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October  2018, 23(8): 3237-3274. doi: 10.3934/dcdsb.2018241

Partial control of chaos: How to avoid undesirable behaviors with small controls in presence of noise

 1 Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain 2 Institute for New Economic Thinking at the Oxford Martin School, Mathematical Institute, University of Oxford, Walton Well Road, Eagle House OX2 6ED, Oxford, UK 3 Department of Applied Informatics, Kaunas University of Technology, Studentu 50-415, Kaunas LT-51368, Lithuania 4 Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA

Received  December 2017 Revised  March 2018 Published  August 2018

The presence of a nonattractive chaotic set, also called chaotic saddle, in phase space implies the appearance of a finite time kind of chaos that is known as transient chaos. For a given dynamical system in a certain region of phase space with transient chaos, trajectories eventually abandon the chaotic region escaping to an external attractor, if no external intervention is done on the system. In some situations, this attractor may involve an undesirable behavior, so the application of a control in the system is necessary to avoid it. Both, the nonattractive nature of transient chaos and eventually the presence of noise may hinder this task. Recently, a new method to control chaos called partial control has been developed. The method is based on the existence of a set, called the safe set, that allows to sustain transient chaos by only using a small amount of control. The surprising result is that the trajectories can be controlled by using an amount of control smaller than the amount of noise affecting it. We present here a broad survey of results of this control method applied to a wide variety of dynamical systems. We also review here all the variations of the partial control method that have been developed so far. In all the cases various systems of different dimensionality are treated in order to see the potential of this method. We believe that this method is a step forward in controlling chaos in presence of disturbances.

Citation: Rubén Capeáns, Juan Sabuco, Miguel A. F. Sanjuán. Partial control of chaos: How to avoid undesirable behaviors with small controls in presence of noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3237-3274. doi: 10.3934/dcdsb.2018241
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Transient chaos. The figure illustrates the chaotic transient behavior of a trajectory. Starting in the red point, the trajectory falls in the chaotic region where it remains for a while. After a finite amount of time, the trajectory escapes from the chaotic region towards an external attractor (blue fixed point). The transient chaotic behavior is produced by the presence of a chaotic saddle in phase space. This invariant set is a non-attractive chaotic set and this is the reason why the trajectory eventually escapes. The goal of applying control is to sustain the chaotic behavior forever, avoiding the escape of the trajectories
Example of the set needed to partially control the Lorenz system. The figure shows an example of a safe set in the phase space computed for the partially controlled Lorenz system in the transient chaotic regime. The blue set represents the points of the phase space that satisfy the control condition defined by the partial control method. The red set is a subset of the blue set, and represents the asymptotic region where the controlled dynamics converges
Graphical process used by the Sculpting Algorithm to obtain the safe set. The denoted set $Q_i$ is fattened by the thickness $u_0$. The fattened set is displayed in red. Then, the new set is shrunk or contracted by a distance $\xi_0$, obtaining the set denoted $Q_i+u_0-\xi_0$ (in green). Finally we remove the grid points $q \in Q_i$ whose image $f(q)$ falls outside $Q_i+u_0-\xi_0$. Notice that $Q_{i+1} \subset Q_i$
Sequence for computing the safe set in the Hénon map. The map ($x_{n+1} = 2.16-0.3y_{n}-x^{2}_{n};~ y_{n+1} = x_{n}$). The initial region $Q_0$ is the square $[-5, 5]\times[-5, 5]$, (a grid of $3000\times 3000$ points), and the values $\xi_0 = 0.3$ (green circle) and $u_0 = 0.18$ (yellow circle). At each step, part of $Q_n$ is removed (magenta region) while the blue region remains. After $12$ steps the safe set converges
Safe set of the Hénon map. In this figure we can see the result of applying the Sculpting Algorithm to the Hénon map, $x_{n+1} = 2.16-0.3y_{n}-x^{2}_{n}; y_{n+1} = x_{n}$. The safe set appears in blue. The minimum control allowed so that it exits a safe set is $u_0 = 0.18$ for the given disturbance of $\xi_0 = 0.3$. This is equal to a ratio of $\rho = 0.6$
Periodic attractors of the Duffing oscillator. (a) In this figure we show the complex structure of the phase space for the Duffing oscillator $\ddot{x}+0.15\dot{x}-x+x^{3} = 0.245\sin(t)$. In this system are present three different basins of attraction (magenta, blue and green) and the system has the Wada property. The invariant unstable manifold associated to the chaotic saddle appears in yellow. (b) This figure reproduces the periodic attractors: two period-1 attractors and one period-3 attractor. We also show with circles of radius $0.2$ the region of the phase space that we want to avoid, whatever the disturbances. (c) We use a grid of $6000\times 6000$ points in the square $[-2, 2]\times[-2, 2]$ as our initial set, but removing the zones that we want to avoid, that is the circles. Applying the Sculpting Algorithm over several iterations, we will obtain the desired safe set. We let $\xi_0 = 0.08$ be the maximum size of the vector perturbation
Application of the Sculpting Algorithm to the Duffing oscillator. This figure shows the sequence for creating the safe set in the Duffing oscillator. At each step, part of $Q_n$ is removed. The blue color represents the part of the set that remains and the magenta the part that is to be removed. Only 12 iterations out of the 15 appear. The small green and yellow circle represents the intensities of the control and disturbance used ($u_0 = 0.0475$ and $\xi_0 = 0.08$)
Safe set of the Duffing oscillator. In this figure we can see the result of applying the Sculpting Algorithm to the Duffing oscillator $\ddot{x}+0.15\dot{x}-x+x^{3} = 0.245\sin(t)$. The safe set appears in blue. The minimum control allowed, so that it exits a safe set is $u_0 = 0.0475$ (yellow circle), with a maximum disturbance of $\xi_0 = 0.08$ (green circle). This is equal to a safe ratio $\rho\approx 0.59$
Dynamics of the extended McCann-Yodzis model proposed by Duarte et al from Eqs. (3). Depending on the values of the parameters $(K, \sigma)$ different dynamics are possible. Fixing $K = 0.99$, the boundary crisis appears at $\sigma_c = 0.04166$. (a) Before the boundary crisis ($K = 0.99$, $\sigma = 0$), there are two possible attractors depending on the initial conditions: one chaotic attractor where the three species coexist, and one limit cycle where only the resources and consumers coexist. (b) After the boundary crisis ($K = 0.99$, $\sigma = 0.07$), the limit cycle is the only asymptotic attractor. (c) Time series of the predators population corresponding to the case $(b)$, where the chaotic transient before the extinction is shown
Return map $P_{n+1} = f(P_n)$ map obtained by using the successive local minima of the time series $P(t)$. Notice that below $P^{*} = 0.589$ the trajectory asymptotes to zero. In order to keep the trajectory in the region $Q$ indicated we compute the safe set. In the lower part it is shown the steps of the Sculpting Algorithm that converges to the final safe set. The horizontal black bars helps us to visualize the process and represent the points $P_n$ that satisfy the condition to be a safe point at each step. In this case, the upper bound disturbance and control used are $\xi_0 = 0.0114$ and $u_0 = 0.0076$, respectively
Final safe set. The safe set is composed of different subsets obtained with the Sculpting Algorithm using $\xi_0 = 0.0114$ and $u_0 = 0.0076$. We also indicate the group of subsets where the dynamics remains trapped, that is, the asymptotic safe set
Controlled time series controlled. Red line: Time series of the predators population without control exhibiting a escape towards zero, what implies the extinction of the predators. Blue line: Controlled time series of the predators population where the extinction is avoided. At every minimum, the value of $P$ is evaluated and if necessary a small control is applied. This time series corresponds to $50$ iterations in the return map $P_{n+1} = f(P_n)$. A zoom of one of the minima of the time series of $P(t)$ is also shown on the right in order to see how the noise is introduced and how the corresponding control is applied
Dynamics of the Lorenz system. We select the transient chaotic regime with $\sigma = 10$, $b = 8/3$ and $r = 20$. On the left, the trajectory is deterministic. On the right, the trajectory is affected by some disturbances. The disturbances here, were enlarged in order to help the eye. Almost all trajectories eventually spiral to one of the two attractors ($C^{+}$ or $C^{-}$). Here both trajectories spiral to $C^{+}$
The 1D safe set. The black curve is the 1D map built with the successive maxima of $z$. We take as initial set $Q_0$ (upper segment in blue) the region where transient chaos occurs. The map is affected by disturbances with an upper bound $\xi_0 = 0.080$, while we choose the upper bound of the control as $u_0 = 0.055$, (the bounds are the width of the bars displayed in the upper left side). The figure shows the successive steps computed by the Sculpting Algorithm, from an initial region $Q_0$ until it converges to the subset $Q_4 = Q_\infty \subset Q_0$. We use a grid of $4000$ points in the interval $z_n \in [26.8, 30.8]$, that corresponds to a resolution of $0.001$
Time series of the variable $z$ for the Lorenz system with $r = 20$. The figure shows a comparison between an uncontrolled trajectory that escapes from chaos (red line) and a partially controlled trajectory (black line). Starting with the same initial condition, the uncontrolled trajectory eventually decays to $C^{+}$ or $C^{-}$, which physically means a steady rotation of the fluid flow. On the other hand the partially controlled trajectory is maintained in the chaotic transient regime, that is, the rotation of the fluid flow remains chaotic forever
The Lorenz system with $r = 20$ (transient chaos). The figure shows an uncontrolled trajectory in phase space crossing a square with $x\in[-3, 3]$ and $y\in[-3, 3]$ in the plane $z = 19$. To built the map, we use a grid of initial conditions in the plane, and evaluate the images of the trajectories when they cross again the plane. The goal of the control will be to keep the trajectories in this plane, avoiding the escape to one of the attractors $C^+$ or $C^-$, placed outside
The 2D safe set and how it is used to control the trajectory. (a) The safe set obtained using the map built with the plane displayed in Fig. 16. We show in blue the computed safe set $Q_\infty$ for $\xi_0 = 0.09$ and $u_0 = 0.06$ ($u_0<\xi_0$). The grid size used is $1201 \times 1201$ points. The radius of the balls in the lower left side indicates the bounds of the disturbance, $\xi_0 = 0.09$ (green) and the control $u_0 = 0.06$ (yellow). (b) A partially controlled trajectory in phase space for case. Each time that the trajectory crosses the safe set plane (placed in $z = 19$), the control is applied pushing the trajectory onto the set avoiding the escape from chaos. (c) Zoom of how the control is applied in the safe set
A choice of 3D set $Q$. The 3D set $Q$ is the cube $x\in[-20, 20]$, $y\in[-20, 20]$, $z\in[0, 40]$ except that the balls of radius $4$, centered in $C^+ = (7.12, 7.12, 19)$ and $C^- = (-7.12, -7.12, 19)$ are removed from $Q$. We want trajectories to stay in $Q$ and not fall to these attractors. A trajectory is plotted to show the chaotic transient behavior in this region
The 3D safe set and how it is used to control the trajectory. (a) In blue the 3D safe set $Q_\infty$ for Fig. 18, obtained after applying the Sculpting Algorithm. We set $\Delta t = 1.2$, $\xi_0 = 1.5$ ($\xi_0 =$ radius of the green ball) and $u_0 = 1.0$ ($u_0 =$ yellow ball's radius). In red the asymptotic safe set which is a subset of the safe set. This is the region in which the controlled trajectories eventually lie. (b) The asymptotic safe set alone. Partially controlled trajectories converge rapidly to this region. (c) A cut-away section of the asymptotic safe set in order to see a partially controlled trajectory (with $\Delta t = 1.2$) displayed in black. The controls (yellow segments inserted in the trajectory) are applied every $\Delta t = 1.2$. As a result, the trajectory is kept in the chaotic region and the attractors $C^+$ and $C^-$ are avoided. (d) Zoom in the small cube displayed in Fig. 19(c). Only few lines are displayed for a better visualization. The controls (yellow segments) are applied to move the trajectories (in black) into the asymptotic safe set (in red)
Different safe set for different values of $\Delta$ time. (a) The asymptotic safe set computed for $\Delta t = 1.8$. To compute this set we have taken $\xi_0 = 1.5$ (green ball) and $u_0 = 1.0$ (yellow ball).(b) A half section of the asymptotic safe set (red) and a partially controlled trajectory (in black). In this case the controls (yellow segments inserted in the trajectory) are applied every $\Delta t = 1.8$
Comparison of the three controlled trajectories of the $z$ variable obtained with the 3D, 2D and 1D map respectively. The marks indicate the moment where the control is applied. Only in the 3D case are the controls time periodic
Scheme of the parametric partial control. The red arrow shows the mapping of a point $q$, under the application of a random map in which a parameter $p$ is affected by a bounded disturbance $|\xi_n|<\xi_0$. The green arrow shows the mapping of a point $q$, once the control $u_n$ was applied to the parameter to keep the point in the blue region. Given the upper values of the disturbance $\xi_0$ and the control $u_0<\xi_0$, the partial control method removes the points of the blue region that need a control $|u_n|>u_0$ for some possible $|\xi_n|<\xi_0$. For every point we have to evaluate all possible disturbances $|\xi_n|<\xi_0$. Once the "bad" points are removed, a new region $Q_1\subset Q_0$ is obtained. Iterating this process until it converges, we get a final region $Q_k\subset...\subset Q_1\subset Q_0$. We call this region, the parametric safe set
Logistic map where the parameter $r$ is affected by disturbances. (a) Logistic map $x_{n+1} = r x_n(1-x_n)$ where the parameter $r = 5$ is affected by disturbances with upper bound $\xi_0 = 0.6$. The black wide curve is obtained for all possible values of the parameter, $r\in[5-\xi_0, 5+\xi_0]$, of the logistic map. In red, we show an example of an uncontrolled trajectory that after a chaotic motion in $Q_0$, escapes to minus infinity. (b) We apply the partial control method to the logistic map, with $\xi_0 = 0.6$ and $u_0 = 0.5$ and a grid resolution of $0.001$, to obtain the parametric safe set which is shown with the wide blue segments to help the visualization. The orbits starting in this set, remain there after applying a control $u_n\leq 0.5$ every iteration. In red, we show an example of a partially controlled trajectory. We are plotting only 50 iterations
The Hnon map where the parameter $b$ is affected by disturbances. (a) An uncontrolled trajectory in the Hnon random map with $a = 2.16$ and $b = 0.3$. The parameter $b$ is affected by disturbances with upper bound $\xi_0 = 0.20$. The blue square $[-4, 4] \times [-4, 4]$ is the region $Q_0$. In absence of an external control, the trajectories in $Q_0$ escape outside the square after a very short chaotic transient. An example of an uncontrolled trajectory is displayed with the red points connected by the green lines. (b) The partial control method has been applied to keep trajectories in $Q_0$ forever. The upper bound of control is $u_0 = 0.15$. The grid resolution taken is $0.01$ and the parameter resolution is $0.005$. As a result, the parametric safe set (in blue) is obtained. All the orbits of the map starting in the blue set, remain there after the application of controls smaller than $u_0 = 0.15$. The red points display a partially controlled trajectory, where 20000 iterations of the trajectory have been plotted. (c) For this case the upper value of control is $u_0 = 0.036$, the grid resolution used is $0.001$ and the parameter resolution $0.0005$. In we compare it with the previous figure, we see that the appearance of the parametric safe set is more complex, due to fact that the disturbance value is smaller
Controlled trajectory in the Duffing oscillator with $\xi_0 = 0.020$ and $u_0 = 0.014$. Numbers indicates the three attractors of the system, two period-1 and one period-3. The aim of applying control is to avoid trajectories falling in these attractors. After removing the holes, corresponding to the attractors, the safe set (in blue) was computed with a grid of $1000\times1000$, (grid resolution $0.004$, parameter resolution $0.0002$). The red dots represent a controlled trajectory made up of $30000$ iterations in the stroboscopic map
Conceptual framework. From left to right. Step $1$: data acquisition from a chaotic system. We assume here that only one variable is observable. Step $2$: using embedding and parametric reconstruction techniques, construct a delay-coordinate map. The term $\xi_n$ represents a disturbance term that encloses all possible deviations from the real dynamics. Step $3$: apply the partial control method introducing and additive control term $u_n$ acting on the observable variable. In this work we assume that we already possess the knowledge of the delay-coordinate map
Dynamics in $Q_0$ and $Q_\infty$. The left side shows an example of a $2D$ region $Q_0$ (in blue) in which we want to keep the dynamics described by $x_{n} = f(x_{n-1}, x_{n-2})+\xi_n+u_n$. We say that $|\xi_n| \leq \xi_0$ is a bounded disturbance affecting the map, and $u_n$ is the control chosen so that $q_{n+1}$ is again in $Q_0$. Notice that disturbance and control arrows are drawn parallel to current state of the variable since only the present state is affected by them. To apply the control, the controller only needs to measure the state of the disturbed system, that is $[f(x_{n-1}, x_{n-2})+\xi_n]$. The knowledge of $f(x_{n-1}, x_{n-2})$ or $\xi_n$ individually is not required. The right side of the figure, shows the region $Q_\infty \subset Q_0$ (in blue), called the safe set, where each $(x_{n-1}, x_{n-2}) \in Q_\infty$ has $(x_{n}, x_{n-1}) \in Q_\infty$ for some control $|u_n|\leq u_0$, which is chosen depending on $\xi_n$. Notice that the removed region does not satisfy $|u_n|\leq u_0$
Time series of the two-dimensional cubic map for different disturbances. (a) Time series of variable $x_n$ with no disturbance affecting it. (b) Time series with $|\xi_n|\leq\xi_0 = 0.02$ affecting the map. (c) Time series with $|\xi_n|\leq\xi_0 = 0.20$ affecting the map. After some iterations the trajectory escapes towards $-\infty$
Safe set and controlled dynamics in the two-dimensional delayed cubic map ($\bf{x_{n} = ax_{n-1}-bx_{n-2}-x_{n-1}^3}$). (a) In blue the initial region $Q_0$ where we want to keep the trajectories. (b) The safe set obtained with the values of disturbance $\xi_0 = 0.020$ and control $u_0 = 0.015$. A grid of $1000 \times 1000$ points has been used. The red dots represent $1000$ iterations of a partially controlled trajectory. (c) In the top it is represented an uncontrolled time series affected with $\xi_n\leq\xi_0 = 0.020$. In the bottom the controlled time series corresponding to the red dots shown in case b
Safe set and controlled dynamics for the 3D delayed Hnon map ($\bf{z_{n} = 1-az_{n-1}^2+bz_{n-2}-cbz_{n-3}}$). (a) The safe set computed for the parameter values $a = 1.1$, $b = 0.3$, $c = 1$. A grid of $1000\times1000\times1000$ was taken in the box $[-2, 2] \times [-2, 2]\times [-2, 2]$ that represents the initial region $Q_0$. Taking the upper bound of the disturbance $\xi_0 = 0.12$ and the control $u_0 = 0.08$, the safe set converges after 15 iterations. (b) The safe set is represented in transparent blue to see the controlled trajectory inside (red dots). The variable $z_n$ is affected by a random disturbance with upper bound $\xi_0 = 0.12$ and control $u_0 = 0.08$. (c) Comparison between an uncontrolled trajectory and a controlled one in the 3D delayed Hnon. In black, the uncontrolled trajectory which after some iterations escapes to $-\infty$. In red, the controlled trajectory. For a fair comparison, both trajectories start with the same initial condition and are affected by the same sequence of random disturbances
Controlled dynamics in the logistic map. The black line in figures (a) and (c), represents the logistic map for the parameter $r = 4.1$. For this value, transient chaos appears and orbits starting in the interval $[0, 1]$ eventually escape to $-\infty$. In order to apply the control, the safe set was computed for the value of disturbance $\xi_0 = 0.03$ and control value $u_0 = 0.02$. The safe set is showed with thick blue bars to improve the visualization. In the first $20$ iterations (green points) the control is applied to return the orbit to the safe set. After that, in figure (a) the orbit is free to escape (no control is applied). However in figure (c) the orbit is forced to escape (red points). In figures (b) and (d) the corresponding time series are displayed. Notice that, by inducing the escape, the time to abandon the interval $[0, 1]$ is greatly reduced
Average escape times. The figure represents the interval $[0, 1]$ and the safe set (in blue) for the same conditions as the previous figure. The upper red line shows the average escape time when the orbit abandons the interval $[0, 1]$ without the application of any perturbation. The lower black line shows the average escape time when the orbits are forced to escape by applying small controls. In this way, the trajectory escapes about $2.5$ times faster than without control
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