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This work demonstrates an innovative numerical method for counting andlocating eigenvalues with the Evans function. Utilizing the geometric phase inthe Hopf bundle, the technique calculates the winding of the Evans functionabout a contour in the spectral plane, describing the eigenvalues enclosed bythe contour for the Hocking-Stewartson pulse of the complex Ginzburg-Landauequation. Locating eigenvalues with the geometric phase in the Hopf bundle wasproposed by Way, and proven by Grudzien, Bridges & Jones. Way demonstrated hisproposed method for the Hocking-Stewartson pulse, and this manuscriptredevelops this example as in the proof of the method, modifying his numericalshooting argument, and introduces new numerical results concerning the phasetransition.

The Instability of the Hocking-Stewartson Pulse and its Geometric Phase in the Hopf Bundle