By Carmen Charles Chicone
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Extra resources for Linearization via the lie derivative
72) linearizes the vector field X if and only if the pair of functions α, β satisfies the system of partial differential equations DαX − Aα = −f, DβX − Bβ = −g. 73) 0 provided that the improper integral converges. Similarly, the second equation is equivalent to the differential equation d −tB e β(φt (x, y)) = −e−tB g(φt (x, y)) dt and has the solution ∞ β(x, y) = e−tB g(φt (x, y)) dt. 73), is a C 1 function. The proof for β is similar. By using a smooth bump function as in Section 2, there is no loss of generality if we assume that X is bounded on Rn .
9 is a corollary of the following more general result. 10. If X is a C 1,1 vector field on Rn such that X(0) = 0 and DX(0) satisfies Hartman’s (µ, ν)-spectral condition, then X is locally C 1 conjugate to its linearization at the origin. 10 has two main ingredients: a change of coordinates into a normal form where the stable and unstable manifolds of the saddle point at the origin are flattened onto the corresponding linear subspaces of Rn in such a way that the system is linear on each of these invariant subspaces, and the application of a linearization procedure for systems in this normal form.
Because the real parts of the eigenvalues of C lie in the interval [−c, −d] and λ can be chosen anywhere in the interval (0, d), the numbers µ and ϑ can be chosen so that the positive quantity (b − c)d − µ(1 − ϑ) bc is as small as we wish. We will choose µ(1 − ϑ), the H¨ older exponent, as large as possible under the constraints imposed by the spectral gap condition and the inequality µ(1 − ϑ) < (b − c)d . 1). At the first step of the finite induction on the dimension of the “unlinearized” part of the system, we artificially introduce a scalar equation z˙ = −cz where 0 < c < b1 .