In a recent article,^{1} Ong described how saddle-node bifurcations arise in a simple system comprised of two identical springs connected in a symmetrical V-shaped configuration to a mass *m*, such as that depicted in Fig. 1. Readers may be interested to note, however, that the same spring-mass system can also be used to study both perfect and imperfect pitchfork bifurcations.^{2,3}

*y*in Ong's system are given by solutions to

^{1}

*y*is the vertical equilibrium position of the mass,

*f*is a forcing parameter (which includes gravity), 2

*x*is the horizontal separation between the points securing the springs,

*k*is the spring constant, and

*l*

_{0}is each spring's natural length (see Fig. 1). Ong studied

*saddle-node bifurcations*by exploring how the number of possible equilbrium configurations changes if the forcing

*f*is varied, with

*x*,

*k*, and

*l*

_{0}held fixed. Here, however, we study

*pitchfork bifurcations*by exploring how the number of possible equilibrium configurations changes if the separation

*x*is varied instead, with

*f*,

*k*, and

*l*

_{0}held fixed. As we shall see, pitchfork bifurcations arise in two different ways depending on the value of the forcing

*f*.

*f*= 0, then Eq. (1) may be solved to give either

*x*=

*x*is called a

_{c}*perfect pitchfork bifurcation*, as shown schematically in Fig. 2. Notice that if $ x > l 0$, then the springs are under tension, such that only one equilibrium is possible, with the mass at the centre

*y*= 0. If $ x < l 0$, however, then three equilibria are permitted: two stable, symmetric equilibria $ y = \xb1 [ l 0 2 \u2212 x 2 ] 1 / 2$, with the springs relaxed at the natural length $ l 0 = x 2 + y 2$; and one unstable equilibrium

*y*= 0, with the springs pushing against each other. Observe from Fig. 2 that transitions between the stable equilibria are continuous at

*x*=

*x*.

_{c}*f*> 0, then it is not easy to solve Eq. (1) for

*y*, and the situation becomes more complicated. One can, however, demonstrate that the number of

*possible*equilibria changes from three to one (or vice versa) whenever

*x*passes through the critical value

^{3}

*x*=

*x*is called an

_{c}*imperfect pitchfork bifurcation*(see Fig. 3).

*x*through

*x*(e.g., by pulling the points of attachment apart from one-another) can lead to the system shifting from a stable equilibrium configuration $ y < \u2212 y c < 0$ (with the mass above the springs) to the one with $ 0 < y \u2264 y s$ (mass below the springs), where

_{c}^{3}

*x*=

*x*(see Fig. 3). Note that once the system has shifted from $ \u2212 y c$ to

_{c}*y*, decreasing

_{s}*x*through

*x*(e.g., by pushing the points of attachment closer together) will simply lead to the mass sinking further on the

_{c}*y*> 0 equilibrium; it will not make the mass “jump” back to the $ y < \u2212 y c < 0$ configuration. In this respect, the transition at

*x*is “non-reversible.”

_{c}Ultimately, it may be shown that the saddle-node bifurcations discussed by Ong,^{1} and the pitchfork bifurcations described above, are both features of a *cusp catastrophe*,^{2,3} meaning that the system exhibits a wider range of threshold phenomena than it has been possible to summarise in our *Comment* here (see, e.g., our supplementary analysis published in another context).^{3} Given such rich behaviour, therefore, the simplicity of Ong's system makes it an ideal candidate for developing undergraduate practical work on non-linear effects more generally. Indeed, the presence of bi-stability suggests that if friction is considered, then driving the system may even lead to chaotic behaviour analogous to that exhibited by a Duffing oscillator.^{4} We look forward to investigate such possibilities in future publications.