^{1}, Jesús M. Seoane

^{1}and Miguel A. F. Sanjuán

^{1}

### Abstract

Suppression of chaos is a relevant phenomenon that can take place in nonlinear dynamical systems when a parameter is varied. Here, we investigate the possibilities of effectively suppressing the chaotic motion of a dynamical system by a specific time independent variation of a parameter of our system. In realistic situations, we need to be very careful with the experimental conditions and the accuracy of the parameter measurements. We define the suppressibility, a new measure taking values in the parameter space, that allows us to detect which chaotic motions can be suppressed, what possible new choices of the parameter guarantee their suppression, and how small the parameter variations from the initial chaotic state to the final periodic one are. We apply this measure to a Duffing oscillator and a system consisting on ten globally coupled Hénon maps. We offer as our main result tool sets that can be used as guides to suppress chaotic dynamics.

Chaotic dynamical systems present sensitivity to initial conditions, what makes their evolution to become unpredictable for long enough times. Nonetheless, in many situations, we need to deal with systems displaying predictable behavior. A possible way to suppress chaos is by adding suitable small perturbations to a system, which typically depend on one or more parameters. If we have a thorough knowledge of the precision in the measurement of these parameters and the response of the system to their variations, we can use them to improve the predictability. This task can be performed with the help of the

*chaotic parameter set*, which in parameter space informs us about the periodicity of a certain dynamical system. Our main goal is to quantify the possibilities that a system offers to make transitions from a chaotic regime to a regular one under some given conditions. For that purpose, we introduce a new concept, defined by means of the chaotic parameter set, that we call

*suppressibility*. The suppressibility tells us for which regions in the parameter space the dynamics of a system can be more easily and in more ways suppressed. We also present two different sets that can be used as guides to suppress chaotic dynamics by the variation of a parameter: the

*suppression parameter set*and the

*set of the total accessible transitions*. The suppressibility measure is numerically tested for both, a flow and a discrete dynamical system, showing the generality of this technique.

This work was supported by the Spanish Ministry of Science and Innovation under Project No. FIS2009‐09898.

I. INTRODUCTION

II. MODEL DESCRIPTION

III. CHAOTIC PARAMETER SET

IV. EFFECTIVE SUPPRESSIBILITY OF CHAOS

V. ELIMINATING SPURIOUS PIXELS

VI. CONCLUSIONS AND DISCUSSION

## Figures

Chaotic attractor for the Duffing oscillator , with initial conditions .

Chaotic attractor for the Duffing oscillator , with initial conditions .

Chaotic parameter set for the Duffing oscillator in the parameter space. The color bar shows the value of the largest Lyapunov exponent, computed for a grid of 720 × 720 points, and using as initial condition . Both periodic (gray colored) and chaotic (non-gray colored) motions are displayed.

Chaotic parameter set for the Duffing oscillator in the parameter space. The color bar shows the value of the largest Lyapunov exponent, computed for a grid of 720 × 720 points, and using as initial condition . Both periodic (gray colored) and chaotic (non-gray colored) motions are displayed.

(a) A suppressing line of eight pixels with an uncertainty in the measurement of the suppressing parameter F of seven pixels. Two unsafe transitions are shown among all the possible. In this case, no matter which black pixel we are in, no transition to any white pixel guarantees suppression, since the distance to the closest black pixel is always smaller than the uncertainty. (b) Same suppressing line with an uncertainty of one pixel. Now only transitions to white pixels one pixel away from any black one are unsafe. (c) The suppressing line with an uncertainty of a little less than half a pixel, for which all transitions are safe.

(a) A suppressing line of eight pixels with an uncertainty in the measurement of the suppressing parameter F of seven pixels. Two unsafe transitions are shown among all the possible. In this case, no matter which black pixel we are in, no transition to any white pixel guarantees suppression, since the distance to the closest black pixel is always smaller than the uncertainty. (b) Same suppressing line with an uncertainty of one pixel. Now only transitions to white pixels one pixel away from any black one are unsafe. (c) The suppressing line with an uncertainty of a little less than half a pixel, for which all transitions are safe.

(a) Chaotic attractor for the Duffing oscillator . (b) Same chaotic attractor, though letting the driving amplitude fluctuate randomly in the interval [6.76, 6.84], what corresponds to an uncertainty . This attractor looks a bit thicker, but preserves the shape.

(a) Chaotic attractor for the Duffing oscillator . (b) Same chaotic attractor, though letting the driving amplitude fluctuate randomly in the interval [6.76, 6.84], what corresponds to an uncertainty . This attractor looks a bit thicker, but preserves the shape.

(a) A white pixel has its center in a periodic window. (b) A white pixel whose center lands in the beginning of a period doubling cascade leading to a chaotic attractor. (c) A zoom in a white pixel containing plenty of black pixels.

(a) A white pixel has its center in a periodic window. (b) A white pixel whose center lands in the beginning of a period doubling cascade leading to a chaotic attractor. (c) A zoom in a white pixel containing plenty of black pixels.

A suppressing line with unaccessible regions due to precision limitations in the measurement of a parameter denoted by question marks. Pixels at a distance to the boundary smaller than the uncertainty in the measurement of the suppressing parameter are marked in orange. Transitions involving these pixels are unaccessible.

A suppressing line with unaccessible regions due to precision limitations in the measurement of a parameter denoted by question marks. Pixels at a distance to the boundary smaller than the uncertainty in the measurement of the suppressing parameter are marked in orange. Transitions involving these pixels are unaccessible.

Suppression parameter set for in the parameter space, with suppressing parameter F, uncertainty and , where is the Heavyside function. This set shows all the safe chaotic events for which chaos can be suppressed, and which ones offer better possibilities of being suppressed, according to the conditions imposed by w. It is obtained by computing the suppressibility for every safe black pixels (i, j) and assigning each chaotic event a color depending on its value. The color bar goes from cold colors to hot ones, corresponding, respectively, to the lower (1) and higher (56) values of the suppressibility measure.

Suppression parameter set for in the parameter space, with suppressing parameter F, uncertainty and , where is the Heavyside function. This set shows all the safe chaotic events for which chaos can be suppressed, and which ones offer better possibilities of being suppressed, according to the conditions imposed by w. It is obtained by computing the suppressibility for every safe black pixels (i, j) and assigning each chaotic event a color depending on its value. The color bar goes from cold colors to hot ones, corresponding, respectively, to the lower (1) and higher (56) values of the suppressibility measure.

(a) First, second, and third order transitions in a suppressing line. Since longer transitions contribute less or equal than shorter ones, we have . This explains the monotonic decreasing character of w. (b) Two suppressing lines having the same total suppressibility but different number of black pixels or chaoticity. Note that .

(a) First, second, and third order transitions in a suppressing line. Since longer transitions contribute less or equal than shorter ones, we have . This explains the monotonic decreasing character of w. (b) Two suppressing lines having the same total suppressibility but different number of black pixels or chaoticity. Note that .

A plot of the total suppressibility χ (black line) together with chaoticity κ (red line). The former reaches its maximum for , where chaos is better spread. The more alternation of chaotic and periodic events there is, the higher the total suppressibility. This implies that the closer the chaoticity is to 0.5, the higher the total suppressibility. In this case, chaoticity reaches a maximum close to 0.4, near the value of the damping for which the maximum total suppressibility is obtained. For very high values of the damping, mainly periodic events appear, so there is little chaos to be suppressed and either χ or κ take low values.

A plot of the total suppressibility χ (black line) together with chaoticity κ (red line). The former reaches its maximum for , where chaos is better spread. The more alternation of chaotic and periodic events there is, the higher the total suppressibility. This implies that the closer the chaoticity is to 0.5, the higher the total suppressibility. In this case, chaoticity reaches a maximum close to 0.4, near the value of the damping for which the maximum total suppressibility is obtained. For very high values of the damping, mainly periodic events appear, so there is little chaos to be suppressed and either χ or κ take low values.

(a) Chaotic parameter set for in the parameter space with suppressing parameter F, uncertainty and . Unsafe regions due to uncertainty are colored orange. The thin horizontal line represents the value of the damping for which a maximum total suppressibility is obtained. (b) The set of the total accessible transitions for that maximum. The plot displays transitions involving different values of the suppressing parameter F. In the x axis, the values of the forcing for the initial state indicated with the superscript c, while in the y axis the values of the forcing for the final state in the transition, denoted by p. If the starting pixel corresponds to chaotic (c) motion and the transition leads to periodic (p) regime, the point is colored black. Otherwise the transitions are left uncolored. Note the great alternation of white and black.

(a) Chaotic parameter set for in the parameter space with suppressing parameter F, uncertainty and . Unsafe regions due to uncertainty are colored orange. The thin horizontal line represents the value of the damping for which a maximum total suppressibility is obtained. (b) The set of the total accessible transitions for that maximum. The plot displays transitions involving different values of the suppressing parameter F. In the x axis, the values of the forcing for the initial state indicated with the superscript c, while in the y axis the values of the forcing for the final state in the transition, denoted by p. If the starting pixel corresponds to chaotic (c) motion and the transition leads to periodic (p) regime, the point is colored black. Otherwise the transitions are left uncolored. Note the great alternation of white and black.

(a) Chaotic parameter set in space for ten globally coupled Hénon maps. Note the dusty regions, where chaotic (black) and periodic (white) asymptotic motion are very interspersed. (b) Same chaotic parameter set after the application of the algorithm. Spurious white pixels are marked in red.

(a) Chaotic parameter set in space for ten globally coupled Hénon maps. Note the dusty regions, where chaotic (black) and periodic (white) asymptotic motion are very interspersed. (b) Same chaotic parameter set after the application of the algorithm. Spurious white pixels are marked in red.

(a) Cleaned chaotic parameter set in the space for ten globally coupled Hénon maps. Unaccessible regions due to spurious pixels are marked in red, while those due to a lack of precision in the measurement of a parameter are marked in orange. (b) Total suppressibility χ in black together with chaoticity κ in red.

(a) Cleaned chaotic parameter set in the space for ten globally coupled Hénon maps. Unaccessible regions due to spurious pixels are marked in red, while those due to a lack of precision in the measurement of a parameter are marked in orange. (b) Total suppressibility χ in black together with chaoticity κ in red.

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