Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control

For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragas. A recent paper by Fiedler et al. Phys. Rev. Lett. 98, 114101 (2007) uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al. for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragas-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragas control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al. example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.

Figure 1

(Color online) The figures show the curves $\tau ={\tau }_P\left(\lambda \right)$ (dashed curve) and $\tau ={\tau }_H\left(\lambda \right)$ (solid curve) for three values of $b_0$. The remaining parameters in Eq. (1) are $\beta =\pi ∕4$ and $\gamma =-10$. In (a), portions of four of the Hopf curves are shown, but in (b) and (c) we show only the curve which passes through $(\lambda ,\tau )=(0,2\pi )$. In all three cases, the two curves cross at $\lambda =0$ (shown by a solid dot). In (a) (with $b_0>b_0^c$), the curves cross again at $\lambda <0$ (shown by an empty dot), and in (c), $(b_0<b_0^c)$ the curves cross again at $\lambda >0$. Case (b) has $b_0=b_0^c$ and the two curves are tangent at $\lambda =0$. The origin is stable (unstable) in those regions marked by an s (u).

Figure 2

In (a), the two Hopf bifurcation curves $\lambda =0$ and $b_0=b_0^{\text{Hopf}}\left(\lambda \right)$ (solid bold lines), and the transcritical (TC) bifurcation curve (dashed bold line) divide the $\lambda \text{−}{b}_0$ plane into five regions. The curves intersect at $(\lambda ,b_0)=(0,b_0^c)$. The Pyragas orbit is stable in the shaded region. (b) Schematic representation of the solutions as a path $C$ is traversed anticlockwise around the origin. Solid lines represent stable solutions and dashed lines represent unstable solutions.

Figure 3

(Color online) The top figure is a bifurcation diagram of the subcritical Hopf bifurcation in Eq. (7), showing the fixed point at zero and the bifurcating branch of unstable periodic orbits. Solid lines indicate stable solutions and dashed lines indicate unstable solutions. The lower figure shows the period, $T$, of the bifurcating periodic orbit, as a function of $\rho$.

Figure 4

(Color online) The figure shows the curves $\tau ={\tau }_H\left(\rho \right)$ (solid lines) and $\tau ={\tau }_P\left(\rho \right)$ (dashed lines) for $b_0=1.2>b_0^c$ and $b_0=0.1<b_0^c$. Remaining parameters are $\sigma =10$, $\alpha =8∕3$, and $\beta =\pi ∕4$. Compare with Fig. 1. The Hopf bifurcation at $\rho ={\rho }_h\approx 24.74$ is shown with a solid dot. In (a) the curves additionally cross in $\rho <{\rho }_h$ (shown by an empty dot). The origin is stable (unstable) in those regions marked with an s (u). These figures were produced using DDE-BIFTOOL [25].

Figure 5

(Color online) (a) Curves of Hopf bifurcations from the zero solution in the $\rho \text{−}{b}_0$ plane for the Lorenz system with delay. The vertical line is the bifurcation to the Pyragas periodic orbits, and the second curve is the bifurcation to the delay-induced periodic orbits. The curve of transcritical bifurcations of periodic orbits is not shown, but it can be seen in Fig. 6. The dotted ellipse is the curve traversed to generate the bifurcation diagram in (b). Here, solid lines indicate stable solutions, and dashed lines indicate unstable solutions. Parameter values are $\sigma =10$, $\alpha =8∕3$, $\beta =\pi ∕4$. The ellipse is parametrized by $\theta$: $\rho -{\rho }_h=1.4\cos \theta$, $b_0-b_0^c=0.03\sin \theta$, and encloses the codimension-two point. This figure was generated using DDE-BIFTOOL. Compare with Fig. 2.

Figure 6

The figure shows a contour plot of the modulus of the largest Floquet multipliers for the Pyragas orbit as $\rho$ and $b_0$ are varied. The shaded area indicates the region where the periodic orbit is stable. The codimension-two point is marked with a dot. The boundary of the stable region which emanates from this point is the transcritical bifurcation of periodic orbits. The left boundary of the stable region corresponds to a period-doubling bifurcation. This figure was produced using DDE-BIFTOOL.

Figure 7

(Color online) The figure shows time integration of the Lorenz equations with feedback which is switched off at $t=50$ (indicated by a dashed vertical line). Parameter values are $\sigma =10$, $\alpha =8∕3$, $\beta =\pi ∕4$, $\rho =23$, and $\tau ={\tau }_P\left(\rho \right)=0.7191$. $b_0=1.2$ for $t⩽50$ and $b_0=0$ for $t>50$. The Pyragas orbit is initially stable, but without feedback, the trajectory decays back to the origin. The data was computed using the MATLAB routine dde23 for integrating delay-differential equations.

Figure 8

(Color online) (a) and (b) Two time integrations of the Lorenz equations, (a) has no feedback terms, and in (b) the feedback is “turned on” at $t=5$. Without feedback, the well-known chaotic strange attractor is stable, but with the feedback, a delay-induced periodic orbit becomes the stable solution. The stability of the periodic orbit has been confirmed using DDE-BIFTOOL. Parameters are $\rho =24.8388$, $b_0=0.22$, and $\tau =0.6494$. The stable periodic orbit has a period of 0.6537. (c) The time plot of (b). The data was computed using the MATLAB routine dde23.

Figure 9

(Color online) Curves of Hopf bifurcations from the zero solution of Eq. (8) with $\Gamma =b_{0}I$. Curves are shown for $b_0=0.1$, 0.05, and $-0.1$ (from right to left). The zero solution is stable to the left of these curves and unstable to the right of these curves. The dashed curve is the curve $\tau ={\tau }_P\left(\rho \right)$. For each value of $b_0$, the curve lies between $\rho ={\rho }_h$ and $\rho ={\rho }^{\star }\left(b_0\right)$.

Figure 10

The figure shows a contour plot of the magnitude of the largest Floquet multipliers for the Pyragas periodic orbit, as $\rho$ and $b_0$ are varied. The periodic orbit is unstable for all parameters shown. This figure was produced using DDE-BIFTOOL.