A bifurcation occurs when the qualitative phase portrait changes as a parameter crosses a critical value. Nonhyperbolicity is necessary for a local equilibrium bifurcation: hyperbolic equilibria persist smoothly.
Persistence of hyperbolic equilibria
Theorem. If \(f(x_0,\mu_0)=0\) and \(D_xf(x_0,\mu_0)\) is invertible, then near \(\mu_0\) there is a unique smooth equilibrium branch \(x=x(\mu)\), with unchanged stable and unstable dimensions.
Proof. The implicit function theorem gives the branch. Eigenvalues vary continuously, so none cross the imaginary axis in a sufficiently small neighborhood.
Saddle-node bifurcation
The normal form \[ \dot x=\mu-x^2 \] has no equilibria for \(\mu<0\), one nonhyperbolic equilibrium at \(\mu=0\), and two equilibria \(\pm\sqrt\mu\) for \(\mu>0\), one stable and one unstable.
Theorem (scalar saddle-node conditions). If \[ f(0,0)=0,\quad f_x(0,0)=0,\quad f_\mu(0,0)\ne0,\quad f_{xx}(0,0)\ne0, \] then smooth coordinate and parameter changes reduce the local equilibrium equation to the saddle-node form up to a nonzero factor.
The conditions express a simple fold: the equilibrium equation loses rank in one direction, varies transversely with the parameter, and has nonzero quadratic curvature.
Transcritical and pitchfork bifurcations
Transcritical normal form \[ \dot x=\mu x-x^2 \] has branches \(x=0\) and \(x=\mu\) exchanging stability. Pitchfork normal form \[ \dot x=\mu x-x^3 \] has a symmetry-forced central branch and two new stable branches for \(\mu>0\). Without the symmetry, a pitchfork is typically unfolded into saddle-node behavior; it is not generic among arbitrary one-parameter systems.
Hopf bifurcation
Theorem (Hopf, schematic). Suppose an equilibrium has a simple conjugate pair \[ \lambda_{1,2}(\mu)=\alpha(\mu)\pm i\omega(\mu), \qquad \alpha(0)=0,\quad\alpha'(0)\ne0,\quad\omega(0)\ne0, \] while all other eigenvalues have nonzero real part. If the first Lyapunov coefficient \(l_1\ne0\), a branch of small periodic orbits bifurcates. Its stability and the side on which it exists are determined by the signs of \(\alpha'\) and \(l_1\).
Center-manifold reduction and normal-form transformations yield the radial equation \[ \dot r=r\{\beta\mu+l_1r^2+O(\mu^2+\mu r^2+r^4)\}, \] from which the periodic-orbit branch follows.
Normal forms and universality
Near-identity coordinate changes remove nonresonant nonlinear terms. The remaining resonant coefficients determine local qualitative behavior. Normal forms are asymptotic models, not globally accurate equations; higher orders and global invariant manifolds determine what happens away from the bifurcation neighborhood.