Circulant Ring Systems

2.

′

x

i

(t)

x

i

(t)(-

x

i-2

(t)-

x

i

(t)-

x

i+1

(t)+1)

x

i

(0) = 0.4+0.3sin

2πi

N

38

t

388.

view

{

x

1

,

x

2

}

all

x

i

's

Consider a set of elements linked to form a ring. Examples might be masses joined by springs, nodes in a circular communication system, or circuit components. The equation for the state of the

th

i

element depends on the state of its nearby neighbors, for instance the ones at positions

i-2

,

i-1

, and

i+1

in the first equation. (The position of an element is taken modulo

N

.) This construction forms a circulant ring system analogous to systems arranged linearly, such as the Lotka–Volterra systems of equations or equations arising from neural networks, cubic oscillators, and so on. Starting from an initial condition for the state of each element, the trajectories of the elements are often chaotic.

Using the view "all

x

i

's," the point

(a,b)

is color-coded according to the value of

x

a

(b)

.