Saddle-node, transcritical, and pitchfork bifurcations

Let x dot = f(x,mu), with f sufficiently smooth near the parameterized equilibrium under study. For a scalar saddle-node at (0,0), require f=0 and its x derivative zero there, while its second x derivative and its parameter derivative are both nonzero. These conditions say that the equilibrium equation has a genuine quadratic tangency that the parameter unfolds transversely. After local coordinate and scale changes, its basic portrait is represented by x dot = mu−x squared, with parameter orientation chosen for convenience.

For positive mu, this form has roots at plus and minus square root of mu. The derivative is minus twice x, so the positive branch attracts and the negative branch repels. At zero the branches collide. The remaining equilibrium is semistable: positive nearby points approach it, while negative ones move away. For negative mu no equilibrium remains and the arrows point downward everywhere. Slow passage near the vanished equilibrium is a bottleneck, but the normal form does not specify every distant feature of the original field.

The transcritical normal form is x dot = mu x−x squared. Its branches x=0 and x=mu exist on both sides of zero. At the origin branch the derivative is mu; at the other branch it is minus mu, so stability is exchanged. This pattern is natural when a constraint preserves the zero branch, such as an extinction state in a population model. A generic arbitrary perturbation need not preserve that branch and can replace the crossing by a different local picture.

The supercritical pitchfork x dot = mu x−x cubed has an odd symmetry under x changing sign. For negative mu only zero is an equilibrium and it attracts. For positive mu, zero repels while two attracting branches appear at opposite square roots of mu. Symmetry forces the paired alternatives and removes even powers from the normal form. A small constant forcing breaks that symmetry and generally unfolds the pitchfork rather than leaving a perfect three-branch meeting. Genericity must therefore be understood within the stated symmetry-preserving class.

For a general odd pitchfork family shifted to the origin, require f_x(0,0)=0, a nonzero mixed derivative f_xmu(0,0), and a nonzero cubic derivative f_xxx(0,0). The mixed derivative makes the linear coefficient cross zero, while the cubic derivative supplies the leading saturation or destabilization. Changing the cubic sign reverses the local criticality. Likewise a transcritical calculation needs the persistent equilibrium branch and nonzero mixed parameter-state and quadratic terms after shifting that branch to zero. Vanishing leading coefficients create degenerate cases requiring higher-order analysis. In systems of several variables, a single zero eigenvalue can permit reduction to a one-dimensional center manifold when the standard smoothness and spectral assumptions hold, but that reduction is a theorem rather than deletion of other coordinates.

An equilibrium diagram plots state against parameter; it is not a trajectory in time. Stable branches can be marked with solid curves and unstable ones with dashed curves, but actual parameter sweeps add another timescale. Delayed transitions, noise-driven escapes, and hysteresis involving distant attractors require more information than a local scalar drawing. Always distinguish a fixed-parameter stability result from a prediction for a time-varying control. This is especially important when an application retains physical constraints such as nonnegative populations, which can exclude mathematical branches from the allowed state space.