Relation between the extended time-delayed feedback control algorithm and the method of harmonic oscillators

In a recent paper [Phys. Rev. E 91, 012920 (2015)] Olyaei and Wu have proposed a new chaos control method in which a target periodic orbit is approximated by a system of harmonic oscillators. We consider an application of such a controller to single-input single-output systems in the limit of an infinite number of oscillators. By evaluating the transfer function in this limit, we show that this controller transforms into the known extended time-delayed feedback controller. This finding gives rise to an approximate finite-dimensional theory of the extended time-delayed feedback control algorithm, which provides a simple method for estimating the leading Floquet exponents of controlled orbits. Numerical demonstrations are presented for the chaotic Rössler, Duffing, and Lorenz systems as well as the normal form of the Hopf bifurcation.

Figure 1

The relations (16) between the parameters of the MHO and ETDFC algorithms.

Figure 2

The amplitude $|H_{E,N}\left(i\omega \right)|$ and the angle $\text{arg}\left[H_{E,N}\left(i\omega \right)\right]$ of the transfer functions of the ETDFC (red dashed curves) and the MHO controllers (blue solid curves) for $R=0.4, K=1$, and two different numbers of oscillators: (a) $N=10$ and (b) $N=20$. The parameters of the MHO controller are obtained from relations (17).

Figure 3

The zeros and the poles of the transfer functions of the ETDFC (4) (red color) and the MHO (11) (blue color) controllers. The zeros are marked by circles (ETDFC) and points (MHO), while the poles are marked by crosses (ETDFC) and plus signs (MHO). The parameters are $R=0.4$ and $N=20$.

Figure 4

The real parts of two leading FEs for the period-two UPO of the Rössler system (18) as functions of the feedback gain $K$ at three different values of the memory parameter: (a) $R=0$, (b) $R=0.4$, and (c) $R=0.8$. The red curves show the results derived from the ETDFC equation (5). The blue circles are obtained from the MHO with $N=20$ using Eqs. (12) and relations (17). The controlled orbit is stable where the real part of the nonzero FE is negative.

Figure 5

The real part of the leading FE as a function of the feedback gain $K$ for the period-one UPO of the Duffing oscillator (19). The red curve shows the result of the ETDFC, while the blue circles represent the MHO with $N=20$. The value of the memory parameter is $R=0.4$.

Figure 6

The real parts of two leading FEs for the subcritical Hopf bifurcation equation (20) as functions of the feedback gain $\left|K\right|$. The red curves show the results derived from the ETDFC equation (21). The blue circles are obtained from the MHO with $N=20$ using Eqs. (12) and relations (17). The values of the parameters are $\alpha =\pi /4, \gamma =-10, {\lambda }_s=-0.02$, and $R=0.4$.

Figure 7

The real parts of two leading FEs for the period-one UPO of the Lorenz system (22) as functions of the feedback gain $K$. The red curves correspond to the half-period ETDFC (23) with $R=-0.4$, while the blue circles show the results of the MHO consisting of odd harmonics with $N=21$. The parameters of the MHO are obtained from relations (26).