# Stuart-Landau Oscillator

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## WHAT IS IT?

The Stuart-Landau oscillator model

## HOW IT WORKS

Each node (turtle) represents one oscillator: a single variable 'z' which is complex (i.e. z = a + i*b).

The nodes coordinates represent both real part and imaginary part of 'z' (i.e. xcor is the real part and ycor is the imaginary part).

Each node follows the following equation of motion:

dz/dt = (lambda + i*omega - |z|^2)*z

Where lambda and omega are control parameters. |z| represent the modulus of the complex number z.

## HOW TO USE IT

Choose the number of nodes (note that in this model the nodes do not interact with each other, so you can simply put one node) with the 'nb-node slider'. Putting more than one node helps to explore the behaviour of the whole system (e.g. to see if there is an attractor).

Choose the values of lambda and omega.

Set dt (I usually use 0.001). This is the increment of time in the model. For more details about 'dt' see Euler algorithm for numerical simulation of derivative equations.

Choose if you want to see the trajectories of the nodes.

Push 'setup' and 'go' :)

## THINGS TO NOTICE

Drawing the trajectories helps to see the the stable region (here a cycle).

## THINGS TO TRY

Move lambda and omega to see their influence on the behaviour of the nodes.

## EXTENDING THE MODEL

You could add links between nodes and make them interact...

## NETLOGO FEATURES

## RELATED MODELS

## CREDITS AND REFERENCES

I used the code for complex operations directly from the "Mandelbrot model". Thanks, it was really helpful.

See http://www.scholarpedia.org/article/Periodic_orbit if you want to know more about oscillators in general.

Find mode about the "Stuart-Landau" oscillator in Handbook of Chaos Control, edited by E. Schoell and H. G. Schuster (Wiley-VCH, Weinheim, 2008), second completely revised and enlarged edition.

## Comments and Questions

breed [nodes node] links-own [ weight ] to setup clear-all ;; creates all the nodes set-default-shape nodes "circle" create-nodes nb-nodes [ setxy random-xcor random-ycor set size 0.25 ] ;; set simulation time to 0 reset-ticks end to go ifelse draw-trajectories [ ;; draw the trajectories? ask nodes [pen-down] ][ ask nodes [pen-up] ] move-nodes tick end to move-nodes ask nodes [ let z-real xcor let z-imag ycor ;; f(z) = (lambda + i*omega - |z|^2)*z let mod-z-sq (modulus z-real z-imag) * (modulus z-real z-imag) ;; modulus(z) square let fz-real (rmult (lambda - mod-z-sq) omega z-real z-imag) let fz-imag (imult (lambda - mod-z-sq) omega z-real z-imag) ;; euler algorithm setxy (xcor + (fz-real) * dt) (ycor + (fz-imag) * dt) ] end ;;; Real and Imaginary Arithmetic Operators ;;; real part of the multiplication (a+ib)*(c+id) = (ac-bd) + i(ad+cb) ;;; returns the real part (ac-bd) to-report rmult [real1 imaginary1 real2 imaginary2] report real1 * real2 - imaginary1 * imaginary2 end ;;; imaginary part of the multiplication (a+ib)*(c+id) = (ac-bd) + i(ad+cb) ;;; returns the imaginary part (ad+cb) to-report imult [real1 imaginary1 real2 imaginary2] report real1 * imaginary2 + real2 * imaginary1 end ;;; returns the modulus of a complex number a+ib to-report modulus [real imaginary] report sqrt (real ^ 2 + imaginary ^ 2) end

There are 3 versions of this model.

## Attached files

File | Type | Description | Last updated | |
---|---|---|---|---|

Stuart-Landau Oscillator.png | preview | Preview for 'Stuart-Landau Oscillator' | about 6 years ago, by julien siebert | Download |

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