Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control

We investigate the normal form of a subcritical Hopf bifurcation subjected to time-delayed feedback control. Bifurcation diagrams which cover time-dependent states as well are obtained by analytical means. The computations show that unstable limit cycles with an odd number of positive Floquet exponents can be stabilized by time-delayed feedback control, contrary to incorrect claims in the literature. The model system constitutes one of the few examples where a nonlinear time delay system can be treated entirely by analytical means.

Figure 1

(Color online) Left: Two-dimensional bifurcation diagram of Eq. (1) for $\gamma =-10$, $\beta =\pi ∕4$, and $K=0.02$: subcritical Hopf bifurcation [solid red (dark)], supercritical Hopf bifurcation [dashed green (light)], saddle-node (SN) bifurcation of limit cycles (dotted black), transcritical stable Pyragas curve [solid cyan (light)], and transcritical unstable Pyragas curve [dashed blue (dark)]. Right: Same diagram enlarged close to $\lambda =0$, $\tau =2\pi$, and displayed in distorted coordinates so that the Pyragas curve is a straight line.

Figure 2

(Color online) Left: Two-dimensional bifurcation diagram of Eq. (1) for $\gamma =-10$, $\beta =\pi ∕4$, and $K=0.045$: subcritical Hopf bifurcation [solid red (dark)], supercritical Hopf bifurcation [dashed green (light)], saddle-node bifurcation of limit cycles (dotted black), and transcritical stable Pyragas curve [solid cyan (light)]. Right: Same diagram enlarged close to $\lambda =0$, $\tau =2\pi$, displayed in distorted coordinates so that the Pyragas curve is a straight line.

Figure 3

(Color online) Graphical solution of the condition (18b) for $\gamma =-10$, $\beta =\pi ∕4$, $\lambda =-0.02$, $\tau =8$, and different values of the control amplitude: $K=\left(\mathrm{a}\right)$ 0.04, (b) 0.0604, and (c) 0.08. Dotted blue, left-hand side of Eq. (18b); dashed green (solid red), right-hand side of Eq. (18b) such that the inequality (18a) is valid (invalid) (cf. Fig. 6).

Figure 4

(Color online) Bifurcation diagram of Eq. (1) for $\gamma =-10$, $\beta =\pi ∕4$ in distorted coordinates (see Fig. 1) for different values of the control amplitude $K=\left(\mathrm{a}\right)$ 0.019 and (b) 0.016. Subcritical Hopf bifurcation [solid red (dark)], supercritical Hopf bifurcation [dashed green (light)], saddle-node bifurcation of limit cycles (dotted black), transcritical stable Pyragas curve [solid cyan (light)], and transcritical unstable Pyragas curve [dashed blue (dark)].

Figure 5

(Color online) Unfolding of the transcritical bifurcation for $\gamma =-10$, $\beta =\pi ∕4$, and $K=0.019$ [cf. Fig. 4a]. Blue (green) [black (dark)] surface: parameter dependence of the periodic orbits according to Eq. (18). Yellow (light) surface: orthogonal projection of the Pyragas orbit onto the parameter plane. Black line: saddle-node bifurcation line as a projection of the fold. Blue (cyan) line: Pyragas line as a projection of the unstable (stable) Pyragas orbit onto the parameter plane.

Figure 6

(Color online) Left: Two-dimensional bifurcation diagram of Eq. (1) for $\gamma =-10$, $\beta =\pi ∕4$, and $\lambda =-0.02$ in the control parameter plane: subcritical Hopf bifurcation [solid red (dark)], supercritical Hopf bifurcation [dashed green (light)], saddle-node bifurcation of limit cycles (dotted black), transcritical stable Pyragas curve [solid cyan (light)], and transcritical unstable Pyragas curve [dashed blue (dark)]. Right: Enlarged section. Labels indicate the number of harmonic solutions of the system.

Figure 7

(Color online) Dependence of the radius $R$ of the periodic orbits upon the control amplitude $K$ for $\gamma =-10$, $\beta =\pi ∕4$, $\lambda =-0.02$, and $\tau$ according to Eq. (3), i.e., along the Pyragas curve. Solid cyan (light), stable limit cycle; dashed blue (dark), unstable periodic orbit; solid red (dark), stable trivial fixed point; dashed green (light), unstable trivial fixed point. Exchange of stability at the transcritical bifurcation and the subcritical Hopf bifurcation are clearly visible (cf. Fig. 6).

Figure 8

(Color online) Asymptotic control domain in reduced Cartesian control amplitudes ${\kappa }_x=K\tau \cos \left(\beta \right)$, ${\kappa }_y=K\tau \sin \left(\beta \right)$ for different values of $\gamma$, in the limit $\mid \lambda \mid ⪡1$. Solid line: upper boundary caused by Hopf instability [cf. Eq. (21)], dashed lines: lower boundary caused by transcritical instability for different values of $\gamma$ [cf. Eq. (20)]. The different control domains overlap.

Figure 9

(Color online) Bifurcation diagram of the characteristic equation (B2) in a two-dimensional parameter plane for $\gamma =-10$, $\beta =\pi ∕4$, and $\lambda =-0.02$. Solid blue (cyan): Transcritical bifurcation, Eq. (B6) [unstable eigenvalue occurs left (right) of the line]. Dashed red (green): Hopf bifurcation, Eq. (B8) [unstable eigenvalues occur above (below) the line]. Dotted black: Bifurcation path traced by the control scheme [cf. Eq. (B10)]. Labels indicate the number of unstable eigenvalues.

Figure 10

(Color online) Bifurcation diagrams for the characteristic equation (B2) in a two-dimensional parameter plane for $\gamma =-10$, $\beta =\pi ∕4$, and different values of $\lambda =\left(\mathrm{a}\right)$ $-0.064$ and (b) $-0.068$. Solid blue (cyan): transcritical bifurcation, Eq. (B6) [unstable eigenvalue occurs left (right) of the line]. Dashed red (green): Hopf bifurcation, Eq. (B8) [unstable eigenvalues occur above (below) the line]. Dotted black: bifurcation path traced by the control scheme [cf. Eq. (B10)]. Labels indicate the number of unstable eigenvalues.

Figure 11

Floquet spectrum (real part of the exponents) in dependence on the control amplitude for $\gamma =-10$, $\beta =\pi ∕4$, and two different values of $\lambda =\left(\mathrm{a}\right)$ $-0.064$ and (b) $-0.068$ (cf. Fig. 10).