We consider the differential equation \[\frac{\dd y}{\dd t}= 3y-y^3+b\] If \(b=0\) then we find the equilibria by detecting the roots of the cubic polynomial \(-y^3+3y\). These are \(y=0\) and \(y=\pm\sqrt{3}\). By local linearization one can verify (do it yourself!) that \(y=0\) is a repeller, and that \(y=-\sqrt{3}\) and \(y=\sqrt{y}\) are attractors. Because of the latter property we speak of a bistable differential equation.

The parameter \(b\) shifts the cubic polynomial \(-y^3+3y\) upwards or downwards. In such a translation, three roots of the cubic polynomial remain as long as \(-2<b<2\).

If \(b=2\) then \[\frac{\dd y}{\dd t}= -(y-2)(y+1)^2\] and there are two equilibria: a semi-stable equilibrium \(y=-1\) and a stable equilibirum \(y=2\) (check the nature of the equilibria yourself!).

If \(b=-2\) then \[\frac{\dd y}{\dd t}= -(y-2)(y+1)^2\] and there are two equilibria: a semi-stable equilibrium \(y=1\) and a stable equilibrium \(y=-2\) (check the nature of the equilibria yourself!).

This implies that the two bifurcations \((-2,1)\) and \((2,-1)\) are saddle-node bifurcations.

With a larger translation \(|b|>2\) there exists only one root of the polynomial \(3y-y^3+b\) and this leads to a repelling equilibrium.

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We explore what happens when we start with \(b=0\) in the equilibrium \(y=-\sqrt{3}\) and then gradually increase the parameter value \(b\) with small steps and wait after each change until a new equilibrium has settled. With increasing \(b\), the point \((b,y)\) first remains in the vicinity on the lower branch of the stable equilibria; the solution goes in the course of time to a new equilibirum in the neighbourhood of where it came form along the branch in bifurcation diagram where the point is present. This continues until we reach \(b=2\) at a saddle-node bifurcation; as soon as \(b\) is slightly greater than \(2\), than the solution moves away to the stable equilibrium in the upper branch of the stable equilibria. When we now decrease the parameter value step by step to \(b=0\), then the point \((b,y)\) remains on the upper branch of the stable equilibria and we arrive for \(b=0\) at the equilibrium \(y=\sqrt{3}\). The parameter value \(b\) returned to its original value, but the equilibrium \(y\) not! This phenomenon of lack of reversibility is called hysteresis. It is visualized in the figure below.