Floquet multipliers of symmetric rapidly oscillating periodic solutions of the differential delay equation x(t) = αf(x(t), x(t − 1)) with the symmetries f(−x, y) = f(x, y) = −f(x,−y) are described in terms of zeroes of a characteristic function. A relation to the characteristic function of symmetric slowly oscillating periodic solutions is found. Sufficient conditions for the existence of at least one real multiplier outside the unit disc are derived. An example with a piecewise linear function f is studied in detail, both analytically and numerically.
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