arXiv subjects

Rubén Capeáns

Publications and source records attributed to Rubén Capeáns.

12 recordsLinked to original sources

arXiv2025

AI-Driven Control of Chaos: A Transformer-Based Approach for Dynamical Systems

Chaotic behavior in dynamical systems poses a significant challenge in trajectory control, traditionally relying on computationally intensive physical models. We present a machine learning-based algorithm to compute the minimum control bounds required to confine particles within a region indefinitely, using only samples of orbits that iterate within the region before diverging. This model-free approach achieves high accuracy, with a mean squared error of $2.88 \times 10^{-4}$ and computation times in the range of seconds. The results highlight its efficiency and potential for real-time control of chaotic systems.

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arXiv2025

Stability enhanced by transient attractors in a memristive chaotic map

The introduction of a memristor in a chaotic system can significantly modify its dynamical behavior. In this paper, we couple a discrete memristor model with a chaotic map to investigate the memristor's impact on the stability of chaotic attractors. Our results reveal that introducing the memristor substantially enlarges the basin of attraction of a given chaotic attractor, thereby enhancing its stability. This phenomenon arises as a direct consequence of the memory and switching effects introduced by the memristor, which create new paths to the chaotic attractors.

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arXiv2025

Two-Player Yorke's Game of Survival in Chaotic Transients

We present a novel two-player game in a chaotic dynamical system where players have opposing objectives regarding the system's behavior. The game is analyzed using a methodology from the field of chaos control known as partial control. Our aim is to introduce the utility of this methodology in the scope of game theory. These algorithms enable players to devise winning strategies even when they lack complete information about their opponent's actions. To illustrate the approach, we apply it to a chaotic system, the logistic map. In this scenario, one player aims to maintain the system's trajectory within a transient chaotic region, while the opposing player seeks to expel the trajectory from this region. The methodology identifies the set of initial conditions that guarantee victory for each player, referred to as the winning sets, along with the corresponding strategies required to achieve their respective objectives.

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arXiv2025

A simple method to enlarge a basin of attraction using a memristive function

This study presents an innovative approach to chaotic attractor stabilization introducing a memristor in discrete dynamical systems. Using the Hénon map as a test case, we replace a system parameter with a memristive function governed by a sigmoid activation function. The method relies on leveraging attractors with larger basins of attraction to attract the orbits and guide them towards the desired chaotic attractor. The effectiveness of the method is confirmed through numerical simulations, showing substantial enhancement in attractor stability without requiring explicit parameter control.

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arXiv2024

Systematic search for islets of stability in the standard map for large parameter values

In the seminal paper (Phys. Rep. 52, 263, 1979), Boris Chirikov showed that the standard map does not exhibit a boundary to chaos, but rather that there are small islands (islets) of stability for arbitrarily large values of the nonlinear perturbation. In this context, he established that the area of the islets in the phase space and the range of parameter values where they exist should decay following power laws with exponents -2 and -1, respectively. In this paper, we carry out a systematic numerical search for islets of stability and we show that the power laws predicted by Chirikov hold. Furthermore, we use high-resolution 3D islets to reveal that the islets volume decays following a similar power law with exponent -3.

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arXiv2022

Controlling the bursting size in the two-dimensional Rulkov model

We propose to control the orbits of the two-dimensional Rulkov model affected by bounded noise. For the correct parameter choice the phase space presents two chaotic regions separated by a transient chaotic region in between. One of the chaotic regions is the responsible to give birth to the neuronal bursting regime. Normally, an orbit in this chaotic region cannot pass through the transient chaotic one and reach the other chaotic region. As a consequence the burstings are short in time. Here, we propose a control technique to connect both chaotic regions and allow the neuron to exhibit very long burstings. This control method defines a region Q covering the transient chaotic region where it is possible to find an advantageous set $S \in Q$ through which the orbits can be driven with a minimal control. In addition we show how the set S changes depending on the noise intensity affecting the map, and how the set S can be used in different scenarios of control.

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arXiv2022

Controlling two-dimensional chaotic transients with the safety function

In this work we deal with the Hénon and the Lozi map for a choice of parameters where they show transient chaos. Orbits close to the chaotic saddle behave chaotically for a while to eventually escape to an external attractor. Traditionally, to prevent such an escape, the partial control technique has been applied. This method stands out for considering disturbances (noise) affecting the map and for finding a special region of the phase space, called the safe set, where the control required to sustain the orbits is small. However, in this work we will apply a new approach of the partial control method that has been recently developed. This new approach is based on finding a special function called the safety function, which allows to automatically find the minimum control necessary to avoid the escape of the orbits. Furthermore, we will show the strong connection between the safety function and the classical safe set. To illustrate that, we will compute for the first time, safety functions for the two-dimensional Hénon and Lozi maps, where we also show the strong dependence of this function with the magnitude of disturbances affecting the map, and how this change drastically impacts the controlled orbits.

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arXiv2021

Beyond partial control: Controlling chaotic transients with the safety function

Partial control is a technique used in systems with transient chaos. The aim of this control method is to avoid the escape of the orbits from a region Q of the phase space where the transient chaotic dynamics takes place. This technique is based on finding a special subset of Q called the safe set. The chaotic orbit can be sustained in the safe set with a minimum amount of control. In this work we develop a control strategy to gradually lead any chaotic orbit in Q to the safe set by using the safety function. With the technique proposed here, the safe set can be converted into a global attractor of Q.

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arXiv2021

Forcing the escape: Partial control of escaping orbits from a transient chaotic region

A new control algorithm based on the partial control method has been developed. The general situation we are considering is an orbit starting in a certain phase space region Q having a chaotic transient behavior affected by noise, so that the orbit will definitely escape from Q in an unpredictable number of iterations. Thus, the goal of the algorithm is to control in a predictable manner when to escape. While partial control has been used as a way to avoid escapes, here we want to adapt it to force the escape in a controlled manner. We have introduced new tools such as escape functions and escape sets that once computed makes the control of the orbit straightforward. We have applied the new idea to three different cases in order to illustrate the various application possibilities of this new algorithm.

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arXiv2019

A new approach of the partial control method in chaotic systems

We present here a new approach of the partial control method, which is a useful control technique applied to transient chaotic dynamics affected by a bounded noise. Usually we want to avoid the escape of these chaotic transients outside a certain region $Q$ of the phase space. For that purpose, there exists a control bound such that for controls smaller than this bound trajectories are kept in a special subset of $Q$ called the safe set. The aim of this new approach is to go further, and to compute for every point of $Q$ the minimal control bound that would keep it in $Q$. This defines a special function that we call the safety function, which can provide the necessary information to compute the safe set once we choose a particular value of the control bound. This offers a generalized method where previous known cases are included, and its use encompasses more diverse scenarios.

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arXiv2017

Partial control of delay-coordinate maps

Delay-coordinate maps have been widely used recently to study nonlinear dynamical systems, where there is only access to the time series of one of their variables. Here, we show how the partial control method can be applied in this kind of framework in order to prevent undesirable situations for the system or even to reduce the variability of the observable time series associated with it. The main advantage of this control method, is that it allows to control delay-coordinate maps even if the control applied is smaller than the external disturbances present in the system. To illustrate how it works, we have applied it to three well-known models in Nonlinear Dynamics with different delays such as the two-dimensional cubic map, the standard map and the three-dimensional hyperchaotic Hénon map. For the first time we show here how hyperchaotic systems can be partially controlled.

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arXiv2016

Partially controlling transient chaos in the Lorenz equations

Transient chaos is a characteristic behavior in nonlinear dynamics where trajectories in a certain region of phase space behave chaotically for a while, before escaping to an external attractor. In some situations the escapes are highly undesirable, so that it would be necessary to avoid such a situation. In this paper we apply a control method known as partial control that allows one to prevent the escapes of the trajectories to the external attractors, keeping the trajectories in the chaotic region forever. To illustrate how the method works, we have chosen the Lorenz system for a choice of parameters where transient chaos appears, as a paradigmatic example in nonlinear dynamics. We analyze three quite different ways to implement the method. First, we apply this method by building a 1D map using the successive maxima of one of the variables. Next, we implement it by building a 2D map through a Poincaré section. Finally, we built a 3D map, which has the advantage of using a fixed time interval between application of the control, which can be useful for practical applications.

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