Table 4.

Common types of bifurcation points

Name	Defining characteristics
Saddle-node (SN)	The coalescence of a stable node and a saddle point, leading to the annihilation of both steady states, forcing the system to evolve to some other attractor, e.g., a different stable node. Examples: Fig. 3C at S≈0.04 and S≈0.67; Fig. 3D at S≈1.30 and S≈3.64 (where the saddle point coalesces with an unstable node). Close to an SN point, the speed at which the dynamical system departs from the vicinity of the recently annihilated steady states is very slow; an experimentally noticeable trait called ‘critical slowing down’.
Pitchfork bifurcation (PF)	The point where a steady-state solution exchanges stability with a pair of alternative steady-state solutions. Close to PF point the loci of the three steady states has the appearance of a ‘pitchfork’. Examples: Fig. 3C at S≈0.28 and S≈0.43. PF bifurcations are indicative of ‘symmetries’ in the dynamical system’s governing equations. If the symmetry is broken, the pitchfork splits into a continuous branch of steady states and a SN bifurcation connecting the two other ‘tines’ of the pitchfork.
Hopf bifurcation (HB)	The point where a steady state changes from slowly damped oscillations to oscillations of increasing amplitude that are ‘captured’ by a periodic limit cycle oscillation. Close to an HB point, the amplitude of the periodic solutions is small, and the frequency of the oscillations is nearly constant. Example: Fig. 3D at S≈0.825 (in this case the amplitude does indeed start off at zero but rapidly increases as S increases above 0.825).
Saddle-node on an invariant circle (SNIC)	The coalescence of a stable node and a saddle point that are connected by a trajectory (an ‘invariant circle’) that proceeds out of the saddle point, makes a long loop through the state space and comes back to the saddle-node along one of its attracting directions. Example: Fig. 3D at S≈1.30 (where the large amplitude limit cycle oscillations end precisely at the SN bifurcation point). Close to a SNIC bifurcation, the amplitude of the limit cycle oscillations is large and nearly constant, and the frequency of the oscillations is small (i.e., the period approaches infinity). Hence, the ‘signatures’ of limit cycle oscillations arising from HB’s and SNIC’s are precisely opposite to each other.