Attractors are fixed points or cycles that the synchronous dynamics cannot leave.

highlighted = computed this step

Attractors are states the system cannot leave

An attractor is a closed part of the transition graph. A fixed point maps to itself; a cycle returns to its starting state after several synchronous updates.

attractor=fixed point or cycle\text{attractor}=\text{fixed point or cycle}
Toggle state tableEach state maps to its synchronous successor.StateNextType0011cycle0101fixed point1010fixed point1100cycle

The toggle has stable memory states

The toggle has 2 fixed points: 01 and 10. It also has a cycle of length 2: 00 to 11 and back.

fixed points=2,cycle length=2\text{fixed points}=2,\quad \text{cycle length}=2
Toggle state tableEach state maps to its synchronous successor.StateNextType0011cycle0101fixed point1010fixed point1100cycle

A larger loop can oscillate longer

A three gene repression loop has 2 attractors, including one cycle of length 6.

long cycle length=6\text{long cycle length}=6
Repressilator state tableA three gene repression loop has a longer cycle.StateNextType000111cycle001011cycle010110cycle011010cycle100101cycle101001cycle110100cycle111000cycle