Persistence of chaos in a time-delayed-feedback controlled Duffing system

This paper concerns global phase structures of a time-delayed-feedback controlled two-well Duffing system. The remains of a global stretch and fold structure along an unstable manifold, which develops from an unstable fixed point in function space, reveals that the global chaotic dynamics is inherited from the original system by the controlled system. The remains of the original chaotic dynamics causes a highly complicated domain of attraction for target orbits and a long chaotic transient before convergence.

Figure 1

Unstable period-$2\pi$ orbits embedded in original chaotic attractor [23]. ${}^1\mathsf{I}$ and ${}^1\mathsf{I}^{\prime }$ denote inversely unstable periodic orbits of the target, which are stabilized under sufficient large amplitude of feedback gain. ${}^1\mathsf{D}$ is a directly unstable periodic orbit, which cannot be stabilized because of the odd number condition. In Figs. 2c, 2d, 4, 5 below, the notation ${}^1\mathsf{I}$ and ${}^1\mathsf{I}^{\prime }$ is replaced by ${}^1\mathsf{S}$ and ${}^1\mathsf{S}^{\prime }$, respectively, to reflect the stability change of the target orbits.

Figure 2

Unstable manifold of ${}^1\mathsf{D}$ (projection on a two-dimensional stroboscopic plane). In (a) and (b), the target orbits are unstable and the chaotic attractor is generated, as shown by gray stroboscopic points. In (c) and (d), the targets are stable and stroboscopic points show transient behavior before convergence to a target orbit. The notation of target orbits is changed to ${}^1\mathsf{S}$ and ${}^1\mathsf{S}^{\prime }$ because of their stability change. Arrows in (c) and (d) indicate the same stroboscopic point at which the control is activated.

Figure 3

Transient behavior in the time domain for $K=0.75$. Scale of time axis (not renormalized) is the same in (a) displacement and (b) control input. Controlled trajectory irregularly goes back and forth between two target orbits before final convergence to ${}^1\mathsf{S}$, corresponding to chaotic motion of stroboscopic points in Fig. 2c. The same time scale is used in Fig. 6 below.

Figure 4

Domain of attraction for target orbits at $K=0.75$. Black and gray points show convergence to ${}^1\mathsf{S}$ and ${}^1\mathsf{S}^{\prime }$, respectively. They are finely mixed with one another all over the original chaotic attractor.

Figure 5

Domain of attraction for target orbits at $K=1.1$. Black and gray points show convergence to ${}^1\mathsf{S}$ and ${}^1\mathsf{S}^{\prime }$, respectively. Most of the controlled trajectories converge to the target orbit in their initial side.

Figure 6

Transient behavior for $K=1.1$. Time scale (not renormalized) is the same as in Fig. 3. Corresponding to the simple global structure in Fig. 2d, the controlled trajectory quickly converges to ${}^1\mathsf{S}^{\prime }$ without any irregular behavior.