Bifurcation, stability, and critical slowing down in a simple mass–spring system

We analyse a simple mass–spring system as an accessible context for showcasing how continuous changes to system parameters can lead to critical transitions ('tipping points'). Two kinds of transition are explored in particular: saddle–node bifurcations, due to changes in a mass forcing parameter a; and pitchfork bifurcations, due to changes in a spring separation parameter X. Both types of bifurcation arise as features of a cusp catastrophe characterised in X−a parameter space by the critical curve X2/3+a2/3=1, leading to hysteresis cycles, as described by C. Ong (2021), and non-reversible pitchfork catastrophes, which are discussed here for the first time. In each case we demonstrate critical slowing down of the oscillation period τ→∞ as the system approaches bifurcation.