Archive for the 'Chaos & oscillations' Category

Bifurcations from Strogatz’ Nonlinear Dynamics and Chaos

Published on October 5, 2011 in Chaos & oscillations, Dynamics, Published and Vensim. 0 Comments Tags: , , .

The following models are replicated from Steven Strogatz’ excellent text, Nonlinear Dynamics and Chaos.

These are just a few of the many models in the text. They illustrate bifurcations in one-dimensional systems (saddle node, transcritical, pitchfork) and one two-dimensional system (Hopf). The pitchfork bifurcation is closely related to the cusp catastrophe in the climate model recently posted.

Spiral from a point near the unstable fixed point at the origin to a stable limit cycle after a Hopf bifurcation (mu=.075, r0 = .025)

These are in support of an upcoming post on bifurcations and tipping points, so I won’t say more at the moment. I encourage you to read the book. If you replicate more of the models in it, I’d love to have copies here.

These are systems in normal form and therefore dimensionless and lacking in physical interpretation, though they certainly crop up in many real-world systems.

3-1 saddle node bifurcation.mdl

3-2 transcritical bifurcation.mdl

3-4 pitchfork bifurcation.mdl

8.2 Hopf bifurcation.mdl

Climate Catastrophe

Published on September 15, 2011 in Chaos & oscillations, Climate, Published and Vensim. 0 Comments Tags: , , , , .

This is an interesting, simple model of global ice age dynamics, from:

“A Catastrophe Model of the Paleoclimate”, Douglas R MacAyeal, Journal of Glaciology, Vol 24 No 90, 1979

It illustrates a pitchfork bifurcation as a slice through a cusp catastrophe. It’s conceptually related to earlier models by Budyko and Weertmans that demonstrated hysteresis in temperature and ice sheet dynamics.

The model is used qualitatively in the paper. I’ve assigned units of measure and parameter values that reveal the behavior of the catastrophe, but there’s no guarantee that they are physically realistic.

The .vpm package includes several .cin (changes) files that reproduce interesting tests on the model. The model runs in PLE, but you may want to use the Model Reader to access the .cin files in SyntheSim.

Catastrophe.vpm

 

Gumowski-Mira Attractor

Published on September 2, 2011 in Chaos & oscillations, Discrete, Dynamics, Toy Model and Vensim. 0 Comments Tags: .

I became aware of this neat model via the Vensim forum. I have no idea what the physical basis is, but the diverse and beautiful output it generates is quite amazing.

Interestingly, if you only looked at time series of this sequence, you’d probably never notice it.

This runs in any version of Vensim. gumowski mira.mdl

The Economic Long Wave

Published on July 6, 2011 in Chaos & oscillations, Economics, Published and Vensim. 0 Comments

This is John Sterman’s model of long waves (long-duration economic cycles), driven by capital accumulation dynamics. This version is replicated from a JEBO article,

STERMAN, J. D. (1985) A Behavioral Model of the Economic Long Wave. Journal of Economic Behavior and Organization, 6, 17-53.

There’s some interesting related literature (including other economic models in this library). From Sterman’s publications list:

STERMAN, J. D. & MOSEKILDE, E. (1994) Business Cycles and Long Waves: A Behavioral, Disequilibrium Perspective. IN SEMMLER, W. (Ed.) Business Cycles: Theory and Empirical Methods. Boston, Kluwer Academic Publishers.

STERMAN, J. D. (1994) The Economic Long Wave: Theory and Evidence. IN SHIMADA, T. (Ed.) An Introduction to System Dynamics. Tokyo.

STERMAN, J. D. (2002) A Behavioral Model of the Economic Long Wave. IN EARL, P. E. (Ed.) The Legacy of Herbert Simon in Economic Analysis. Cheltenham, UK, Edward Elgar.

STERMAN, J. D. (1985) An Integrated Theory of the Economic Long Wave. Futures, 17, 104-131.

RASMUSSEN, S., MOSEKILDE, E. & STERMAN, J. D. (1985) Bifurcations and Chaotic Behavior in a Simple Model of the Economic Long Wave. System Dynamics Review, 1, 92-110.

STERMAN, J. D. (1983) The Long Wave. Science, 219, 1276.

KAMPMANN, C., HAXHOLDT, C., MOSEKILDE, E. & STERMAN, J. D. (1994) Entrainment in a Disaggregated Economic Long Wave Model. IN LEYDESDORFF, L. & VAN DEN BESSELAAR, P. (Eds.) Evolutionary Economics and Chaos Theory. London, Pinter.

MOSEKILDE, E., LARSEN, E. R., STERMAN, J. D. & THOMSEN, J. S. (1993) Mode Locking and Nonlinear Entrainment of Macroeconomic Cycles. IN DAY, R. & CHEN, P. (Eds.) Nonlinear Economics and Evolutionary Economics. New York, Oxford University Press.

MOSEKILDE, E., THOMSEN, J. S. & STERMAN, J. D. (1992) Nonlinear Interactions in the Economy. IN HAAG, G., MÜLLER, U. & TROITZSCH, K. (Eds.) Economic Evolution and Demographic Change. Berlin, Springer Verlag.

THOMSEN, J. S., MOSEKILDE, E. & STERMAN, J. D. (1991) Hyperchaotic Phenomena in Dynamic Decision Making. IN SINGH, M. G. & TRAVÉ-MASSUYÈS, L. (Eds.) Decision Support Systems and Qualitative Reasoning. Amsterdam, Elsevier Science Publishers.

THOMSEN, J. S., MOSEKILDE, E., LARSEN, E. R. & STERMAN, J. D. (1991) Mode-Locking and Chaos in a Periodically Driven Model of the Economic Long Wave. IN EBELING, W. (Ed.) Models of Self Organization in Complex Systems. Berlin, Akademie Verlag.

STERMAN, J. D. (1988) Nonlinear Dynamics in the World Economy: The Economic Long Wave. IN CHRISTIANSEN, P. & PARMENTIER, R. (Eds.) Structure, Coherence, and Chaos in Dynamical Systems. Manchester, Manchester University Press.

STERMAN, J. D. (1987) Debt, Default, and Long Waves: Is History Relevant? IN BOECKH, A. (Ed.) The Escalation in Debt and Disinflation: Prelude to Financial Mania and Crash? Montreal, BCA Publications.

STERMAN, J. D. (1987) An Integrated Theory of the Economic Long Wave. IN WANG, Q., SENGE, P., RICHARDSON, G. P. & MEADOWS, D. H. (Eds.) Theory and Application of System Dynamics. Beijing, New Times Press.

STERMAN, J. D. (1987) The Economic Long Wave: Theory and Evidence. IN VASKO, T. (Ed.) The Long Wave Debate. Berlin, Springer Verlag.

RASMUSSEN, S., MOSEKILDE, E. & STERMAN, J. D. (1987) Bifurcations and Chaotic Behavior in a Simple Model of the Economic Long Wave. IN WANG, Q., SENGE, P., RICHARDSON, G. P. & MEADOWS, D. H. (Eds.) Theory and Application of System Dynamics. Beijing, New Times Press.

And from Christian Kampmann,

“The Role of Prices in Long Wave Entrainment” (with C. Haxholdt, E. Mosekilde, and J.D. Sterman), International System Dynamics Conference, Stirling, U.K. and at the Spring 1994 ORSA/TIMS conference, Boston, MA. 1994.
“Disaggregating a simple model of the economic long wave” International Conference of the System Dynamics Society, Keystone, CO, 1985.
The long wave model was the guine pig for Kampmann’s interesting ’96 conference paper that combined a graph-theoretic identification of a set of feedback loops having independent gains with eigenvalue analysis,
Kampmann, Christian E.   Feedback Loop Gains and System Behavior
There also used to be a nifty long wave game, programmed on NEC minicomputers (32k memory?), but I’ve lost track of it. I’d be interested to here of a working version.

Economic Cycles: Underlying Causes

Published on July 6, 2011 in Business, Chaos & oscillations, Economics, Published, Thesis and Vensim. 0 Comments Tags: , , .

Nathaniel Mass’ model of economic cycles, replicated from his 1975 book, Economic Cycles: An Analysis of Underlying Causes, which unfortunately seems to have disappeared from the Productivity Press site (though you can still find used copies).

I haven’t checked, but I’m guessing that the model is quite similar to that in his PhD thesis, which you can get from MIT libraries here. Here’s the abstract:


The models: mass2.mdl mass2.vpm

These don’t have units defined, unfortunately – I’d love to have a copy with units if you’re so inclined.

The Dynamics of Commodity Production Cycles

Published on July 5, 2011 in Business, Chaos & oscillations, Economics, Published, Resources, Thesis and Vensim. 0 Comments Tags: , , .

These classic models are from Dennis Meadows’ dissertation, the Dynamics of Commodity Production Cycles:

While times have changed, the dynamics described by these models are still widespread.

These versions should work in all recent Vensim versions:

DLMhogs2.vpm DLMhogs2.mdl

DLMgeneric2.vpm DLMgeneric2.mdl

 

Lotka-Volterra predator-prey system

Published on June 16, 2011 in Biology, Chaos & oscillations, Economics, Published and Vensim. 0 Comments Tags: , , .

The Lotka-Volterra equations, which describe a predator-prey system, must be one of the more famous dynamic systems. There are many generalizations and applications outside of biology.

Wikipedia has a nice article, which I used as the basis for this simple model.

Continue reading ‘Lotka-Volterra predator-prey system’

A System Zoo

Published on April 8, 2011 in Biology, Business, Chaos & oscillations, Climate, Delays, Discrete, Dynamics, Economics, Energy, Engineering, Environment, External, Health, Population, Published, Resources, Social and Vensim. 0 Comments Tags: .

I just picked up a copy of Hartmut Bossel’s excellent System Zoo 1, which I’d seen years ago in German, but only recently discovered in English. This is the first of a series of books on modeling – it covers simple systems (integration, exponential growth and decay), logistic growth and variants, oscillations and chaos, and some interesting engineering systems (heat flow, gliders searching for thermals). These are high quality models, with units that balance, well-documented by the book. Every one I’ve tried runs in Vensim PLE so they’re great for teaching.

I haven’t had a chance to work my way through the System Zoo 2 (natural systems – climate, ecosystems, resources) and System Zoo 3 (economy, society, development), but I’m pretty confident that they’re equally interesting.

You can get the models for all three books, in English, from the Uni Kassel Center for Environmental Systems Research, http://www.usf.uni-kassel.de/cesr/. Follow the Download link and choose the Software category to obtain a .zip archive of the zoo models for the whole series, in Vensim .mdl format.

To tantalize you, here are some images of model output from Zoo 1. First, a phase map of a bistable oscillator, which was so interesting that I built one with my kids, using legos and neodymium magnets:

Continue reading ‘A System Zoo’

Boiling Water Reactor Dynamics

Published on March 14, 2011 in Chaos & oscillations, Energy, Engineering, Published and Vensim. 0 Comments Tags: , , , .

Replicated from “Hybrid Simulation of Boiling Water Reactor Dynamics Using A University Research Reactor” by James A. Turso, Robert M. Edwards, Jose March-Leuba, Nuclear Technology vol. 110, Apr. 1995.

This is a simple 5th-order representation of the operation of a boiling water reactor around its normal operating point, which is subject to interesting limit cycle dynamics.

The original article documents the model well, with the exception of the bifurcation parameter K and a nonlinear term, for which I’ve identified plausible values by experiment.

TursoNuke1.mdl

Oscillation from a purely positive loop

Published on November 22, 2010 in Chaos & oscillations, Delays, Dynamics, Structural analysis, Thesis, Toy Model and Vensim. 0 Comments Tags: , , , , , .

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