Synergetics - Lorenz Field Dynamics

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WHAT IS IT?

This is a model of a lattice field dynamics whose order parameters follow the Lorenz system.

The model exemplifies the application of Stuttgart School's Synergetics, in particular, it provides an example of Haken's slaving principle in a lattice field model.

HOW IT WORKS

A lattice field variable is introduzed u(i,j,t) (where i is the pxcor (the horizontal axis lattice coordinate) and j the pycor (the vertical axis lattice coordinate)).

The lattice field dynamics is, in turn, given by the following field equations:

u(i,j,t) = epsilon * sin(k * i * x(t) - z(t)) + (1 - epsilon) * cos(k * j * x(t) - z(t)) + delta * w(i,j,t)

dx(t)/dt = sigma * (y(t) - x(t))
dy(t)/dt = x(t) * (rho - z(t)) - y
dz(t)/dt = x(t) * y(t) - beta * z(t)
w(i,j,t) ~ U([0,1])

(By U([0,1]) we understand the uniform distribution over the unit interval.)

Thus, we have N lattice sites and the field state dynamics is decomposed in terms of a local random fluctuation and local sum of periodic functions that depend upon three order parameters, whose dynamics is described by the Lorenz dynamical equations.

Therefore, three collective degrees of freedom drive the deterministic part of the field's fluctuations.

The user can control the strength of the random fluctuations versus the deterministic part, simulating, in this way, the strength to which the slaving dynamics takes place.

HOW TO USE IT

Using the sliders you can change the control parameters for the Lorenz system as well as the parameters for the field: k, epsilon and delta. Changing the field control parameters, in particular k, leads to interesting qualitative changes, namely, raising k you may get turbulence at the macroscopic variable Sum_i,j(u(i,j,t)).

Three plots are provided for the order parameter dynamics and one plot is provided for the sum of the field values Sum_i,j(u(i,j,t)).

NETLOGO FEATURES

This model uses Netlogo's system dynamics procedures coupling it to the distributed agent-based framework.

RELATED MODELS

Another model dealing with complex field dynamics, but using the Brussels-Austin School's conceptual framework is:

Also, for complex field dynamics with application in finance see:

Although the current model addresses classical chaos and field theory, it provides for an example that may be adapted to econophysics (classical and quantum) with useful elements for risk science, namely, the emergence of turbulent dynamics linked to the local computation of order parameter dynamics.

In the global economic system, some of the notions from Synergetics may become relevant tools to address the coevolving dynamics between emerging macroscopic degrees of freedom and microscopic fluctuations.

CREDITS AND REFERENCES

Haken, H. (1977). "Synergetics: An Introduction". Germany, Springer.

Haken, H. (1985). "Towards a Quantum Synergetics: Pattern Formation in Quantum Systems far from Thermal Equilibrium". Phys. Scr. 32 274.

Regarding Synergetics, one may wish to consult the website:

http://www.scholarpedia.org/article/Synergetics

Also consult the following website, for further details on the umbrella research and lecture project on "Complex Quantum Systems Science and Risk Mathematics" that underlies the current model and the three modeling commons projects MC Project #51: Complex Quantum Systems ( http://modelingcommons.org/projects/51 ); MC Project #40: Econophysics ( http://modelingcommons.org/projects/40 ) and MC Project #50: Risk Governance ( http://modelingcommons.org/projects/50 ):

https://sites.google.com/site/quantumcomplexity/

A video of the model is available at:

http://youtu.be/iIMWeHomtJE

Comments and Questions

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Click to Run Model

patches-own [ u M c-red c-green c-blue w]

to setup
  ca
  system-dynamics-setup  
end  

to go
  system-dynamics-go  
  ask patches 
  [ set w random-float 1.000
    set u (1 - delta) * ( epsilon * sin(k * pxcor * x - z ) + (1 - epsilon) * cos(k * pycor * y - z) ) + delta * w ]
  ask patches 
  [ set c-red 250 * (u + 1) / 2
    set c-green 250 * (u + 1) / 2
    set c-blue 250 * (u + 1) / 2 ]
  
  ask patches [ set pcolor (list c-red c-green c-blue) ]
  
  if ticks >= 2
  
[ set-current-plot "x vs y"
  plotxy x y
  
  set-current-plot "x vs z"
  plotxy x z
  
  set-current-plot "y vs z"
  plotxy y z
  
  
  set-current-plot "Sum"
  plot sum [u] of patches ]
end 

There is only one version of this model, created over 6 years ago by Carlos Pedro S. Gonçalves.

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Synergetics - Lorenz Field Dynamics.png preview Preview for 'Synergetics - Lorenz Field Dynamics' almost 6 years ago, by Carlos Pedro S. Gonçalves Download

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