# Coupled oscillators

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## WHAT IS IT?

The Stuart-Landau oscillator model. Several oscillators are coupled.

## HOW IT WORKS

Each node (turtle) represent a single oscillator: a 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 + sigma*c

Where lambda and omega are control parameters.

|z| represents the modulus of the complex number z.

sigma is the strength of the coupling (interaction) between linked nodes.

c represents the coupling and corresponds to the weighted average of distance between nodes (some kind of center of mass).

All the nodes are linked together via a "Watts and Strogatz Small World" where blue links are weighted +1 (excitatory links) and red links are weighted -1 (inhibitory links)

## 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'.

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 or if you want to see the link.

Choose the K and beta parameters for the Watts and Strogatz Small World algorithm.

Choose sigma, the strength of the coupling.

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. Try with only two or three nodes to see the impact of the coupling.

## EXTENDING THE MODEL

You could change the way the nodes are coupled.

## NETLOGO FEATURES

## RELATED MODELS

Stuart-Landau model

## 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.

For the Watts and Strogatz algorithm, see http://en.wikipedia.org/wiki/Watts*and*Strogatz_model

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 0.125 ] ;; link the nodes link-nodes ;; set simulation time to 0 reset-ticks end to go ifelse draw-trajectories [ ;; draw the trajectories? ask nodes [pen-down] ][ ask nodes [pen-up] ] ifelse draw-links [ ;; show the trajectories? ask links [show-link] ][ ask links [hide-link] ] move-nodes tick end to move-nodes ask nodes [ ;; x and y coordinates of the 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) ;; interaction of the neighbors: [SUM_j (xj - xi) * wj]/nb_links let coupling-real (sum [weight * [xcor - z-real] of other-end] of my-links) / (count my-links) let coupling-imag (sum [weight * [ycor - z-imag] of other-end] of my-links) / (count my-links) ;; euler algorithm setxy (xcor + (fz-real + sigma * coupling-real) * dt) (ycor + (fz-imag + sigma * coupling-imag) * dt) ] end ;; create the links among nodes (Watts and Strogatz algorithm), add weight and color to link-nodes ;; create a regular lattice ask turtles [ let n 1 while [ n <= K ] [ ifelse who < (count turtles - K) [ ifelse (random-float 1) < beta [ random-link ] [ normal-link n ] ] [ normal-link n ] set n (n + 1) ] ] end to normal-link [n] create-link-with turtle ((who + n) mod count turtles) [ set color blue set weight 1 ] end ;; create a link with another 'not-yet linked' node whose id > caller id to random-link let myid who let other-node one-of other turtles with [(who > myid) and (not link-neighbor? turtle myid)] if other-node != nobody [ create-link-with other-node [ set color red set weight -1 ] ] 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 is only one version of this model, created about 6 years ago by julien siebert.

## Attached files

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

Coupled oscillators.png | preview | Preview for 'Coupled oscillators' | about 6 years ago, by julien siebert | Download |

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