the
model equations, otherwise the above statement cannot be true.
Looking forward to any comments!
Peter Fleissner
Department for Design and Assessment of Technology/Social Cybernetics
Vienna University of Technology
Moellwaldplatz 5
A-1040 Vienna
Austria
Tel: *431-504-11-86-11
FAX: *431-504-11-88
E-mail: peter@iguwnext.tuwien.ac.at (Peter Fleissner)
WWW-URL: http://iguwnext.tuwien.ac.at/
-------------------------------------------------------
From: Bob Eberlein
Normally in system dynamics the concentration is on continuous time systems
and their behavior. For example it is well known that first order negative
feedback loop in a difference equation can cause oscillation. For a
differential equation, no matter how nonliner, this is not true. A first
order negative loop is always goal seeking.
Thus chaos and instability that results from the finiteness of computation -
the solution technique used to solve a differential equation numerically is
not of that much interest. This is not a statement about reality. The people
who have worked most seriously on Choas in system dynamics are interested in
systems for which the exact solution to the differential equation is chaotic.
This is tougher to prove, but more of a statement about reality.
Bob Eberlein
vensim@world.std.com
I agree to George P. Richardsons comment on the impossibility to produce
chaos by first order LINEAR systems, but I disagree to this statement in
general. Here is a counter-example:
Try the Verhulst-equation (it is a discrete version of the differential
equation
dx/dt = a.x.(b-x)
generating the logistic function):
x(t) = a . x(t-1) . (1 - x(t-1))
It is able to produce nice chaos for some values of the parameter a.
Chaos comes up if the difference equation (or the system of difference
equations) is bounded and thus nonlinear, and some particular solution is
unstable (meaning that the initial error compared with the original
starting point will be amplified, or, what means the same, the
Liapunov-multiplier is greater than 1).
The property of the system to be unstable or chaotic w. r. t. a certain
trajectory (or particular solution) cannot change if the equation is not
changed, or, what is the same, it cannot change if the parameter values are
not changed.
Thus, in my opinion, exerting "stress" to any equation must mean that there
is some change in the values of the parameters or in the structure of the
model equations, otherwise the above statement cannot be true.
Looking forward to any comments!
Peter Fleissner
Department for Design and Assessment of Technology/Social Cybernetics
Vienna University of Technology
Moellwaldplatz 5
A-1040 Vienna
Austria
Tel: *431-504-11-86-11
FAX: *431-504-11-88
E-mail: peter@iguwnext.tuwien.ac.at (Peter Fleissner)
WWW-URL: http://iguwnext.tuwien.ac.at/
-------------------------------------------------------
From: Bob Eberlein
Normally in system dynamics the concentration is on continuous time systems
and their behavior. For example it is well known that first order negative
feedback loop in a difference equation can cause oscillation. For a
differential equation, no matter how nonliner, this is not true. A first
order negative loop is always goal seeking.
Thus chaos and instability that results from the finiteness of computation -
the solution technique used to solve a differential equation numerically is
not of that much interest. This is not a statement about reality. The people
who have worked most seriously on Choas in system dynamics are interested in
systems for which the exact solution to the differential equation is chaotic.
This is tougher to prove, but more of a statement about reality.
Bob Eberlein
vensim@world.std.com