## Delay Sandbox

There’s a handy rule of thumb for estimating how much of the input to a first order delay has propagated through as output: after three time constants, 95%. (This is the same as the rule for estimating how much material has left a stock that is decaying exponentially – about a 2/3 after one lifetime, 85% after two, 95% after three, and 99% after five lifetimes.)

I recently wanted rules of thumb for other delay structures (third order or higher), so I built myself a simple model to facilitate playing with delays. It uses Vensim’s DELAY N function, to make it easy to change the delay order.

Here’s the structure:

## Fibonacci Rabbits

This is a small, discrete time model that explores the physical interpretation of the Fibonacci sequence. See my blog post about this model for details.

Fibonacci2.vpm This runs with Vensim PLE, but users might want to use the Model Reader in order to load the included .cin file with non-growing eigenvector settings.

## Oscillation from a purely positive loop

Replicated by Mohammad Mojtahedzadeh from Alan Graham’s thesis, or created anew with the same inspiration. He created these models in the course of his thesis work on structural analysis through pathway participation matrices.

Alan Graham, 1977. Principles on the Relationship Between Structure and Behavior of Dynamic Systems. MIT Thesis. Page 76+

These models are pure positive feedback loops that don’t exhibit exponential growth (under the right initial conditions). See my blog post for a discussion of the details.

These are generic models, and therefore don’t have units. All should run with Vensim PLE, except the generic gain matrix version which uses arrays and therefore requires an advanced version or the Model Reader.

The original 4th order model, replicated from Alan’s thesis: PurePosOscill4.vpm – note that this includes a .cin file with an alternate stable initialization.

My slightly modified version, permitting initialization with different gains at each level: PurePosOscill4alt.vpm

Loops of different orders: 3.vpm 6.vpm 8.vpm 12.vpm (I haven’t spent much time with these. It appears that the high-order versions transition to growth rather quickly – my guess is that this is an artifact of numerical precision, i.e. any tiny imprecision in the initialization introduces a bit of the growth eigenvector, which quickly swamps the oscillatory signal. It would be interesting to try these in double precision Vensim to see if I’m right.)

Stable initializations: 2stab.vpm 12stab.vpm

A generic version, representing a system as a generic gain matrix, so you can use it to explore any linear unforced variant: Generic.vpm