Pseudospectral approximation reduces delay differential equations (DDE) to ordinary differential equations (ODE). Next one can use ODE tools to perform a numerical bifurcation analysis. By way of an example we show that this yields an efficient and reliable method to qualitatively as well as quantitatively analyze certain DDE. To substantiate the method, we next show that the structure of the approximating ODE is reminiscent of the structure of the generator of translation along solutions of the DDE. Concentrating on the Hopf bifurcation, we then exploit this similarity to reveal the connection between DDE and ODE bifurcation coefficients and to prove the convergence of the latter to the former when the dimension approaches infinity.

We explore stability and instability of rapidly oscillating solutions x(t) for the hard spring delayed Duffing oscillatorFix T > 0. We target periodic solutions x n (t) of small minimal periods p n = 2T /n, for integer n → ∞, and with correspondingly large amplitudes. Simultaneously, for sufficiently large n ≥ n 0 , we obtain local exponential stability for (−1) n b < 0, and exponential instability for (−1) n b > 0, provided thatWe interpret our results in terms of noninvasive delayed feedback stabilization and destabilization of large amplitude rapidly periodic solutions in the standard Duffing oscillator. We conclude with numerical illustrations of our results for small and moderate n which also indicate a Neimark-Sacker-Sell torus bifurcation at the validity boundary of our theoretical results.

In the spirit of the well-known odd-number limitation, we study the failure of Pyragas control of periodic orbits and equilibria. Addressing the periodic orbits first, we derive a fundamental observation on the invariance of the geometric multiplicity of the trivial Floquet multiplier. This observation leads to a clear and unifying understanding of the odd-number limitation, both in the autonomous and the non-autonomous setting. Since the presence of the trivial Floquet multiplier governs the possibility of successful stabilization, we refer to this multiplier as the determining center. The geometric invariance of the determining center also leads to a necessary condition on the gain matrix for the control to be successful. In particular, we exclude scalar gains. The application of Pyragas control on equilibria does not only imply a geometric invariance of the determining center but surprisingly also on centers that resonate with the time delay. Consequently, we formulate odd-and any-number limitations both for real eigenvalues together with an arbitrary time delay as well as for complex conjugated eigenvalue pairs together with a resonating time delay. The very general nature of our results allows for various applications.

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The complex Ginzburg-Landau equation serves as a paradigm of pattern formation and the existence and stability properties of Ginzburg-Landau m-armed spiral waves have been investigated extensively. However, most spiral waves are unstable and thereby rarely visible in experiments and numerical simulations. In this article we selectively stabilize certain significant classes of unstable spiral waves within circular and spherical geometries. As a result, stable spiral waves with an arbitrary number of arms are obtained for the first time. Our tool for stabilization is the symmetry-breaking control triple method, which is an equivariant generalization of the widely applied Pyragas control to the setting of PDEs.