Linearization and hyperbolic equilibria

Suppose f is continuously differentiable and f(p)=0. With u=x−p, differentiability writes the system as u dot = Au + R(u), where A is the Jacobian at p and the remainder is smaller than the size of u near zero. The equilibrium is hyperbolic when A has no eigenvalue with zero real part. Stable and unstable generalized eigenspaces collect negative and positive real parts respectively. Jordan blocks can add polynomial factors in time, but strict exponential signs still determine eventual contraction or expansion.

If every real part is negative, the nonlinear equilibrium is locally asymptotically stable, and in fact locally exponentially stable under these smooth finite-dimensional assumptions. If a hyperbolic equilibrium has any positive real part, it is unstable; mixed signs create a saddle. In the plane, a negative-real-part complex pair produces a spiral sink in the linear coordinates. A repeated eigenvalue needs its eigenspace structure for a detailed portrait, so reading only the trace can miss important geometric information.

The Hartman–Grobman theorem gives a local homeomorphism between the nonlinear flow near a hyperbolic equilibrium and its linear flow, with the flow relation holding while the relevant trajectories remain in their local neighborhoods. This preserves the qualitative organization of nearby trajectories and the stable and unstable dimensions. A homeomorphism is continuous with a continuous inverse; it need not be differentiable. Distances, angles, curvature, and exact numerical contraction rates cannot be read from that topological statement alone.

At a nonhyperbolic equilibrium the theorem makes no such conclusion. The equations x dot = −x cubed and x dot = x cubed have the same zero Jacobian at the origin, yet the first attracts nearby points and the second repels them. Likewise, a purely imaginary linear pair can correspond to a nonlinear center, a weak attracting spiral, or a weak repelling spiral. Higher-order terms, an invariant quantity, or a Lyapunov argument must decide these cases. Inconclusive linearization is information about the method, not a verdict of neutral stability.

An equilibrium with invertible Jacobian continues locally under a small smooth parameter change by the implicit function theorem. Hyperbolic eigenvalues remain away from the imaginary axis under sufficiently small changes, so the local stable and unstable dimensions persist. A bifurcation can occur when hyperbolicity fails, although failure alone does not force a particular bifurcation. A numerical eigenvalue close to zero also demands care: rounding may obscure the sign that determines the local classification.

The basin of a sink is the set of states whose forward solutions approach it. A local sink theorem only guarantees a neighborhood inside that basin. Farther away, nonlinear terms can create other equilibria, separating cycles, escape, or finite-time blow-up. Two fields with identical Jacobians can therefore have radically different global behavior. When a phase portrait seems to extrapolate a local spiral across the whole page, test a radial derivative, invariant region, or other global quantity before trusting that extrapolation.