@MISC{Hasselblatt_phaseportraits, author = {Boris Hasselblatt}, title = {PHASE PORTRAITS OF LINEAR SYSTEMS}, year = {} }

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Abstract

For our purposes phase portraits of linear second-order systems are of interest primarily when we might use the Hartman–Grobman Theorem [1, page 353]. This means that we can restrict attention to those cases where no eigenvalue has zero real part. 1. REAL EIGENVALUES If none of the two eigenvalues is zero then there are 3 cases: Both eigenvalues are negative, both are positive, or there is one each. 1.1. One positive and one negative eigenvalue. This is described in Example 4.2.1 of [1]. Here the origin is a saddle: unstable but neither a repeller nor an attractor. Draw the two eigenlines (the lines defined by eigenvectors). Put two “outward ” arrows on the eigenline that corresponds to the positive eigenvalue and two “inward” arrows on the eigenline for the negative eigenvalue. Fill in the “quadrants ” between eigenlines with “hyperbolic ” curves with arrows that are consistent with the ones on the eigenlines. See Figures 4.8 and 4.9 in [1]. 1.2. Two positive eigenvalues. There are two possibilities here: The eigenvalues are