Abstract
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.
Highlights
In a dynamical system given by the ordinary differential equation x(t) = f(x(t)), x ∈ RN, unstable periodic orbits can be stabilized using additive control terms of the formK (x(t) − x(t − T)) . (1)Here, T > 0 is the time delay, and K ∈ RN×N is the weight of the control term, which we call the gain matrix
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
Summary
The odd-number limitation states that in non-autonomous periodic ordinary differential equations, hyperbolic periodic orbits with an odd number of real Floquet multipliers larger than one cannot be stabilized using Pyragas control. Our main results Theorems 1 and 2 show that the geometric multiplicity of the Floquet multiplier 1, or eigenvalue zero, is invariant under control Whether such a Floquet multiplier 1, or eigenvalue 0, is present in the uncontrolled system or not decides whether the periodic orbit can in principle be stabilized via Pyragas control. All results are of a qualitative nature and apply to any ordinary differential equation (ODE) subject to Pyragas control They do not give any quantitative restrictions on the Floquet multipliers or on the time delay..
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