Exploring partial control of chaotic systems

In this paper we make a thorough exploration of the technique of partial control of chaotic systems. This control technique allows one to keep the trajectories of a dynamical system close to a chaotic saddle even if the control applied is smaller than the effects of environmental noise in the system, provided that the chaotic saddle is due to the existence of a horseshoelike mapping in phase space. We state this here in a mathematically precise way using the Conley-Moser conditions, and we prove that they imply that our partial control strategy can be applied. We also give an upper bound of the control-noise ratio needed to achieve this goal, and we describe how this technique can be applied for large noise values. Finally, we study in detail the effect of imperfect targeting in our control technique. All these results are illustrated numerically with the paradigmatic Hénon map.

Figure 1

A pictorial representation of the conditions C1 and C2. Condition C1 says that inside the square $Q$ there are at least two vertical strips $V_1$ and $V_2$ that are mapped to two horizontal strips $H_1$ and $H_2$, in such a way that horizontal bounds of $H_i$ (gray) are mapped into horizontal bounds (gray) of $V_i$, and the same applies to vertical bounds (black) of $H_i$ and $V_i$. The minimum distance between the top or the bottom of $Q$ and the strips $H_1$ and $H_2$ is noted $\Delta$. Condition C2 says that if we apply the map $\mathbf{f}$ to a horizontal strip $H$ and we look at the intersection of $\mathbf{f}\left(H\right)$ with $H_1$ and $H_2$, we will find two horizontal strips thinner than $H$. A similar phenomenon occurs when we apply ${\mathbf{f}}^{-1}$ to a vertical strip $V$ and we look at the intersection of ${\mathbf{f}}^{-1}\left(V\right)$ with $V_1$ and $V_2$.

Figure 2

The action of the Hénon map $x_{n+1}=6-0.4y_n-x_n^2$, $y_{n+1}=x_n$ in a square $Q$ on the plane. This map satisfies conditions C1 and C2, so our technique can be applied.

Figure 3

The safe sets for the Hénon map $S^0$ (thick line), $S^1$ (black line), $S^2$ (gray line), and $S^3$ (light gray line). The square $Q$ (−) and its image under the map (thick gray) are also plotted for the sake of clarity. Note that they fulfill properties (i)–(iii): The set $S^k$ consists of $2^k$ vertical curves, each of them having two adjacent vertical curves of $S^{k+1}$ closer to it than any other curve of $S^k$, and the distance with these adjacent curves decreases with $k$.

Figure 4

The action of the map $\mathbf{f}$ on the square $Q$ (dashed line). If we assume that points to the right of the right-hand side of $Q$ settle after iterates into a periodic attractor $A_1$, and that points to the left of the left-hand side of $Q$ settle after iterates into a periodic attractor $A_2$, it can be shown that this type of mapping implies the appearance of fractal boundaries between the basins of attractions of those attractors. But this type of geometrical action would also allow us to build safe sets that can keep trajectories inside $Q$ in the presence of noise with $r_0<u_0$.

Figure 5

A trajectory starting in ${\mathbf{p}}_{\mathbf{0}}$ belonging to $S^3$ (light gray line) is mapped to $\mathbf{f}\left({\mathbf{p}}_{\mathbf{0}}\right)$, that belongs to $S^2$ (gray line). The noise action can either deviate it to the region between the two adjacent curves belonging to $S^3$, as ${\mathbf{u}}_{\mathbf{0}}^{\prime }$, or take it outside this region, as ${\mathbf{u}}_{\mathbf{0}}$. In any case, trajectories can be placed again on $S^3$ by applying a perturbation (either ${\mathbf{r}}_{\mathbf{0}}^{\prime }$ or ${\mathbf{r}}_{\mathbf{0}}$) bounded by a $r_0$ such that $r_0<u_0$, and this can be repeated forever.

Figure 6

Example of application of our strategy to the Hénon map for $u_0=0.25$. (a) A plot of 1000 points or the orbit, that lie on $S^3$, the suitable safe set for this value of noise. The square $Q$ and its image under the map (dashed line) are also plotted for the sake of clarity. The inset plot shows a zoom of the square, showing that the points of the trajectory always fall on the curves of $S^3$. (b) The value of the control applied for each iteration is to keep the trajectories on $S^3$ and thus on $Q$. Note that it is always smaller than $u_0=0.25$ (marked with a dashed line), as expected.

Figure 7

Control-noise ratio needed to partially control the Hénon map, computed numerically taking 1000 trajectories of 1000 iterations (dots). We also plot (solid) the upper bounds for this ratio given by the expression derived in the text. In order to use those expressions, numerical estimates of the values of ${\delta }_{\max }^k$ and ${\delta }_{\min }^k$ are needed.

Figure 8

A possible situation arising when $u_0>\Delta$. A trajectory starting in ${\mathbf{p}}_{\mathbf{0}}$ belonging to $S^3$ (light gray line) is mapped to $\mathbf{f}\left({\mathbf{p}}_0\right)$, that belongs to $S^2$ (gray line), a point that is inside the horizontal strip $H_1$ (bounds in thick gray lines). If $u_0>\Delta$ the point $\mathbf{f}\left({\mathbf{p}}_0\right)+{\mathbf{u}}_0$ can lie out of $Q$, whose lower bound is also represented here (dashed). But if we have chosen adequately the set $S^k$, we can see here that with a control ${\mathbf{r}}_0$ bounded by a $r_0$ such that $r_0<u_0$, the trajectory can be lead to $S^k$, and this can be repeated forever.

Figure 9

Example of application of our strategy to the Hénon map for $u_0=0.25$, with a control noise such that $\Delta r_0\approx 0.1r_0$. (a) A plot of 1000 points or the orbit, that lie on $S^3$, the suitable safe set for this value of noise. The square $Q$ and its image under the map (dashed line) are also plotted for the sake of clarity. The inset plot is a zoom of the square showing that now the points do not fall exactly on the safe set $S^3$, due to the presence of control noise. (b) The value of the control applied to each iteration to keep the trajectories on $S^3$ and thus on $Q$. Note that it is always smaller than $u_0=0.25$ (marked with a dashed line), as expected, in spite of the presence of control noise.