Non-linear rate equations

[Weaver Elise A eweaver WPI.EDU]

Posted by ""Weaver, Elise A"" <eweaver@WPI.EDU>

I have started reading a little about complexity and non-linear dynamics as applied to social systems. It took me a while to understand that when people say ""non-linear"" they are referring to equations for rates. Now I also understand that phase diagrams are sometimes plots of incremental relative changes in two dimensions rather than absolute values.

I now see that I have been using exclusively linear models, except for a short introduction to Lorenz equations (Lorenz of chaos theory, not the Lorentz of special relativity; Edward, not Konrad Lorenz with the ducks).

Anyway, how frequently do system dynamicists model non-linear systems? (with their attendant bifurcation points, chaotic regions, attractors, and so on).

Are we missing a lot of behavior if we stick to linear systems?

Elise
Posted by ""Weaver, Elise A"" <eweaver@WPI.EDU>
posting date Sun, 18 Sep 2005 20:56:25 -0400

[Weaver Elise A eweaver WPI.EDU]

Posted by ""Weaver, Elise A"" <eweaver@WPI.EDU>
I have to retract my last comments. I was thinking the definition of ""linear rate equation"" referred only to the immediate equation of the rate.

I have been reminded to think not just about the immediate equation for the rates, but the equation that incorporates the entire loop (with all extra stocks and so forth). I apologize for the temporary myopia.

Are there then any fundamental differences between SD and complexity theory when you render them in differential equation form?

Elise
Posted by ""Weaver, Elise A"" <eweaver@WPI.EDU>
posting date Mon, 19 Sep 2005 21:21:35 -0400

[Jay Forrest systems jayforrest.c]

Posted by ""Jay Forrest"" <systems@jayforrest.com>
>From my perspective there is tremendous difference.

First, I see the term complexity including a broad range of phenomena many of which are primarily mathematical relationships without any sense of causality. SD is as much as anything else a shorthand and philosophy for describing situations as systems and for exploring implications of the perceived structure/relationships. The magic of SD is using well defined, fundamental concepts from control theory in electrical engineering to fuzzy situations where quantitative modeling would normally have not been utilized
- such as social and business problems.

SD, as stated previously, has two flavors -
1) predictive - where the model is used to make relatively fine predictions. This works best where the system is relatively independent (isolated with clear boundaries) and where the relationships of the elements of the model are relatively clear/definable.
2) As a learning tool - where the model is used to gain insight into the bahavior of a system.

Over time and with exposure to many systems one develops deep appreciation for the difficulty humans face in interpreting the implications of (in
particular) stocks, positive (reinforcing) feedback loops, and negative
(balancing) loops and how they convert seemingly straightforward problems into ones that are very difficult to analyze.

Ultimately, a lot of SD is about stock dynamics, exponential growth due to positive feedback, and shifting of ""controlling loops"" when growth (or
decline) hits a limit. One question that is often addressed is how to intervene within the system to achieve desired behavior.

As models are applied to less ""firm and concise systems (such as social) the ability of SD to provide clear and reliable predicitons decline and the development and analysis process tends to become more about refining perceptions and less about specific mathematical outcomes.

All that said, I can build chaos models in SD for I have the mathematical ability to. However, even simple SD models often generate (or explain) ""baffling"" behavior because of the loops and such - but chaos is hardly the intent or philosophy of SD.

I can see describing SD as residing within the realm of chaos related modeling approaches but it is to me only a tiny element of chaos and way off to one side. In my work I scoured chaos related systems approaches (including binary networks, mathematical biology, etc.) for structurally related insights that should facilitate generating stronger mental models for understanding the future.

And again, all that said, once rendered as equations and limits (such as in DYNAMO programs) it may be very difficult to distinguish between an SD model and at least some range of chaos related models. But the process in getting there and the learnings may be very different.

Hope that helps?????
Jay
Posted by ""Jay Forrest"" <systems@jayforrest.com>
posting date Tue, 20 Sep 2005 09:30:58 -0500

[Weaver Elise A eweaver WPI.EDU]

Posted by ""Weaver, Elise A"" <eweaver@WPI.EDU>

One more question:

What is the order of a system like this:

dx/dt = x2

Is it first order, because there is only 1 stock or is it 2nd order because of the exponent?

It is certainly non-linear.

Elise
Posted by ""Weaver, Elise A"" <eweaver@WPI.EDU>
posting date Tue, 20 Sep 2005 06:09:01 -0400

[Joel Rahn jrahn sympatico.ca]

Posted by Joel Rahn <jrahn@sympatico.ca>
dx/dt = x2 is a first-order dynamic system in the state (stock, level) variable x. The flow or rate is a second-order polynomial function of the state variable, because of the exponent. If the rate has more than one polynomial term, and only polynomial terms, the order is given by the value of the largest exponent. Joel Rahn Posted by Joel Rahn <jrahn@sympatico.ca> posting date Wed, 21 Sep 2005 08:31:57 -0400