Theory

Bifurcations & routes to chaos

A bifurcation is a qualitative change in a dynamical system’s long-term behaviour as a parameter is varied. A handful of codimension-1 bifurcations cover most of the territory.

Codimension-1 bifurcations

• Saddle-node (fold): pair of fixed points collides and annihilates.
• Transcritical: two fixed points cross, exchanging stability.
• Pitchfork (sub/supercritical): symmetric branching; one fixed point splits into three.
• Hopf (sub/supercritical, Andronov-Hopf): fixed point loses stability to an orbiting limit cycle.
• Period-doubling (flip): stable period-T orbit becomes unstable; stable period-2T orbit appears nearby.
• Neimark-Sacker: discrete-time Hopf; fixed point of a map becomes an invariant torus.
• Homoclinic / heteroclinic: trajectory connects a saddle to itself or another; horseshoes nearby.
• Crisis: an attractor collides with a basin boundary or another attractor.

Period-doubling cascade (Feigenbaum)

Iterate: fixed point loses stability, period-2 appears; period-2 loses stability, period-4 appears; and so on, accumulating at a finite parameter value r∞, beyond which chaos starts. The ratios of successive bifurcation intervals tend to the universal Feigenbaum constant δ = 4.6692016091… for any unimodal map with a quadratic maximum. The scaling factor between successive doublings of the orbit pattern is α = 2.5029078750….

Watch this in action at /tools/bifurcation.

Quasi-periodic route (Ruelle-Takens-Newhouse)

Successive Hopf bifurcations produce a 2-torus, then a 3-torus; generic small perturbations on the 3-torus produce a strange attractor (Ruelle-Takens 1971, Newhouse-Ruelle-Takens 1978).

Intermittency (Pomeau-Manneville)

Just past a critical parameter, the system spends long stretches in an almost-periodic phase, punctuated by chaotic bursts. Three flavours:

• Type I: triggered by a saddle-node bifurcation of the Poincaré map.
• Type II: subcritical Hopf bifurcation.
• Type III: inverse period-doubling.

On-off intermittency: an invariant subspace becomes transversely unstable; trajectories bounce between near-laminar and chaotic.

Crisis-induced chaos

Boundary crisis: chaotic attractor collides with the boundary of its basin and disappears, replaced by a chaotic transient. Interior crisis: two co-existing attractors merge. Crisis-induced intermittency: bursts of large-amplitude chaos in a previously small-amplitude chaotic regime.

Edge of chaos & self-organised criticality

Langton (1990) and Wolfram (1984) observed maximum computational capacity at the boundary between ordered (class I, II) and chaotic (class III) dynamics. Bak, Tang, Wiesenfeld (1987) showed that certain dissipative systems self-organise to a critical state with power-law statistics; sandpile model is the canonical example.

References: Strogatz, Nonlinear Dynamics and Chaos; Guckenheimer & Holmes, Nonlinear Oscillations; Kuznetsov, Elements of Applied Bifurcation Theory.