Hartman–Grobman theorem

The Hartman–Grobman theorem states that near a hyperbolic fixed point, a nonlinear dynamical system is topologically conjugate to its linearization. The conjugacy preserves orbit structure but need not preserve distances or differentiability.

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Hyperbolic fixed point: A fixed point whose linearization has no eigenvalues with zero real part. Hyperbolicity is the condition that lets the linearized flow determine nearby qualitative behavior.
Dynamical system: A rule describing how a state changes over time, either continuously or in discrete steps. The theorem compares a nonlinear system’s local evolution with a linear one.
Linearization: The approximation of a nonlinear system near a point by its first-order linear part. The theorem justifies linearization for local orbit structure under hyperbolicity, but not for quantitative accuracy.
Philip Hartman: An American mathematician known for work in differential equations and dynamical systems. Hartman independently established a form of the local equivalence between nonlinear systems and their linearizations.
Topological conjugacy: A homeomorphism that maps the orbits of one dynamical system to those of another while preserving time evolution. The theorem’s conclusion is a local topological conjugacy between nonlinear and linear systems.
Fixed point: A state left unchanged by a dynamical system’s evolution rule. The theorem studies trajectories near a state where the system does not move.
Lyapunov stability: A stability property in which sufficiently small initial perturbations remain small for all future time. Topological equivalence does not by itself preserve metric notions such as Lyapunov stability in every setting.
Dmitriĭ Grobman: A Soviet mathematician who proved a local linearization result for systems near hyperbolic equilibria. Grobman’s independent proof gave the theorem its other eponym.
Jacobian matrix: The matrix of first derivatives of a differentiable map, describing its best linear approximation at a point. At the fixed point, this matrix supplies the linear system used in the theorem.
Eigenvalue: A scalar by which a linear transformation stretches an associated nonzero vector. The eigenvalues of the Jacobian determine whether the fixed point is hyperbolic.