When equilibria appear and exchange stability

Consider the scalar normal form $\dot x=\mu-x^2$. For positive parameter there are two fixed points: the positive root attracts locally and the negative root repels. At zero they meet; for negative parameter no equilibrium remains nearby. The exact signs depend on the chosen coordinates and direction of the parameter. What survives those conventions is pair creation or annihilation, with one stable and one unstable scalar branch on the side where the pair exists.

For a smooth scalar family $f(x,\mu)$, a generic saddle-node at the origin requires $f=f_x=0$ while $f_{xx}\ne0$ and $f_\mu\ne0$ there. The first two equalities identify the degenerate equilibrium; the inequalities ensure a quadratic state dependence and an effective parameter crossing. A zero derivative without those inequalities can describe a higher degeneracy. In higher-dimensional systems a single zero eigenvalue and suitable complementary stability permit a reduced one-dimensional description, but those additional assumptions must be checked.

The supercritical pitchfork $\dot x=\mu x-x^3$ is invariant under replacing x by its negative. Its central branch exists for every parameter. Below zero it attracts; above zero it repels and two attracting states appear at opposite signs. This makes a useful model of a symmetric device choosing either orientation. The two alternatives are related by the symmetry, not by a random term inserted into the deterministic equation; imperfect initial preparation can select which branch is reached.

That symmetry also explains fragility. Adding a small constant bias destroys the exact oddness and generally unfolds the pitchfork into a different arrangement with folds. A subcritical cubic sign instead produces unstable nearby nonzero branches on the opposite side. Stable distant states and hysteresis may require a higher-order saturating term; the destabilizing cubic alone does not prove them. Normal forms describe a neighborhood, so do not infer the entire global response curve from a truncated polynomial near threshold.

Close to an attracting branch with a very small negative derivative, disturbances decay slowly. This critical slowing can make a gradual parameter sweep appear to lag behind its equilibrium diagram. A saddle-node can also leave a bottleneck just beyond disappearance: trajectories move slowly through the place where equilibria nearly existed. An observed long pause is therefore not necessarily a surviving fixed point. Estimate rates and observe a long enough interval before labeling a state stationary.

To use these ideas in an experiment, distinguish the controlled parameter from the observed coordinate, sweep in both directions, and record the disturbance size and settling time. Bias, noise, finite sweep rate, and hidden degrees of freedom can blur a diagram. A bifurcation is a statement about the model's invariant structures as parameters change, while a measured trace includes the protocol. Matching those two requires checking symmetry and nondegeneracy, rather than assigning a familiar name to any branching picture.