Delayed feedback control of chaos: Bifurcation analysis

We study the effect of time delayed feedback control in the form proposed by Pyragas on deterministic chaos in the Rössler system. We reveal the general bifurcation diagram in the parameter plane of time delay $\tau$ and feedback strength $K$ which allows one to explain the phenomena that have been discovered in some previous works. We show that the bifurcation diagram has essentially a multileaf structure that constitutes multistability: the larger the $\tau$, the larger the number of attractors that can coexist in the phase space. Feedback induces a large variety of regimes nonexistent in the original system, among them tori and chaotic attractors born from them. Finally, we estimate how the parameters of delayed feedback influence the periods of limit cycles in the system.

Figure 1

(Color online) Dots: chaotic attractor of Rössler system Eq. (2) without delayed feedback $(K=0)$, gray line (red online): period-1 unstable periodic orbit, black line (black online): period-2 unstable periodic orbit.

Figure 2

(Color online) Bifurcation diagram of the Rössler system with delayed feedback in the plane of feedback parameters: time delay $\tau$ and feedback strength $K$. Bifurcation lines of all leaves are shown. A scheme of the leaf structure is given in Fig. 3. For bifurcations on separate leaves see Fig. 5. Black areas mark the existence of a stable fixed point, which never coexists with any other attractors. Black solid lines mark Hopf bifurcations of the fixed point, at which a periodic orbit (stable or unstable) is born. White circles mark points of self-intersection of Hopf bifurcation line. On gray (red online) solid lines an unstable limit cycle acquires stability via a subcritical Neimark-Sacker bifurcation. On gray (red online) dashed lines a stable torus is born from the stable limit cycle via a supercritical Neimark-Sacker bifurcation. Dotted lines show period-doubling bifurcations of stable limit cycles. On dash-double-dotted lines the attractors undergo crises and cease to exist. Hatched areas denote chaos born as a result of period-doubling bifurcations or of smoothness loss by the invariant torus.

Figure 3

(Color online) Sketch of multiple leaf structure of bifurcation diagram of system (2) on the parameter plane $(K,\tau )$. Letters “FP” mark the stability regions for the only fixed point of system (2).

Figure 4

(Color online) Illustration of how the leaves of the bifurcation diagram are formed. In each plot the spread of the $x$ variable $x_{\mathit{max}}-x_{\mathit{min}}$ as a measure of the size of the period-1 orbit vs $\tau$ is given for a value of feedback strength $K$ that is indicated to the right of the plot. The width of the leaf in $\tau$ is defined by the range of $\tau$ where the particular orbit has a nonzero size. Inside the shaded areas the orbit is stable.

Figure 5

(Color online) Separate leaves of the bifurcation diagram of the Rössler system with delayed feedback (shown in full in Fig. 2). Shaded gray (green online) areas do not belong to the respective leaf. Lines are denoted in the same way as in Fig. 2.

Figure 6

(Color online) Illustration of torus breakdown as a transition to chaos in the Rössler system with delayed feedback on leaf 2 of the bifurcation diagram (Fig. 5). Poincaré maps of attractors and their respective Fourier power spectral density $S$ calculated from $x\left(t\right)$ are given for three values of $K$ at $\tau =13.5$. Each panel illustrates a particular value of $K$.

Figure 7

One-parameter bifurcation diagrams of system (2), illustration of multistability: for each fixed $K$, $\tau$ is first increased from zero to 25 and then decreased from 25 to zero. (a), (b) $K=0.13$; (c), (d) $K=0.8$; (e), (f) $K=2.0$.

Figure 8

(Color online) Variance of control force $F$ defined by Eq. (3) vs $\tau$ for three different values of $K$: (a), (b) $K=0.13$; (c), (d) $K=0.8$; (e), (f) $K=2.0$. For each fixed value of $K$, $\tau$ is first increased from zero to 25 (black line) and then decreased from 25 to zero (shaded gray, green online). Multistability is clearly visible.

Figure 9

(Color online) Period $\mathrm{\Theta }$ of period-1 orbit (stable or unstable) vs time delay $\tau$ for (a) $K=0.13$; (b) $K=0.8$; (c) $K=2.0$. Black circles denote the period $\mathrm{\Theta }$ calculated numerically, the five gray (red online) solid lines show the estimates of $\mathrm{\Theta }$ using Eq. (9) for $n=0$,1,2,3,4 (from left to right).

Figure 10

(Color online) Period $\mathrm{\Theta }$ (a) of period-2 orbit for $K=0.8$ and (b) of period-4 orbit for $K=2.0$ vs time delay $\tau$. Black circles denote periods calculated numerically, gray (red online) lines show estimates with the formula (10) at $m=2$ and $n=0$,1,2 (from left to right) for period-2 cycles, and at $m=4$ and $n=0$,1 for period-4 cycles.