Controlling dynamical systems using multiple delay feedback control

Multiple delay feedback control (MDFC) with two, three, or four different and independent delay times is used to stabilize steady states of various chaotic dynamical systems. A comparison with delayed feedback control methods that are based on a single (fundamental) delay time [Pyragas’ time delay auto synchronization (TDAS) and extended TDAS] shows that MDFC is more effective for fixed point stabilization in terms of stability and flexibility, in particular for large delay times.

Figure 1

(a), (b) Stability diagram of the controlled Rössler system (15a) with MDFC (15b) for $k_1=k_2=0.2,k_3=k_4=0$. (a) Combinations of delay times ${\tau }_1$ and ${\tau }_2$ where MDFC successfully stabilizes the fixed point are plotted in black. (b) Corresponding stability function $\max [0,-\mathrm{Re}\left(\lambda \right)]$ vs delay times, where $\mathrm{Re}\left(\lambda \right)$ denotes the largest real part of the eigenvalues of the characteristic equation (19). Note that (b) is rotated compared to (a). (c)–(f) Stability diagram for (E)TDAS where the control signal (3) is applied to the Rössler system. (d), (f) Gray-scaled stability function $\min [0,\mathrm{Re}\left(\lambda \right)]$ vs $\tau$ and $k$. (c), (d) $R=0$ (TDAS). (e), (f) $R=0.7$ (ETDAS).

Figure 2

Stability diagram of the controlled Rössler system (15a) for (a),(b) two delay times $(k_1=k_2=0.5,k_3=k_4=0)$ and (c),(d) three delay times $(k_1=k_2=k_3=0.5,k_4=0,{\tau }_3=5.8)$. (a),(c) Combinations of delay times ${\tau }_1$ and ${\tau }_2$ where MDFC successfully stabilizes the fixed point are marked in black. (b),(d) Corresponding stability function $\max [0,-\mathrm{Re}\left(\lambda \right)]$ vs delay times ${\tau }_1$ and ${\tau }_2$. $\mathrm{Re}\left(\lambda \right)$ denotes the largest real part of the eigenvalues of the characteristic equation (19). Note that (b) and (d) are rotated compared to (a) and (c).

Figure 3

(a) Fourier spectrum of the unperturbed Rössler system and transfer function of a high-pass filter with cutoff frequency ${\omega }_0=0.5$. (b)–(d) Transfer functions (21) for different choices of delay times and gain factors: (b) $k_i=1$, solid line: ${\tau }_1=16.5$, dash-dotted line: ${\tau }_1=16.5$ and ${\tau }_2=6$, dotted line: ${\tau }_1=16.5,{\tau }_2=6$, and ${\tau }_3=1.5$, dashed line: ${\tau }_1=16.5,{\tau }_2=6,{\tau }_3=1.5$, and ${\tau }_4=17.7$. No fixed point stabilization is possible if only the first two of these delay times are used, because of notches $[\mid T\left(\omega \right)\mid \approx 0]$ in the range of the main frequency of the free running Rössler system [compare spectrum in Fig. 3a]. (c) $k_i=1$, solid line: ${\tau }_1=6$, dash-dotted line: ${\tau }_1=6$, and ${\tau }_2=4$, dotted line: ${\tau }_1=6,{\tau }_2=4$, and ${\tau }_3=1.5$, dashed line: ${\tau }_1=6,{\tau }_2=4,{\tau }_3=1.5$, and ${\tau }_4=17.7$. For two or more delay times no notches are located in the range of the main frequency of the free running Rössler system. (d) Same delay times as in (c) but different gains: $k_{1a}=1.3,k_{1b}=0.7,k_{2a}=0.4,k_{2b}=1.0,k_{3a}=0.8,k_{3b}=0.5,k_{4a}=0.4$, and $k_{4b}=0.7$.

Figure 4

Stability diagrams of the controlled Colpitts oscillator (22a, 22b) using MDFC with (a), (b) two and (c), (d) three delay times $(k_i=1.35,{\tau }_3=3.1)$. In (a) and (c) combinations of normalized delay times ${\tau }_1∕T$ and ${\tau }_2∕T$ where MDFC successfully stabilizes the fixed point are marked in black. (b) and (d) show the stability function $\max [0,-\mathrm{Re}\left(\lambda \right)]$ vs ${\tau }_1∕T$ and ${\tau }_2∕T$ where $\mathrm{Re}\left(\lambda \right)$ denotes the largest real part of the eigenvalues of the characteristic equation. Note that (b) and (d) are rotated compared to (a) and (c), respectively. (e) $k$ dependence of MDFC along the white dash-dotted line shown in (c) where the control signal (22b) with ${\tau }_1={\tau }_2+1.6,{\tau }_3=3.1,k_1=k_2=k_3=k$, and $k_4=0$ is applied to the Colpitts system. (f) Gray-scaled stability function $\min [0,\mathrm{Re}\left(\lambda \right)]$ vs ${\tau }_2∕T$ for the same parameters as in (e).

Figure 5

Stability diagrams for (a), (b) TDAS and (c), (d) ETDAS $(R=0.7)$ where the control signal (3) is applied to the Colpitts system (22a). In (a) and (c) those parameter combinations that lead to fixed point stabilization are plotted black and (b) and (d) show gray-scaled stability functions $\min [0,\mathrm{Re}\left(\lambda \right)]$ vs $\tau ∕T$ and $k$.

Figure 6

Gray-scaled stability function $\min [0,\mathrm{Re}\left(\lambda \right)]$ of the Colpitts oscillator (22a) controlled using the feedback signal $\dot{u}\left(t\right)=-{\omega }_{0}u\left(t\right)-k{\dot{U}}_{C_1}$. ${\omega }_0$ denotes the cutoff frequency and $k$ the gain of the high pass filter. The darker the gray the more stable is the feedback.

Figure 7

Stability diagram of the controlled Lorenz system (23a, 23b) for (a), (b) $k_1=k_2=0.7,k_3=k_4=0$ and (c), (d) $k_1=k_2=k_3=0.7,k_4=0,{\tau }_3=1.40$. Combinations of delay times ${\tau }_1$ and ${\tau }_2$ where MDFC successfully stabilizes the fixed point are plotted in black. (b), (d) Corresponding stability function $\max [0,-\mathrm{Re}\left(\lambda \right)]$ vs delay times. $\mathrm{Re}\left(\lambda \right)$ denotes the largest real part of the eigenvalues of the characteristic equation (24). Note that (a), (c) and (b), (d) show the ${\tau }_1\text{−}{\tau }_2$ plane from different perspectives.

Figure 8

Stability diagram of the controlled Baier-Sahle system (25a, 25b) for (a),(b) $k_1=k_2=0.5,k_3=k_4=0$, (c),(d) $k_1=k_2=k_3=0.5,k_4=0,{\tau }_3=4.75$, and (e),(f) $k_1=k_2=k_3=k_4=0.5,{\tau }_3=4.75,{\tau }_4=1.43$. Combinations of delay times ${\tau }_1$ and ${\tau }_2$ where MDFC successfully stabilizes the considered fixed point are plotted in black. (b), (d), (f) Corresponding stability function $\max [0,-\mathrm{Re}\left(\lambda \right)]$ vs delay times ${\tau }_1$ and ${\tau }_2$. $\mathrm{Re}\left(\lambda \right)$ denotes the largest real part of the eigenvalues of the characteristic equation. (b), (d), and (f) are rotated compared to (a), (c), and (e).