Topics
Nonlinear phenomena and processes, Stability theory, Mechanical systems, Dynamical systems

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

Fig. 1.
Symmetric V-shaped spring-mass system (Ref. 1); it is assumed that the mass m is constrained to move in the y-direction only (e.g., by sliding on a smooth vertical wire). The y-axis is positive in the “downward” direction.
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Symmetric V-shaped spring-mass system (Ref. 1); it is assumed that the mass m is constrained to move in the y-direction only (e.g., by sliding on a smooth vertical wire). The y-axis is positive in the “downward” direction.

Fig. 1.
Symmetric V-shaped spring-mass system (Ref. 1); it is assumed that the mass m is constrained to move in the y-direction only (e.g., by sliding on a smooth vertical wire). The y-axis is positive in the “downward” direction.
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Symmetric V-shaped spring-mass system (Ref. 1); it is assumed that the mass m is constrained to move in the y-direction only (e.g., by sliding on a smooth vertical wire). The y-axis is positive in the “downward” direction.

Close modal
To see how this works, recall that the equilibrium positions y in Ong's system are given by solutions to1 
(1)
where y is the vertical equilibrium position of the mass, f is a forcing parameter (which includes gravity), 2x is the horizontal separation between the points securing the springs, k is the spring constant, and l0 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 l0 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 l0 held fixed. As we shall see, pitchfork bifurcations arise in two different ways depending on the value of the forcing f.
If f = 0, then Eq. (1) may be solved to give either
(2)
where the latter solutions are possible only if
(3)
This change in the number of possible equilibria when x = xc is called a 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 = ± [ l 0 2 − 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 = xc.
Fig. 2.
Possible equilibrium positions y as a function of x when f = 0 (solid lines depict stable equilibria, dashed lines unstable) (Ref. 3), with the y-axis positive in the “downward” direction (as in Fig. 1). The springs are relaxed at the natural length l0 everywhere on the semi-circle defined by x 2 + y 2 = l 0 2. The perfect pitchfork bifurcation occurs at x = x c = l 0.
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Possible equilibrium positions y as a function of x when f = 0 (solid lines depict stable equilibria, dashed lines unstable) (Ref. 3), with the y-axis positive in the “downward” direction (as in Fig. 1). The springs are relaxed at the natural length l0 everywhere on the semi-circle defined by x 2 + y 2 = l 0 2⁠. The perfect pitchfork bifurcation occurs at x = x c = l 0⁠.

Fig. 2.
Possible equilibrium positions y as a function of x when f = 0 (solid lines depict stable equilibria, dashed lines unstable) (Ref. 3), with the y-axis positive in the “downward” direction (as in Fig. 1). The springs are relaxed at the natural length l0 everywhere on the semi-circle defined by x 2 + y 2 = l 0 2. The perfect pitchfork bifurcation occurs at x = x c = l 0.
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Possible equilibrium positions y as a function of x when f = 0 (solid lines depict stable equilibria, dashed lines unstable) (Ref. 3), with the y-axis positive in the “downward” direction (as in Fig. 1). The springs are relaxed at the natural length l0 everywhere on the semi-circle defined by x 2 + y 2 = l 0 2⁠. The perfect pitchfork bifurcation occurs at x = x c = l 0⁠.

Close modal
If 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 value3 
(4)
In this case, the change in the number of possible equilibrium configurations when x = xc is called an imperfect pitchfork bifurcation (see Fig. 3).
Fig. 3.
Possible equilibrium positions y as a function of x for a fixed value of f > 0 (solid lines depict stable equilibria, dashed lines unstable) (Ref. 3), with the y-axis positive in the ‘downward’ direction. The imperfect pitchfork bifurcation occurs when x = x c ( f ); thus, if the system is initially in a stable equilibrium with y < 0, then increasing x through xc will result in the equilibrium shifting discontinuously from y = − y c to y = ys (arrows). Crossing the semi-circle x 2 + y 2 = l 0 2 (dotted line) corresponds to the springs changing from a compressed state ( x 2 + y 2 < l 0) to an extended state ( x 2 + y 2 > l 0).
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Possible equilibrium positions y as a function of x for a fixed value of f > 0 (solid lines depict stable equilibria, dashed lines unstable) (Ref. 3), with the y-axis positive in the ‘downward’ direction. The imperfect pitchfork bifurcation occurs when x = x c ( f )⁠; thus, if the system is initially in a stable equilibrium with y < 0, then increasing x through xc will result in the equilibrium shifting discontinuously from y = − y c to y = ys (arrows). Crossing the semi-circle x 2 + y 2 = l 0 2 (dotted line) corresponds to the springs changing from a compressed state (⁠ x 2 + y 2 < l 0⁠) to an extended state (⁠ x 2 + y 2 > l 0⁠).

Fig. 3.
Possible equilibrium positions y as a function of x for a fixed value of f > 0 (solid lines depict stable equilibria, dashed lines unstable) (Ref. 3), with the y-axis positive in the ‘downward’ direction. The imperfect pitchfork bifurcation occurs when x = x c ( f ); thus, if the system is initially in a stable equilibrium with y < 0, then increasing x through xc will result in the equilibrium shifting discontinuously from y = − y c to y = ys (arrows). Crossing the semi-circle x 2 + y 2 = l 0 2 (dotted line) corresponds to the springs changing from a compressed state ( x 2 + y 2 < l 0) to an extended state ( x 2 + y 2 > l 0).
View largeDownload slide

Possible equilibrium positions y as a function of x for a fixed value of f > 0 (solid lines depict stable equilibria, dashed lines unstable) (Ref. 3), with the y-axis positive in the ‘downward’ direction. The imperfect pitchfork bifurcation occurs when x = x c ( f )⁠; thus, if the system is initially in a stable equilibrium with y < 0, then increasing x through xc will result in the equilibrium shifting discontinuously from y = − y c to y = ys (arrows). Crossing the semi-circle x 2 + y 2 = l 0 2 (dotted line) corresponds to the springs changing from a compressed state (⁠ x 2 + y 2 < l 0⁠) to an extended state (⁠ x 2 + y 2 > l 0⁠).

Close modal
It may be shown for the imperfect pitchfork bifurcation that increasing x through xc (e.g., by pulling the points of attachment apart from one-another) can lead to the system shifting from a stable equilibrium configuration y < − y c < 0 (with the mass above the springs) to the one with 0 < y ≤ y s (mass below the springs), where3 
(5)
and
(6)
Such transitions between equilibria take the springs from a compressed state (with the downward force supported by the springs' reaction) to an extended state (with the downward force supported by the springs' tension) and are, therefore, discontinuous at x = xc (see Fig. 3). Note that once the system has shifted from − y c to ys, decreasing x through xc (e.g., by pushing the points of attachment closer together) will simply lead to the mass sinking further on the y > 0 equilibrium; it will not make the mass “jump” back to the y < − y c < 0 configuration. In this respect, the transition at xc is “non-reversible.”

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.

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