Bifurcation conditions

Definition (canon)

Three, jointly necessary and individually insufficient. (i) Loop closure — a positive feedback loop contains the node, so that taking a route raises the probability of taking it again. (ii) Saturating nonlinearity — a sufficiently steep saturating element sits somewhere in that loop; a linear loop with gain above one diverges rather than becoming bistable. (iii) Barrier above noise — ΔV > D ln(t_rel/τ₀).

Bifurcation is a property of the loop; steepness and commitment are properties of the node; the barrier is a property of the node in its noise environment. (i) and (ii) decide whether two states exist; (iii) decides whether they can be occupied distinguishably. A node failing (i) or (ii) queues; a node failing only (iii) flickers.

The formula is typeset here in plain text. D is the noise intensity in Kramers form; t_rel is the dwell time before removal; τ₀ is the attempt time (style sheet §1.8).

In plain words

For a node to make a real either-or choice, something must feed its own choice back (i), the feedback must flatten at a ceiling (ii), and the gap between the two outcomes must be larger than the node's own jitter (iii). Miss the first two and the node backs up. Miss only the third and it flickers between two outcomes it cannot tell apart.

It is not

• Three interchangeable ways of saying "tipping point". They fail differently: a failed (i) or (ii) gives a queue, a failed (iii) gives flicker.
• Evidence of a loop from slowing alone. A queue slows too.

What would defeat it

• Follows from the definition. A node that meets (i), (ii) and (iii) and shows no hysteresis; or one that fails (i) or (ii) and shows two distinguishable states with hysteresis.
• Canon. The positive arm of the capacity ⇔ hysteresis test has no assigned node yet (style sheet §1.5), so the conditions stand on derivation, not on a measured node.