On the floquet multipliers of periodic solutions to nonlinear functional differential equations (Q2502046)

The authors consider scalar delay equations of the type \[ \dot x(t) = -\mu x(t) + f(x(t-1)), \] where \( \mu > 0\) and \(f\) is odd. Floquet multipliers of periodic solutions \(x\) with \( x(t + r) = -x(t)\) and with rational period \( 2r\) are characterized by eigenvalues of an associated ODE boundary value problem. Continuing earlier work, this approach is extended to cover also solutions, where the half-period \( r\) is less than 2. Conditions on hyperbolicity of the periodic solutions are obtained (Theorem 4.1), and can be made explicit for a smoothed step function \(f\) (condition (4.28)). (Under these conditions, the eigenvalue \(-1\) of the time-\(r\)-operator is, in particular, simple.) It is shown by an example that equations similar to variational equations along such periodic solutions can well have \(-1\) as a double eigenvalue (of the time-\(r\)-operator). Under hypotheses which are, e.g., satisfied for the eigenvalue \(-1\) in some applications, results for rational periods are extended to solutions with irrational period by approximation. Here, a key result is the Keldysh theorem, an operator-valued version of the Rouché theorem. Periodic solutions with period 5/4 are constructed in the last part of the paper, basically by perturbation of the discontinuous case \( f = \text{ sign}\). The lack of smoothness makes the perturbation arguments delicate; the implicit function theorem can be replaced by use of the intermediate value theorem in some places. Under additional conditions which express `flatness' of \(f\) away from zero, and which are satisfied in the main example, instability of the obtained periodic solutions can be proved by consideration of a limiting characteristic equation (which is actually a polynomial). The paper, which treats a lot of technically difficult problems in great detail, is made self-contained by an appendix containing requisites from functional analysis.