Delayed feedback control and phase reduction of unstable quasi-periodic orbits

Abstract

The delayed feedback control (DFC) is applied to stabilize unstable quasi-periodic orbits (QPOs) in discrete-time systems. The feedback input is given by the difference between the current state and a time-delayed state in the DFC. However, there is an inevitable time-delay mismatch in QPOs. To evaluate the influence of the time-delay mismatch on the DFC, we propose a phase reduction method for QPOs and construct a phase response curve (PRC) from unstable QPOs directly. Using the PRC, we estimate the rotation number of QPO stabilized by the DFC. We show that the orbit of the DFC is consistent with the unstable QPO perturbed by a small state difference resulting from the time-delay mismatch, implying that the DFC can certainly stabilize the unstable QPO.

Statistical test of quasiperiodicity in the presence of dynamical noise

“…The phase response curve is used to analyze the effect of a perturbation in continuous-time periodic orbits [26]. We have extended this method to quasiperiodic systems [11]. The phase response curve dh/dφ is dependent on the phase φ n .…”

“…In the Dicky-Fuller test (and the other methods), the null hypothesis is that the time series is a random walk, i.e., ϕ = 1 in the autoregressive model (11). The alternative hypothesis is that ϕ < 1.…”

“…, O is Landau notation that denotes the error terms, and dh/dφ is the quasiperiodic version of the phase response curve [11]. The phase response curve is used to analyze the effect of a perturbation in continuous-time periodic orbits [26].…”

“…On the other hand, although the existence of quasiperiodicity has been shown at the single-neuron level [10], its application is not necessarily complete. Thus, aiming towards the application of quasiperiodicity to neural systems, we have proposed several methods to control neural networks to be quasiperiodic [11][12][13], along with their implemented circuits [14,15]. The fundamental issue of these works is that it is difficult to conclude whether a time series obtained from the system is quasiperiodic or not, especially in real-world systems such as implemented circuits.…”

Experimental verification of quasi-periodic-orbit stabilization using a switched-capacitor chaotic neural network circuit

“…The second stabilization method [7] is based on delayed feedback control. This technique relies on the almost periodicity of a quasi-periodic orbit; therefore, it is difficult to stabilize the exact quasi-periodic orbit.…”