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be the first-order ODE. Note that is the solution of (9) using as the initial condition and integrating to time i+1. Then is the Jacobian of . The Jacobian measures how changes if is changed by a small amount. The resolvent is the integral of J(t) along the path , and describes how a small perturbation from at time gets mapped to a perturbation from at time . That is, is the solution of the variational equation
where I is the identity matrix. The reason the arguments to R seem reversed is for notational convenience: they satisfy the identity , and so a perturbation at time gets mapped to a perturbation at time by the matrix-matrix and matrix-vector multiplication [30]. Finally, the linear map in the GHYS refinement procedure is .
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where is symmetrical, and has real but not necessarily positive eigenvalues. If is an eigenvalue of , then are eigenvalues of J. If is positive, this gives one expanding and one contracting direction; if is negative, it gives directions that neither expand nor contract, but ``rotate'' in some sense. These are the directions that are non-hyperbolic.
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