Configurations of periodic orbits for equations with delayed positive feedback
In this nicely written paper, the author investigates large-amplitude periodic solutions -- periodic solutions that oscillate about at least two unstable equilibria -- of the delay differential equation \[ \dot{x}(t)=-\mu x(t)+ f(x(t-1)),\tag{1.1} \] where \(\mu>0\) and \(f\) is a nondecreasing \(C^1\) function. The paper extends the results of the work [J. Dyn. Differ. Equations 23, No. 4, 727--790 (2011; Zbl 1247.34120)] of \textit{T. Krisztin} and \textit{G. Vas} which constructed two large-amplitude periodic solutions oscillating about exactly two unstable equilibria. In [Zbl.\ 0849.34056] Mallet-Paret and Sell obtained a Poincaré--Bendixson type result that applies also for (1.1) if \(f'(u)>0\) for all \(u\in\mathbb{R}\). They also proved that the projection \(\pi_2 : C:=C([-1,0],\mathbb{R})\ni \varphi \mapsto (\varphi(0), \varphi(-1))\in \mathbb{R}^2\) maps different periodic orbits onto disjoint simple closed curves in \(\mathbb{R}^2\). Furthermore, they showed that a nonconstant periodic solution \(p\) oscillates about an equilibrium point \(\hat{\chi}\in C\) if and only if \(\pi_2\hat{\chi}=(\chi,\chi)\) is in the interior of \(\pi_2\{p_t:t\in\mathbb{R}\}\). These results give strong restrictions on what periodic solutions the equation may have at the same time for the same nonlinearity \(f\). The main result of the paper proves that all the configurations allowed by Mallet-Paret and Sell exist for suitably chosen feedback function \(f\). More precisely, it is shown that given an arbitrary number \(N\geq 2\) and an equation of the form (1.1), possessing exactly \(N\) unstable equilibria, there is a natural bijection between the set of possible configurations of large-amplitude periodic orbits of (1.1), and the set of possible meaningful parenthetical expressions of \(N\) symbols of fixed order. The constructive proof is supported by illustrative figures and examples. This characterization also yields an exact (but not closed) formula on the number of possible configurations for \(N\) unstable equilibria. In addition, it is also shown that each of the constructed large-amplitude periodic orbits are hyperbolic and have exactly one Floquet multiplier outside the unit circle, which is real, greater than \(1\) and simple. In the last section, some related open problems are discussed. Among others, a generalization of the main result is formulated which shows how arbitrary number of significantly different large-amplitude periodic solutions oscillating about the same set of equilibria can be obtained by choosing the feedback function \(f\) appropriately. Here, two solutions are called significantly different if they cannot be obtained from each other by time translation.
scientific article; zbMATH DE number 6668264
scientific article