Transient behavior in systems with time-delayed feedback

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 21, 023114 (2011) and may be found at https://doi.org/10.1063/1.3581161.We investigate the transient times for the onset of control of steady states by time-delayed feedback. The optimization of control by minimizing the transient time before control becomes effective is discussed analytically and numerically, and the competing influences of local and global features are elaborated. We derive an algebraic scaling of the transient time and confirm our findings by numerical simulations in dependence on feedback gain and time delay. The desire to willingly influence the stability of dynamical systems has led to substantial advances in the field of control theory. A widely used control scheme is time-delayed feedback. The considered systems cover the full range from steady states to chaotic and stochastic dynamics even with spatial degrees of freedom. The large majority of studies has focused on the long-term, asymptotic behavior. Less attention has been paid to the initial stages of the control schemes. Thus, a detailed analysis of the transient behavior toward the target state is still missing. Such an analysis could be used to determine the efficiency of a given control method. For this purpose we investigate the onset of control and analyze the scaling of the transient time. An important aspect for the proposed analysis is the interplay of local dynamics of the system subject to control and global effects by additional delay-induced states.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept