Definition 5 Consider a solution to Equations (16)–(17). Assume for some and that

Let . By the observations made prior to the definition, has smooth expansions in the neighborhood of . In particular converges to a smooth function in and the convergence is exponential in any -norm. We call a non-degenerate true spike if .

The choice of is unimportant. Note that nondegenerate true spikes have punctured neighborhoods with normal expansions.

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