## Tohoku Mathematical Journal 2002 September SECOND SERIES VOL. 54, NO. 3

 Tohoku Math. J. 54 (2002), 419-441

Title FLOQUET MULTIPLIERS OF SYMMETRIC RAPIDLY OSCILLATING SOLUTIONS OF DIFFERENTIAL DELAY EQUATIONS

Author Peter Dormayer, Anatoli F. Ivanov and Bernhard Lani-Wayda

(Received June 27, 2000, revised February 20, 2002)
Abstract. Floquet multipliers of symmetric rapidly oscillating periodic solutions of the differential delay equation $\dot x(t)=\alpha f(x(t),x(t-1))$ 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.

2000 Mathematics Subject Classification. Primary 34K13; Secondary 34K18.

Key words and phrases. Delay equations with symmetry, rapidly oscillating periodic solutions, stability, Floquet multipliers.