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Dynamic Feedback System Models

(c) 2009, Ed Seykota

 

Tutorial on Dynamic Feedback Models:

http://sysdyn.clexchange.org/road-maps/rm-toc.html

 

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Simple Level and Rate Model

 

 

 

We represent  model structure

with a Levels-and-Rates Diagram

 

 

 

We further define the structure

with specific equations.

 

In this case,

In steps up from 0 to 1 at T = 20

and from 1 to 2 at T = 30.

Out steps up from 0 to 1 at T = 20

and from 1 to 2 at T = 30.

 

 

 

We simulate the model

to see how it behaves.

 

In this case,

At T = 0, In and Out are both zero so Level stays constant.

At T = 10, In increases to 1 so Level ramps up from 0 to 10.

At T = 20, Out increases from 0 to 1 so Level stays constant.

At T = 30, Out increases from 1 to 2 so Level ramps down to 0.

At T = 40, In increases from 1 to 2 so Level stays constant.

 

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First-Order Feedback

 

 

 

"First Order" Indicates the number of Levels = 1.

Feedback indicates information from Level

feeds back to control Rate.

 

In this case, Rate = Level / Time_Constant

 

 

 

For TC = -5 Months,

we observe negative feedback,

in which Level drops to 1/e of its original value

every 5 months.

 

This is a basic pattern

that occurs throughout nature and ecomonics.

 

 

 

For TC = +5Months,

we observe positive feedback,

in which Level increases to e times its original value

every 5 months.

 

 

This pattern occurs for some successful businessmen.

 

 

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Pendulum Model

 

 

 

The Structure

 

 

 

 

The Equations

 

 

 

The Simulation

 

In this case,

the pendulum starts off from the "bottom" position with high velocity.

The pendulum "wraps around" a couple times

and then settles down into oscillatory mode

with gradual decay.

 

 

Notes: The period varies with amplitude

in this model and in real-world pendulums.

 

The amplitude-specific property of period

does not generally appear in high school and physics texts,

since solving for this effect with the differential calculus

is a formidable task.

 

The classic solution for the period

P = 2 * pi * sqrt(length/Gravity_Acceleration)

gives P = 2 * 3.14 * sqrt(10/9.8)

P ~ 6-1/2

 

We see this as the approximate period in the model.

 

 

 

Simulation Showing Angle and G-Acceleration

 

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Kondratieff Wave Model

 

 

We can use dynamic feedback modeling

to investigate economics.

 

This model shows one formulation

of the Kondratieff Wave theory

about the capital expansion-contraction cycle.

 

The period is, according to Kondratieff about 60 years.

This matches real world observations and this model.

 

This model shows periods of intense capital formation

and long periods of capital depreciation

in which out-of-date factories

disappear, making room for new ones

during the next expansion cycle.

 

 

Model Structure

 

 

 

 

 

Try a Stimulus Package in the Model

 

 

 In this case,

we add a form of capital stimulus package.

 

The result of this particular form

is that the cycle time lengthens,

and the amount of capital remains generally higher.

 

The stimulus package, in effect, prevents the system

from purging old capital and acquiring new capital.

 

Dynamic Feedback System Models

provide a systematic way to define and test assumptions,

see how behavior flows from assumptions,

learn how systems operate

and investigate sensitivity to various policies.