[Paper Review] A Global Hartman-Grobman Theorem

X. Wang|arXiv (Cornell University)|Feb 14, 2020

Advanced Differential Equations and Dynamical Systems2 references4 citations

TL;DR

This paper extends the classical Hartman-Grobman theorem by constructing a global homeomorphism on the entire region of attraction (or repulsion) of a hyperbolic equilibrium point, transforming a nonlinear system into a linear one via a coordinate change. The transformation is defined using flow integrals involving a cutoff function and is shown to be a global diffeomorphism when all eigenvalues lie in the left- or right-half complex plane.

ABSTRACT

We showed that for any bounded neighborhood of a hyperbolic equilibrium point $x_0$, there is a transformation which is locally homeomorphism, such that the system is changed into a linear system in this neighborhood. If the eigenvalues of $Df(x_0)$ are all located in the left-half complex plane, then there is a homeomorphism on the whole region of attraction such that the nonlinear system on the region of attraction is changed into a linear system under such a coordinate change.

Motivation & Objective

• To generalize the local Hartman-Grobman theorem to global regions of attraction or repulsion for hyperbolic equilibria.
• To establish conditions under which a global coordinate transformation exists that linearizes a nonlinear system.
• To resolve the limitation of local diffeomorphisms by constructing a global homeomorphism in the region of attraction when all eigenvalues are in the left-half plane.
• To provide an explicit construction of the conjugating transformation using flow integrals and a smooth cutoff function.
• To prove that the transformation is a global homeomorphism on the region of attraction, ensuring invertibility and continuity of both the map and its inverse.

Proposed method

• Define a smooth cutoff function α(x) that transitions from 1 to 0 between radii M and M+ε, localizing the nonlinear terms.
• Construct a modified nonlinear vector field Ŵ(x) = W(α(x)x) to localize the perturbation in the bounded neighborhood N_M.
• Define the transformation H(x) as x plus integrals of the form ∫₀^∞ e^{-Ps} Ŵ₁(ϕ_s(x)) ds and ∫_{-∞}^0 e^{-Ns} Ŵ₂(ϕ_s(x)) ds, using the flow ϕ_t(x).
• Prove that H is a local homeomorphism near the origin using the inverse function theorem, since DH(0) = I.
• Show that under H, the nonlinear system ẋ = f(x) transforms into the linear system ẏ = Ay in N_M via verification of the derivative chain rule.
• Extend the transformation globally to the entire region of attraction by proving H is bijective, continuous, and has a continuous inverse using flow continuity and asymptotic stability.

Experimental results

Research questions

• RQ1 Can the Hartman-Grobman theorem be extended beyond a local neighborhood to the entire region of attraction of a hyperbolic equilibrium?
• RQ2 Under what conditions does a global homeomorphism exist that linearizes a nonlinear system on its region of attraction?
• RQ3 Is it possible to construct an explicit global conjugacy map using flow integrals and cutoff functions?
• RQ4 Why does the transformation fail to be a global homeomorphism when multiple equilibria exist in the bounded region?
• RQ5 How do the spectral properties (left- or right-half plane eigenvalues) affect the global linearization of the system?

Key findings

• For any bounded neighborhood N_M of a hyperbolic equilibrium point, there exists a local homeomorphism H such that the nonlinear system ẋ = f(x) is linearized to ẏ = Ay within N_M.
• When all eigenvalues of Df(0) lie in the left-half complex plane, the transformation H extends to a global homeomorphism on the entire region of attraction of the origin.
• The global conjugacy map H is constructed as H(x) = x + ∫₀^∞ e^{-Ps} Ŵ₁(ϕ_s(x)) ds - ∫_{-∞}^0 e^{-Ns} Ŵ₂(ϕ_s(x)) ds, using a smooth cutoff function to localize the nonlinear terms.
• The map H is a global diffeomorphism from the region of attraction onto ℝⁿ, ensuring the nonlinear system is topologically conjugate to its linearization.
• The inverse map H⁻¹ is continuous, and both H and H⁻¹ are continuous with respect to initial conditions due to the continuity of the flow.
• In the special case where ∫₀^∞ e^{-τA}W(ϕ_τ(x)) dτ converges, the transformation simplifies to h(x) = ∫₀^∞ e^{-τA}W(ϕ_τ(x)) dτ, providing a global conjugacy without cutoff.