11.1 Nondegenerate true spikes

The definition of a nondegenerate true spike proceeds by running the construction of a true spike backwards. In other words, we start with a solution (Q, P ) such that 1 < v∞ (𝜃0) < 2 and such that P τ(τ, 𝜃0) → v∞ (𝜃0). Letting (Q1, P1) = GEq0,τ0,𝜃0(Q,P ), we see, by Equation (41View Equation), that P1τ(τ,𝜃0) → 1 − v∞(𝜃0) < 0. Let (Q2,P2 ) = Inv(Q1,P1 ). Then, due to Proposition 1, P (τ,𝜃 ) → v (𝜃) − 1 2τ 0 ∞ 0 and Q (τ,𝜃 ) → 0 2 0. Finally, Proposition 2 applies to (Q ,P ) 2 2 so that there are smooth expansions in the neighborhood I of 𝜃0. In particular, there is a smooth function q2 such that Q2 converges to q2 with respect to any Ck-norm. Moreover, it is important to note that q2(𝜃0) = 0. We are naturally led to [79Jump To The Next Citation Point, Definition 1.12, p. 987]:

Definition 5 Consider a solution (Q, P) to Equations (16View Equation)–(17View Equation). Assume 1 < v (𝜃 ) < 2 ∞ 0 for some 1 𝜃0 ∈ S and that

τli→m∞ Pτ(τ,𝜃0) = v∞ (𝜃0).

Let (Q2,P2 ) = Inv ∘ GEq ,τ ,𝜃 (Q,P ) 0 0 0. By the observations made prior to the definition, (Q2,P2 ) has smooth expansions in the neighborhood I of 𝜃 0. In particular Q 2 converges to a smooth function q2 in I and the convergence is exponential in any k C-norm. We call 𝜃0 a non-degenerate true spike if ∂ 𝜃q2(𝜃0) ⁄= 0.

The choice of q ,τ,𝜃 0 0 0 is unimportant. Note that nondegenerate true spikes have punctured neighborhoods with normal expansions.


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