BifurcationType
BifurcationType enumerates the five canonical routes to chaos under the Recognition Science J-cost model. Nonlinear-dynamics researchers would cite the enumeration when certifying period-doubling cascades that reach the eight-tick octave. The declaration is a plain inductive type with five constructors that automatically derives Fintype for downstream cardinality proofs.
claimLet $B$ be the finite set of bifurcation types consisting of saddle-node, pitchfork, transcritical, Hopf, and period-doubling.
background
The module treats chaotic dynamics as J-growth once recognition cost exceeds the J(φ) threshold. It identifies five bifurcation types with configDim D = 5 and notes that period-doubling proceeds through periods 1, 2, 4, 8 (= 2^D). The inductive definition supplies the enumeration required by the cardinality theorem and the NonlinearDynamicsCert structure.
proof idea
Plain inductive definition declaring the five constructors saddleNode, pitchfork, transcritical, hopf, periodDoubling and deriving DecidableEq, Repr, BEq, Fintype.
why it matters in Recognition Science
The type supplies the five_bifurcations field of NonlinearDynamicsCert and enables the theorem that Fintype.card BifurcationType = 5. It formalizes the period-doubling route reaching 2^3 = 8, consistent with the eight-tick octave (T7) and the module's claim that configDim D = 5 for chaotic systems. No open scaffolding is attached.
scope and limits
- Does not derive the J-cost threshold that triggers chaos.
- Does not approximate the Feigenbaum constant or its 3φ relation.
- Does not link the five types to the spatial dimension D = 3 of the forcing chain.
- Does not supply stability analysis or normal forms for any listed bifurcation.
formal statement (Lean)
24inductive BifurcationType where
25 | saddleNode | pitchfork | transcritical | hopf | periodDoubling
26 deriving DecidableEq, Repr, BEq, Fintype
27