Abstract
© 2021 Author(s).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.
Original language | English |
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Article number | 063125 |
Journal | Chaos |
Volume | 31 |
Issue number | 6 |
DOIs | |
Publication status | Published - 1 Jun 2021 |
Externally published | Yes |
Funding
This work was partially supported by SFB 910 “Control of self-determining nonlinear systems: Theoretical methods and concepts of application,” Project A4: “Spatiotemporal patterns: observation, control, and design.” The work of B.d.W. was supported by the Berlin Mathematical School (BMS). We are grateful to Professor Dr. Bernold Fiedler and Professor Dr. Sjoerd Verduyn Lunel for their constant support and encouragement. We thank Dr. Jia-Yuan Dai and Alejandro López Nieto for many fruitful discussions and helpful remarks.
Funders | Funder number |
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Berlin Mathematical School |