The order of a system

[Niall Palfreyman]

George Richardson schrieb:

> The simplest definition, which I think helps most of us from the most
> lax to the most picky, is that the order of a system is the number of
> INDEPENDENT stocks in the system.
> ...
> A student of system dynamics does not need to know
> anything about differential equations to understand fully the concept
> of "order."

One of the interesting things Im discovering in working on this course
is that by using the mechanical/hydraulic metaphor of SD in teaching
mathematics, Im often forced to think more carefully about the
pedagogical effect of definitions than I otherwise would. The reason is
that the metaphor is so appealing that the students understand the
material far earlier, and so can notice loopholes which I then have to
fill. Ive always believed hand-waving can be used to good effect in
teaching - to relieve students of the necessity to think too early about
complicated things which will be explained later. However now the
hand-waving is spotted, which means Im forced to find ways of
justifying it, which ruins the whole point of using hand-waving.

To a beginning student, the system of differential equations {x=3t;
y=-x} is not obviously a first-order system. However the system which
consists of a single flow from stock x to stock y immediately invites
the question: Isnt there really only one variable in the system? The
result is that topics which were until now firmly positioned at the end
of the course are now already intimated at the beginning, and Im having
to rethink the entire structure of what needs to be known when.

Niall Palfreyman.
n:Palfreyman;Niall
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fn:Prof. Dr. Niall Palfreyman

[Niall Palfreyman]

Hi,

Im just in the process of rewriting my first-year university maths
course, and Ive decided to base it entirely on SD. In the first chapter
Ive introduced the students to general dynamical systems via a simple
exponential decay problem, and now in the second chapter I want to look
at a number of examples of common functions with which the students
should be familiar - linear, parabola, sin, x^2 + y^2 = R^2, exp, log.
In thinking about this I came up against some problems with my thinking
about the order of a system. For example:

1. These systems are all of the form x = x(t). They contain no
derivatives, and so I would call them zero-order systems, Nth-order
systems being systems with only Nth-order derivatives. Yet while "x =
A.exp(t)" is in this terminology a zero-order system, its corresponding
differential equation is "dx/dt = x", which is a first-order equation.
So the system itself does not have an order, but rather the mathematical
description. Or am I mixing things up somewhere?

2. Im looking for a simple definition of the order of a system for
someone who does not yet (necessarily) know what a derivative is. At
this stage the students know about the idea of a state variable (stock)
and a flow variable, which they know is also sometimes called a
derivative. My provisional definition of the order of a system is the
following: "The order of a dynamical system is the maximum number of
flows in the system which are linked together in the following way: "
And then I give graphical examples of stocks linked via information
flows to material flows. Im not at all happy with this definition - can
anyone come up with something better?

Thanks,
Niall.
n:Palfreyman;Niall
org:Weihenstephan University of Applied Sciences;Bioinformatics
adr:;;Am Hofgarten 10;Freising;;85350;Germany
email;internet:Niall.Palfreyman@fh-weihenstephan.de
title:Deacon of Studies
fn:Prof. Dr. Niall Palfreyman

["Guenther Ossimitz"]

Hello Nial,

You are right: the order is not a property of the
system, but of its description. Any nth-order
differential equation can be re-written as a system
of n first-order differential equations by a rather elementary
re-definition of variables.

IMHO SD models are something similar to first-order
differential equations. Since any nth-order ordinary
diff-equation can be represented by a system of n
first-order-equations, any nth-oder diff-equation can
be represented (approximately) by some SD-model.

Yet I would say that there is a fundamental
difference in the structure of SD-models and
diff-equations. I encountered this difference the first time,
when I tried to make a simple physics model, which consists
of the locus x(t), the speed v(t) and the acceleration a(t) of
a point moving along a straight line over time t.

We have a(t) = v(t) and v(t) = x(t) (a system of 2 first order D.E:)
or
a(t) = x(t) (a single second-order Diff. Eq.)

If you want to model this with SD technology you have
the ugly necessity that the variable v has to appear twice:
once as a stock, with the acceleration as the associated flow;
and once as a flow, with x(t) as the associated stock.

This is necessary because in SD models stocks refer basically
to time points and flows are basically connected to time intervals.
(The sophisticated argument that a flow might exist at each moment
over the whole simulation interval does not change the fact that a
flow is a fixed number, belonging to the whole time interval of a simulation
time step.)

The big achievement of mathematical calculus is that time intervals
are so to say downsqueezed infinitesimally in such a way, that they
can handled in the same way as time points. So in calculus infinitesimally
the distinction between time point and time step disappears, which is
a necessary prerequisite for higher derivatives and thus for the notion
of the "order" of an diff. equation.

In SD the fundamental distincition between stocks (referring to time points)
and flows (referring to time intervals) does not allow direct access
to higher derivatives (higher orders). One might even say that the relation
between stock and flow is just similar, but not identical to the relation between
a function and its first derivative (but this is not the main point here).

So concluding the whole thing: If you start an SD- related math
course with the question of the order of a differential equation,
you jump right into the core of the subtle differences between
finite SD and infinitesimal Calculus. I think that the issue of
orders of Diff. Eq. is fine for the theory of Diff. Eq., but it is irrelevant
for SD, since SD only has (so to say) first order equations.

Of course one might offer during the course a bridge from SD to the
traditional theory of Diff. Eqs by saying that SD can simulate
any higher-order diff.equ. numerically by transforming it to a system
of n first oder Diff. Equs, and each of this can be represented by a
stock-flow-relationship. I just think that I would not start with this at the
very beginning of an SD-related math course.

Greetings

G. Ossimitz



---
Dr. Guenther Ossimitz
University of Klagenfurt, Univ.str. 65
A-9020 Klagenfurt, Austria
[++43]-(0)463 2700-3132
guenther.ossimitz@uni-klu.ac.at
http://go.just.to/go

[=?iso-8859-1?Q?Andr=E9_Reichel?=]

Hello,

before I start I must admit that my skills in maths are roughly about
average. Nevertheless, the questions asked by Niall made me grab some math
textbooks and the remainders of my SD courses. Here is what I came up with
(which might be completely wrong but maybe good enough to start a discussion
about the mathematics of SD, a topic I cant remember having read much
about):

The order of a differential equation depends on the degree of the
differentiation: x = dx/dt is first-order, x = d(dx/dt)/dt is
second-order. An example from physics might illustrate:

s(t) = 1/2 * a * t^2
s(t) = v(t) = ds/dt = a * t
s(t) = v(t) = d(ds/dt)/dt = a

"Velocity" is the stock in or out of which "acceleration" flows:
acceleration causes velocity to increase/decrease. "Velocity" itself can be
seen as a flow to "stretch" (or "space"): the faster our system moves, the
greater the distance from its starting point in a given time.

This may look basic and, in fact, it is. But I myself had to think about
twice before I realised the meaning of it (in SD-terms):

The order of a system depends on the number of differentiations that are
necessary to reach a constant factor. The differentiation of a constant
factor is const = 0. In the physics example it would not make any sense to
go beyond "acceleration". (unless you give a non-tautological definition of
"a") The same is in x(t) = x^2: x = 2x; x = 2. (x would equal zero)
x^2 is a parabola and what kind of system is needed to produce a
parabola-like behavior? A second-order system with one constant inflow, in
which the parabola-like behavior of the first stock generates an s-shaped
behavior in the second stock.

When we say that the order of a system depends on the number of its stocks
(its integrations/state variables) we are speeking of exactly the same
thing: the last DIFFERENTIATION that brings us to a constant factor. In
another way: every INTEGRATION can only start with a constant (non-zero)
factor.

I really dont know if my conclusions are correct, so maybe someone with a
bit more expertise in SD mathematics could help.

MfG / Kind regards
André Reichel
University of Stuttgart, Germany
A.Reichel@epost.de

["geoff coyle"]

Surely this was all laid out in Jays book 40 years ago. The order of a
system dynamics model is the number of level variables it contains,
including the implicit levels within functions such as DELAY3.

Regards,

Geoff

Professor R G Coyle,
Consultant in System Dynamics and Strategic Modelling,
Telephone +44 (0) 1793 782817, Fax ... 783188
email geoff.coyle@btinternet.com

["J. Pedro Mendes"]

Niall,

Control-theory folks often refer to "system type" to describe how a
feedback system handles disturbances: Type 1 systems handle constant
disturbances, type 2 systems handle linear disturbances, and so
forth. System type increases by adding integrators, but doing so decreases
stability margin. Both type and stability margin can be made arbitrary
when you design policies. I can dig for references if you need. And Id
love to see your text.

Pedro
From: "J. Pedro Mendes" <jpm@ip.pt>

["Hayward J (SoTech - M & S)"]

1. The order of a set of first order differential equations is normally
given as the number of variables MINUS the number of invariants, between
those variables. Take for example the Fisher-Pry diffusion model:

dx/dt = -a x y dy/dt = a x y

x = no. of potential adopters, y = no of adopters

On the face of it looks a second order system. But as x+y = N is a constant
there is one invariant. N is the total number in the fixed population. Thus
it is a first order system. It can be transformed into one first order
differential equation - the logistic equation dy/dt = a(N-y)y.

2. The order of a set of differential equations is important as it will
govern behaviour. Given there is no explicit time dependency a system must
be at least third order to have chaotic behaviour, and at least second order
to oscillate. Thus without even solving or simulating the Fisher-Pry system
you can tell there is no chaos or oscillations.

3. Translating this into system dynamics this definition of order would be
the number of stocks minus the number of conserved flows. Thus a system
whose structure is:

__ __
|__|----------O-------------> |__| (Use Courier font)

will be first order, even though there are two stocks. The flow is
conserved. Thus this system will neither oscillate or have chaos - if it is
intended to be continuous.

4. If the system is discrete - step length 1 with Euler in some software -
things are different. The mathematical equivalent of these systems are
difference equations and even first order systems can oscillate and have
chaos. The classic logistic map x(n+1) = a x(n)(1-x(n)) oscillates 3<a<4
and goes chaotic for a value of a between 3 and 4.

5. If a continuous system has any rigid boundaries, or sudden jumps (in
SD this may be done with "if" statements, graphical relationships) - then
the rule about oscillations and chaos will not hold. Thus, e.g., a pendulum
is second order - distance and velocity - but make it bounce against a wall
and the system can exhibit extreme chaotic behaviour.

John Hayward
Division of Mathematics and Statistics
University of Glamorgan
Wales UK
From: "Hayward J (SoTech - M & S)" <jhayward@glam.ac.uk>

[Niall Palfreyman]

Hi,

First of all, thank you very much for your responses on this question -
theyve helped my thinking a lot. I shall try to gather my reactions to
the various comments, and then draw some provisional conclusions for the
new course. Ill then wait for further responses before finalising the
preliminary draft of the course.

Background:
Some people asked about the course itself, so here is some background to
my query. The course is a general first-year university intro to
mathematics, including general awarenes of important functions (exp,
sin, cos, x^n, etc), differentiation, integration, complex numbers,
simultaneous algebraic equations, vectors, matrices, eigenvectors,
differential equations, phase portraits, partial derivatives, vector
analysis (up to Stokes theorem), Fourier analysis and finally a short
topic on reaction-diffusion systems - this is a course for beginning
bioinformaticists, so Im emphasising the biological links.

It struck me that all these topics have a common thread, and that is
that all are relevant to the analysis of dynamical systems. Linked with
my own interests in SD, and knowing that students often have problems
with many of these topics, I thought I would start the whole course with
a simple exponential decay model, and so provide the SD diagrams and
terminology as a basic visualisation technique which would carry through
the entire course. If it works well I shall extend the method to other
courses next year.

Im now on lecture 2 of the course, which is a general introduction to
the behaviour of various functions like x^n, sin, exp and so on. Lecture
3 will model a simple exponential population and so introduce
derivatives as a way of calculating flows, lecture 4 will deal with
integration as a way of calculating stocks, and lecture 5 will introduce
1st-order differential equations. Many students will not need much of
the information in lectures 2-4, but I need to ensure a common knowledge
base before proceeding to greater things. I also wanted to use these
"foundation-level" lectures as a way of gently sowing the seeds of ideas
like the order of a DE and of a dynamical system which are needed later
in the course. Thats when I noticed that I had a discrepancy in my
thinking. The point to bear in mind is that for pedagogical reasons my
adopted definition of order needs to be easily understandable, and
equivalent across 3 disciplines: SD models, dynamical systems, and
thinking about systems in real-life.

Helpful comments so far:
1. John Sterman, Jay Forrester and Jim Hines all advocate using the
number of stocks in a model as the order of the model, and all warn
against being sidetracked by red herrings like the number of flows or
equivalence to models with fewer stocks. I take these points.

2. Steven Strogatzs book "Nonlinear Dynamics and Chaos" uses the
definition that the order of a dynamical system is the number of phase
variables needed to describe the system as a set of first-order DEs. I
like this definition for its simplicity, but have some difficulty with
it which I shall mention below. Another advantage is that it (almost)
matches the definition (1) above.

3. The definition used by many authors, and mentioned by Guenther
Ossimitz and Andre Reichel, is that the order of a system is the order
of the highest-order DE used to model it. Guenther Ossimitz also makes
the point that this definition can only be applied to SD if we first
"canonicalise" this DE into an equivalen first-order system.

4. Jack Ring further warns that we shouldnt necessarily expect that
"order" means the same thing for both systems and the models that
describe them.

My thoughts:
I like and dislike aspects of all these suggestions. A theme running
through all of them is that the order of a system is the number of
integrations required to solve the system: the stocks integrate flows,
and so the number of stocks is the number of integrations. However a
single, free-standing stock (with no flows) is clearly not performing an
integration. So how about: the order of a model is the (minimal?) number
of stocks which possess a flow? This seems OK, but I still have
reservations.

First - and this is the problem that set my thoughts going on this
subject - what about a model which consists basically of just one
converter:
x = A.sin(kt) + B.cos(kt)
This model has no stocks whatsoever - it isnt really a dynamical system
at all, since it has no integrals, yet it behaves in exactly the same
way as the 2nd-order spring system
x = -k^2.x - (S1)
So is this a zero-order model, or a 2nd-order model, of the system? My
feeling is that this reinforces Jack Rings idea that we shouldnt
expect the order of the model to be the same as the order of the system.
Or are we saying that converters containing sin functions are not
legitimate models?

My second problem with suggestions 1 and 2 above is that Im a bit
uncomfortable calling the following decoupled system a 2nd-order system:
x = ax,
y = by - (S2)
Here x and y have nothing to do with each other - they are just two
first-order systems which happen to have been lumped together. Their
behaviour can never be as interesting as "true" second-order systems.
This is why I tentatively included the coupling of DEs as a criterion
for the order of a system in my original mail.

Request:
I think Ill probably in the end go with the definition that the order
of a system or of a model is the number of stocks connected to a flow.
But I am still unclear on the status of the two problem-cases (S1) and
(S2) given above, and Id welcome any light-shedding anyone can offer.
My tentative solution is that (S1) is a zero-order model of a 2nd-order
system, and that (S2) is simply a decoupled (ie, particularly boring)
2nd-order system.

Thanks again,
Niall Palfreyman.
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title:Deacon of Studies
fn:Prof. Dr. Niall Palfreyman

[=?iso-8859-1?Q?Andr=E9_Reichel?=]

Hello,

Niall wrote: "x = A.sin(kt) + B.cos(kt)
This model has no stocks whatsoever - it isnt really a dynamical system
at all, since it has no integrals, yet it behaves in exactly the same
way as the 2nd-order spring system
x = -k^2.x - (S1)
So is this a zero-order model, or a 2nd-order model, of the system?"

I think the described model (x = A.sin(kt) + B.cos(kt)) is in fact of no
order. It is an algebraically expression of the latter. Any stock-flow-model
can be described as a set of first-order differential equations, as some
authors mentioned. It is possible to give analytical solutions to any simple
SD-models.

The sin-function, for example, can be modeled with two stocks and
appropriate delays between them. The function y = tanh x (s-shaped growth)
can also be modeled within a second-order system. The same system structure
can produce different behaviors. That is a strength of SD, though it might
not be necessary to simulate a system which behavior and parameters can be
precisely estimated. In such a case (precise estimation and, therefore, low
uncertainty = low dynamics) an analytical approach is sufficient.

To illustrate that you can think of the e-function y = e^x. Exponential
growth could also be simulated, for example, in a third-order system.
(constant inflow -> linear inflow
-> geometrical inflow = exponential behavior) In this most simple case with
no dynamics or delays, it would be very inefficient to rely on unprecise
simulation outputs gained through numerical integration. Instead you would
use y = e^x and always have the exact values for any given x.

To summarize my thoughts:
The expression x = A.sin(kt) + B.cos(kt) produces the same behavior as the
expression x = -k^2.x -- but without the possibility to research any
dynamic occurences. If you are dealing with complex and changing systems,
the former expression will not generate any insights, for the insights of it
are "frozen" within the sin and cos functions. The quest for an analytical
expression, like y = e^x, in the dynamic case is futile.

I hope that my comments were helpful, though the explanation became a bit
lengthy. The problem in dealing with SD mathematics, as I mentioned some
time ago, is that there seems to be not much literature dealing with the
topic. Maybe this is a general lack within the entire field of SD. So if
anyone knows some good books, articles, web resources etc. I would be
extremely delighted about a posting or eMail.

MfG / Kind regards
André Reichel
University of Stuttgart, Germany
A.Reichel@epost.de

[Niall Palfreyman]

Hi,

Thanks once again for all the feedback. Ive now come round to what for
me feels like a good (although possibly provisional) resolution of my
problems with this topic. The two "Aha!" comments for me were:

> The order of a set of first order differential equations is normally
> given as the number of variables MINUS the number of invariants,
> between those variables. (John Hayward)

Yes!! I knew flows had to come in somewhere. So:

Definition: The order of a system is the number of stocks that are
connected to flows, _minus_ the number of stock pairs which are
connected by conserved flows.

I guess I do still have one issue remaining: What about a system of
three stocks A, B and C, with a conserved flow from A->B, B->C and C->A?
This is either a zero-order system or a first-order system, depending on
the matching of the flows (if theyre all equal, all the stocks are
constant). The above counting method doesnt cope with this. Can anyone
think of a better (simpler would be good, too) counting method, which
copes better with such systems containing invariants?

> The question of the "order" of a behavior is not meaningful, ...
> Indeed, the equation "x = Asin(kt) + Bcos(kt)" isnt a model
> at all ... (John Sterman)

I agree with this first statement, and not with the second, and it was
this conflict which brought me great insight with regard to how to
introduce models in a way which could preceed and support the
introduction of analysis. As I see it, "order" is a term which applies
to a systems structure, rather than to the system itself. And the
systems structure is in the end a causal explanation of the systems
behaviour. Therefore, as John says, the question of the "order" of
behaviour is not meaningful - it is the _explanation_ of that behaviour
which has an "order". The order of a system is the level of causal
explanation provided by the systems structure for its behaviour.

Using this way of thinking, though, the equation "x = Asin(kt) +
Bcos(kt)" is indeed a model. Its just that it is a non-explanatory
model: it describes, but doesnt explain. It is a zero-order system
which provides a zero-level causal explanation of the systems
behaviour. The second-order spring equation, on the other hand, provides
a second-level causal explanation of the same behaviour. In this
terminology, the process of "solving a differential system" is the
process of reducing the explanatory level of a model to the point where
it becomes a pure (non-explanatory) description.

The vague wondering still left in my mind relates to the question: If we
can choose the structural order used to describe a system, how do we
know which is the "canonical" order which we all regard as the
base-level explanation? I assume this relates to Pedro Mendes and Andre
Reichels issue about aiming for constants. I guess ultimately we would
like all the converters in our models to be linear functions (or even
constants?), and this would then provide a canonical level of
explanation. In this regard Im intrigued by Pedro Mendes comment which
was new to me:

> Control-theory folks often refer to "system type" to describe how a
> feedback system handles disturbances: Type 1 systems handle
> constant disturbances, type 2 systems handle linear disturbances,
> and so forth.

Best wishes,
Niall.
n:Palfreyman;Niall
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title:Deacon of Studies
fn:Prof. Dr. Niall Palfreyman

[John Sterman]

Niall Palfreyman asks about the order of a dynamical system, with an
example of the order of the system dx/dt = ax vs its solution x(t) =
x(0)exp(at).

1. The differential equation dx/dt = ax is a description of the
structure of a system, in this case a first-order linear positive
feedback system (for a>0). The behavior of the system (the solution
to the differential equation) is given by x(t) = x(0)exp(at). It is
not meaningful to speak of the "order" of the latter equation as it
is a characterization of the behavior of the underlying structure.
The order of a system describes the number of state variables in the
system and is only meaningful with respect to the structure, not the
behavior.

2. The order of a system or loop is best defined as the number of
state variables (stocks) it contains. The concept of a derivative is
not needed to define or count the order of a system. One needs to
know only how many stocks it contains. In the example above, there
is only one stock (x), so it is a first order system. Any continuous
time in nth order dynamical system (including system dynamics models
in continuous time) can be represented equivalently as a set of n
coupled first-order integral equations or as a set of n coupled
first-order differential equations. And any set of n coupled
first-order differential equations can easily be converted to a
equivalent single nth-order differential equation.

I have found it much easier to explain structure and the concept of
the order of a system or loop as the number of stocks in the system
or loop; it is not necessary to invoke the concept of derivatives at
all.

John Sterman
From: John Sterman <jsterman@MIT.EDU>

["Jay W. Forrester"]

I believe that the "order" of a system is determined by the number of
integrations (stocks or levels). An exponential decay or growth
system with one stock is a first order system. The system with two
stocks in a single loop, which generates a sine wave, is a second
order system. The order is not necessarily related to the number of
flows although sometimes that may be the same as the number of
stocks. The system in my "World Dynamics" book has five stocks and
is a fifth-order system, even though there are many more flows.

--
---------------------------------------------------------
Jay W. Forrester
Professor of Management
Sloan School
Massachusetts Institute of Technology
Room E60-389
Cambridge, MA 02139
From: "Jay W. Forrester" <jforestr@MIT.EDU>

["Jim Hines"]

Most folks simply say that the order of a system is the number of stocks in
it. I think this is really fine, although its true that sometimes you can
find an equivalent system with fewer stocks.

Jim
From: "Jim Hines" <jhines@MIT.EDU>

[John Sterman]

Nialls summary of the responses on the meaning of the order of a
system is interesting. He closes with two concerns and examples:

>First - and this is the problem that set my thoughts going on this
>subject - what about a model which consists basically of just one
>converter:
>x = A.sin(kt) + B.cos(kt)
>This model has no stocks whatsoever - it isnt really a dynamical system
>at all, since it has no integrals, yet it behaves in exactly the same
>way as the 2nd-order spring system
>x = -k^2.x - (S1)
>So is this a zero-order model, or a 2nd-order model, of the system?


The key issue here is the difference between the structure of a
system and its behavior. System structure is the network of stocks,
flows, information feedback, and decision rules from which the
behavior of the system emerges. A dynamic model is a representation
of the structure of some problem issue of concern. That structure
captures assumptions the modeler makes about how the relevant parts
of the world operate, including the important resources (stocks) in
the system, the flows through which they changes, and the feedback
structure (decision rules) governing those flows. From the structure
one can infer the behavior it generates. Dynamic models provide an
endogenous explanation for the behavior of a system - that is, an
explanation for the behavior in terms of the underlying structure of
stocks, flows, feedback loops, and decision rules - and then use that
understanding to design policies (new structures) that will change
the behavior of the system for the better.

In Nialls example, the first equation is a description of the
behavior of the dynamical model given by the second equation. The
second equation (the differential equation) portrays a structure.
The structure generates the behavior. In Nialls example of a mass
on a spring, the structure has two independent stocks, and is
therefore second-order. The two stocks are the position of the mass,
x, which integrates (accumulates) the velocity of the mass, and the
momentum of the mass, which integrates the force applied to it
(equivalently, the second stock can be considered to be the velocity,
which integrates the acceleration of the mass).

The first equation describes the behavior of the system; in this
case, the trajectory followed by the mass through time for a
particular set of initial conditions and parameters. The question
of the "order" of a behavior is not meaningful, just as the question
"what is the order of the base run of the World Dynamics model" is
not meaningful. Indeed, the equation "x = Asin(kt) + Bcos(kt)" isnt
a model at all, in the same way that the base run of World Dynamics
isnt a model - it is the behavior of a model. The question "what is
the order of the World Dynamics model?", and more generally, what
assumptions about how the world works did Forrester make in the World
Dynamics model, are meaningful.

With respect to the second example, x = ax and y = by, Niall is
right that this is a (rather dull) second order system. Its dull
because the two stocks x and y are uncoupled, so the system can be
understood completely as two independent first-order systems.

Finally, Niall is correct that the Strogatz definition of system
order as the number of phase (state) variables needed to describe
the system as a set of 1st order differential equations is equivalent
to the definition that the order of a system is the number of
(independent) stocks it contains.

John Sterman
From: John Sterman <jsterman@MIT.EDU>

[Alan Graham]

sigh....

1. The equation with sines may be a model, but its not a dynamic model.

2. The decoupled system is a second-order system thats indeed boring, but
thats just for the particular policies and causal relationships choosen.
Theres even a body of literature on how to design feedback systems so that
specific control inputs control specific variables only -- deliberately
decoupled systems.

3. An earlier caveat keeps getting lost, namely that the levels/stocks must
be independent. Levels with completely conserved flows wont be
independent. If land passes only between levels of "owned by government"
and "owned by private entities" (and if no new land enters the system),
theres effectively just one level, even though for convenience and clarity,
we may use two. One alternative definition of order might be "the number of
quantities you need to specify to describe the state of the system at a
particular time -- for your purposes." (Speaking of the order of the
underlying system is nearly useless -- its the number of atoms involved, 3
positions, 3 velocities, then any electrons moving between them, etc. etc.
-- always essentially infinite.)

cheers,

alan
From: Alan Graham <Alan.Graham@paconsulting.com>

[Bill Harris]

> >x = A.sin(kt) + B.cos(kt)

> >x = -k^2.x - (S1)

Or, in terms perhaps more matched to Nialls math class, the second is a
differential equation and the first is the solution for that equation,
much as x^2-4=0 is an algebraic equation and x=2 and x=-2 are its
solutions.

Bill
From: Bill Harris <bill_harris@facilitatedsystems.com>
--
Bill Harris 3217 102nd Place SE
Facilitated Systems Everett, WA 98208 USA
http://facilitatedsystems.com/ phone: +1 425 337-5541

[George Richardson]

People wrote:

>Definition: The order of a system is the number of stocks that are
>connected to flows, _minus_ the number of stock pairs which are
>connected by conserved flows.
>
>I guess I do still have one issue remaining: What about a system of
>three stocks A, B and C, with a conserved flow from A->B, B->C and C->A?

The simplest definition, which I think helps most of us from the most
lax to the most picky, is that the order of a system is the number of
INDEPENDENT stocks in the system.

We could get carried away about defining "independence," but it is
easily and quite rigorously thought of as the number of stocks one
could FREELY assign initial values to.

So if you you have an epidemic model with SUSCEPTIBLE, INFECTED, and
CURED populations with no inflows to the SUSCEPTIBLES and no outflow
from the CURED, only two of these three stocks are independent -- the
third could be computed as the total population minus the other two.
So it would be a second-order system even if all three stocks are
shown in the model.

Given the essential duality between integration and differentiation,
one can define the order of a system either in terms of differential
or integral equations. But given the vivid importance and salience
of stocks in dynamic systems, it seems wise to define order in terms
of stocks. A student of system dynamics does not need to know
anything about differential equations to understand fully the concept
of "order."
--
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George P. Richardson gpr@albany.edu
Rockefeller College of Public Affairs and Policy 518-442-3859
University at Albany, Albany, NY 12222 http://www.albany.edu/~gpr
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