Floquet stars

Dimitri Breda, Davide Liessi

Periodic solutions of systems of coupled renewal equations (RE) and retarded functional differential equations (RFDE) are very rarely known explicitly. In order to test the solution operator approach for computing Floquet multipliers, we constructed an ad hoc model with the known periodic solution \((\bar{x}(t), \bar{y}(t)) = (\sin(t), \mathrm{e}^{\sin(t)})\), namely

\[ \left\{ \begin{aligned} & x(t) = - \frac{1}{2} \Bigl[\int_{0}^{\frac{7}{2}\pi} x(t - \sigma) d \sigma - \int_{0}^{\frac{\pi}{2}} \ln(y(t - \sigma)) d \sigma\Bigr], \\ & y'(t) = - \ln\Bigl(y\Bigl(t - \frac{\pi}{2}\Bigr)\Bigr) y(t). \end{aligned} \right.\]

After linearizing the system around the solution and applying eigTMNc to the resulting periodic-coefficient linear system, we were amazed by the picture formed by Floquet multipliers in the complex plane: a seven-pointed star! Moreover, the points of the asterisk are almost aligned with the directions of the seventh roots of unity!

  1. Is this phenomenon essentially due to the coupling?
  2. Does it depend on the periodicity of the coefficients?
  3. Are the seven points related to the \(\frac{7}{2}\pi\) appearing in the first integral as one of the endpoints?
  4. Is it possible to build models producing stars with an arbitrary number of points?

It turns out that the answer is «no» to questions 1 and 2 and seems to be «yes» to questions 3 and 4. Indeed, if \(n\) is odd, the single constant-coefficient linear RE

\[ x(t) = - \frac{1}{2} \int_{0}^{\frac{n}{2}\pi} x(t - \sigma) d \sigma\]

appears to reproduce the same phenomenon, with \(n\) being the number of star points.

Is it possible to prove that the above holds for all odd \(n\) and to predict or describe more precisely the positions of Floquet multipliers? What about even \(n\)?