State-Switching Damped Harmonic Oscillator Model

damping, α

0.5

potential strength, β

4

acceleration strength, γ

10

duration

1

offset

3

transition length scale

10

This Demonstration shows a continuous-time model of a single element,

σ

, in a network. In the discrete version, each

σ

can take the values of 0 or 1, and the elements are updated according to input from their neighbors. In the continuous description,

σ

can take a continuum of values, but in order to approximate the discrete state the values should be close to either 0 or 1. This model implements a damped harmonic oscillator in which a function,

u

, determines when the oscillator switches between states 0 and 1, according to input from the neighbors.

G

is an indicator function that allows switching only when its

argument is positive. The specific form is

G(x)=x

1+tanh(x/L)

2

, where

L

is the transition length scale. Care must be taken in selecting the parameters so that accurate transitions occur between states.