Steering multiattractors to overcome parameter inaccuracy and noise effects

Steering of attractors in multistable systems is used to increase the available parameter domains which lead to stable dynamics in nonlinear physical systems, substantially reducing the undesirable effects of parametric inaccuracy and noise. The procedure proposed here uses time and/or space asymmetric perturbations to move multistable attractors independently in phase space. Applying this mechanism, we increase by around $230\%$ the stable domains in Hénon's map, roughly $85\%$ in the ratchet current described by the Langevin equation, and $60\%$ in Chua's electronic circuit. The proposal is expected to have wide applications in a generic nonlinear complex system presenting multistability, so that related experiments can increase robustness under parametric inaccuracy and noise.

Figure 1

Enlargement of Hénon's stable domains in two-dimensional parameter space ($a,b$) for noise intensities $7\times {10}^{-5}$ (top row) and $2\times {10}^{-4}$ (bottom row).

Figure 2

Two-dimensional parameter space $(K,\chi )$ for the Langevin equation plotting the ratchet current in (a) $F=0.0$ and (b) $F=0.3$, and the number of stable attractors in (g) for $F=0.0$ and (h) for $F=0.3$. (c)–(f) Thermal effects using $T={10}^{-4}$ for the regions selected by colored boxes in (a) and (b). (i) The stable attractors in phase space; (j) the distance $d$ between the black dots in (i), which are the maxima of velocity $\dot{x}$ for the attractors red (dark gray) and green (light gray).

Figure 3

Two-dimensional parameter spaces for Chua's circuit showing the number of stable attractors in (a) $A=0.0$ and (b) $A=0.3$. (c) The two attractors found in the SS indicated by the black arrow in the phase space; (d) the distance in coordinate $z$ between the black points as a function of $A$. (e)–(h) Magnifications of the black box displayed in (a) plotting the largest LE using (e) $A=0.0$, $D=0.0$, (f) $A=0.2$, $D=0.0$, (g) $A=0.0$, $D={10}^{-5}$, and (h) $A=0.2$, $D={10}^{-5}$.