Intermittent feedback induces attractor selection

We present a method for attractor selection in multistable dynamical systems. It involves a feedback term that is active only when the dynamics of the system is in a particular fraction of state space of the attractor. We implement this method first on a simplest symmetric chaotic flow and then on a bistable neuronal system. We find that adding this space-dependent feedback term to the dynamical equations of these systems will drive the dynamics to the desired attractor by annihilating the other. We further demonstrate that the attractor selection due to this feedback term can be used in construction of logic gates, which is one of the practical applications of the proposed method.

Figure 1

Projections of trajectories of the coexisting chaotic attractors onto the (a) $x-y$ (b) $x-z$, and (c) $y-z$ planes generated by Eq. (3) for $a=2.1$ and $c=2$ at $\epsilon =0$. (d) The shaded region of the attractor $A_1$ represents the fraction of state space where the feedback term is active. (e) Desired state for $b=+\sqrt{c}$ at $\epsilon =0.16$. (f) Basin stability $S_{B_{1,2}}$ [37] for coexisting attractors $A_{1,2}$ as a function of feedback strength $\epsilon$. (g) Largest Lyapunov exponent (solid line) as a function of feedback strength $\epsilon$. The dashed line denotes the Lyapunov exponent of the attractor $A_2$ and is drawn for reference.

Figure 2

Values of ${\epsilon }_{th}$ for various feedback active regions (shown by different colors) on attractor $A_1$ of Eq. (3). Here the attractor space is divided into (a) $n=1$, (b) $n=4$, (c) $n=8$, and (d) $n=16$ intervals.

Figure 3

Variation of fixed points of the system described by Eq. (3) with feedback strength $\epsilon$.

Figure 4

(a) The two coexisting periodic attractors of the neuronal system described by Eq. (5). The shaded region shows the feedback active subspace of the state space. (b) The desired state at $\epsilon =0.06$.

Figure 5

Schematic diagram of a bistable chaotic system as a logic unit in the presence of feedback.

Figure 6

Time series of the input (dashed line) and output (solid line) for the (a) nor gate and (b) nand gate and time series of the variable $x$ (solid line) and input (dashed line) for the (c) nor gate (d) nand gate.

Figure 7

(a) Coexisting chaotic attractors for the system defined by Eq. (8) in the presence of noise for $d=0.015$ at $\epsilon =0$. (b) Targeted state for $b=\sqrt{c}$ at $\epsilon =0.2$. (c) The time series of the attractor switching with the start time of feedback is 100 units.