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In this paper we address the problem of controlling chaos by using a selective filter inserted in a negative feedback loop. This has been achieved in driven double-well Duffing oscillator in a parameter region exhibiting both a boundary crisis and generalized bistability. The optimization of the filter response allows us to improve the reduction of the control signal when the periodic solution competing with the chaotic attractor is approached. The advantage in using a selective frequency approach with respect to the Pyragas's method mainly resides in the tunability of the maximum of its amplitude response and zero phase condition. Numerical simulations performed in the temporal domain match with the experiment and make us glimpse the potential of an automatic adaptive strategy adopting filter parameter variations in order to reach an optimal feedback signal reduction.

International audience

International audience; We revisit the model of the laser with feedback and the minimal nonlinearity leading to chaos. Although the model has its origin in laser physics, with peculiarities related to the [Formula: see text] laser, it belongs to the class of the three-dimensional paradigmatic nonlinear oscillator models giving chaos. The proposed model contains three key nonlinearities, two of which are of the type [Formula: see text], where [Formula: see text] and [Formula: see text] are the fast and slow variables. The third one is of the type [Formula: see text], where [Formula: see text] is an intermediate feedback variable. We analytically demonstrate that it is essential for producing chaos via local or global homoclinic bifurcations. Its electronic implementation in the range of kilo Hertz region confirms its potential in describing phenomena evolving on different time scales.

We report experimental evidence of the destabilization of a 3D torus obtained when a small subharmonic perturbation is added to a 2D torus characteristic of a driven relaxation oscillator. The Poincaré sections indicate that the torus breakup is sensitive to the phase difference between the main driving frequency and its first subharmonic perturbing component. The observed transition confirms the Newhouse, Ruelle and Takens quasiperiodic transition to chaos on a 3D torus. Numerical results on a sinusoidally perturbed circle map mirror the experimental results and confirm the key role of the phase difference in the transition between distinct dynamical regimes.

International audience; In 1976, Leon Chua showed that a thermistor can be modeled as a memristive device. Starting from this statement we designed a circuit that has four circuit elements: a linear passive inductor, a linear passive capacitor, a nonlinear resistor and a thermistor, that is, a nonlinear "locally active" memristor. Thus, the purpose of this work was to use a physical memristor, the thermistor, in a Muthuswamy-Chua chaotic system (circuit) instead of memristor emulators. Such circuit has been modeled by a new three-dimensional autonomous dynamical system exhibiting very particular properties such as the transition from torus breakdown to chaos. Then, mathematical analysis and detailed numerical investigations have enabled to establish that such a transition corresponds to the so-called route to Shilnikov spiral chaos but gives rise to a "double spiral attractor". Michael Faraday (1791-1867) is generally well known for his contributions to the study of electromagnetism and electrochemistry. However, according to Orton 1 , while investigating the effect of temperature on the conductivity of "sulphuret of silver" (silver sulfide) in 1833 he discovered what is now considered as a thermistor 2. Thermistor, i.e., thermal resistor, is thus an electrical-resistance element made of a semiconducting material the resistance of which varies with temperature. Thermistor production was then difficult, and applications were limited. One century had to pass before thermistors became commonly used by commercial manufacturers. In the 1930s, Samuel Ruben (1900-1988) invented the first commercial thermistor that he called "electrical pyrometer resistance" (Patents N o 2,021,491). He explained that: This invention relates to an electrical pyrometer and specifically to one utilizing the resistance change of a metallic compound with heat to indicate temperature changes. The characteristic property of the material employed for the temperature indicating resistance element is one having a high negative resistance coefficient.

We propose an electronic implementation to record Poincare sections of dynamical systems exhibiting chaos. Poincare sections are obtained by sampling and holding the maxima of a sequence of pulses of a chaotic relaxation oscillator versus the same temporal sequence shifted by one unit. By using these sections we are able to detail the transition to chaos via torus breakdown.

Dynamical control of entanglement and its connection with the classical concept of instability is an intriguing matter which deserves accurate investigation for its important role in information processing, cryptography and quantum computing. Here we consider a tripartite quantum system made of three coupled quantum parametric oscillators in equilibrium with a common heat bath. The introduced parametrization consists of a pulse train with adjustable amplitude and duty cycle representing a more general case for the perturbation. From the experimental observation of the instability in the classical system we are able to predict the parameter values for which the entangled states exist. A different amount of entanglement and different onset times emerge when comparing two and three quantum oscillators. The system and the parametrization considered here open new perspectives for manipulating quantum features at high temperatures.

Phase control of chaos is a powerful technique but little is known about its physical constraints, relevant for real systems. As a fact, it has not been explored whether this technique can also be applied when the controlling perturbation is not harmonic. Here we apply phase control on a driven double well Duffing oscillator using periodic rectangular pulsed perturbations instead of the classical sinusoidal perturbations. Experimental measurements and numerical simulations show that this kind of perturbation is also able to stabilize the chaotic orbits for an adequate selection of the phase. Furthermore, as the duty cycle of the perturbation (that is, the fraction of the time that the periodically pulsed control is active) is increased, two separate regimes occur. In the first one, the perturbations leading to stabilization of periodic solutions are of constant energy (taken as the product of the duty cycle and the amplitude) and in the second one, a saturation phenomenon occurs, implying that increasing energy values of the perturbations are wasted. Our results unveil the versatility of the pulsed phase control scheme and the importance of energy constraints.

We present an experimental investigation of the complex dynamics of a modulated relaxation oscillator implemented by using a unipolar junction transistor (UJT) showing the transition to chaos through torus breakdown. In a previous paper a continuous model was introduced for the same system, explaining chaos based on analogy with a memristor. We propose here a new approach based on a piecewise linear model with delay considering a measured parasitic delay effect. The inclusion of this delay, accounting for memory effects, increases the dimensionality of the model, allowing the transition to chaos as observed in the experiment. The piecewise delayed model shows analogies with a two-dimensional leaky integrate-and-fire model used in neurodynamics.

Experimental study of a uni junction transistor (UJT) has enabled to show that this electronic component has the same features as the so-called “memristor”. So, we have used the memristor's direct current (DC) current-voltage characteristic for modeling the UJT's DC current-voltage characteristic. This led us to confirm on the one hand, that the UJT is a memristor and, on the other hand to propose a new four-dimensional autonomous dynamical system allowing to describe experimentally observed phenomena such as the transition from a limit cycle to torus breakdown.

We discuss the phase control technique consisting of an applied square pulsed periodic perturbation. We explore the effect of such perturbations to the different terms of the Duffing oscillator. We find that the effect depends sensitively on how the perturbation is applied, indeed, it is specially effective when it modulates the cubic and the linear term and uneffective when applied to the driving term. Our results highlight the highly nontrivial role of the phase when applying a second periodic perturbation to a chaotic system.

Taming chaos arising from dissipative non-autonomous nonlinear systems by applying additional harmonic excitations is a reliable and widely used procedure nowadays. But the suppressory effectiveness of generic non-harmonic periodic excitations continues to be a significant challenge both to our theoretical understanding and in practical applications. Here we show how the effectiveness of generic suppressory excitations is optimally enhanced when the impulse transmitted by them (time integral over two consecutive zeros) is judiciously controlled in a not obvious way. Specifically, the effective amplitude of the suppressory excitation is minimal when the impulse transmitted is maximum. Also, by lowering the impulse transmitted one obtains larger regularization areas in the initial phase difference-amplitude control plane, the price to be paid being the requirement of larger amplitudes. These two remarkable features, which constitute our definition of optimum control, are demonstrated experimentally by means of an analog version of a paradigmatic model, and confirmed numerically by simulations of such a damped driven system including the presence of noise. Our theoretical analysis shows that the controlling effect of varying the impulse is due to a subsequent variation of the energy transmitted by the suppressory excitation., P.J.M. and R.C. acknowledge financial support from the Ministerio de Economía y Competitividad (MINECO, Spain) through FIS2011-25167 and FIS2012-34902 projects, respectively. R.C. acknowledges financial support from the Junta de Extremadura (JEx, Spain) through project GR15146. J.A.C.G. was supported by CNPq, Brazil.

Phase-control techniques of chaos aim to extract periodic behaviors from chaotic systems by applying weak harmonic perturbations with a suitably chosen phase. However, little is known about the best strategy for selecting adequate perturbations to reach desired states. Here we use experimental measures and numerical simulations to assess the benefits of controlling individually the three terms of a Duffing oscillator. Using a real-time analog indicator able to discriminate on-the-fly periodic behaviors from chaos, we reconstruct experimentally the phase versus perturbation strength stability areas when periodic perturbations are applied to different terms governing the oscillator. We verify the system to be more sensitive to perturbations applied to the quadratic term of the double-well Duffing oscillator and to the quartic term of the single-well Duffing oscillator.

We investigate the feedback control of a periodically driven FitzHugh–Nagumo circuit (FHN), which displays both spiking and non-spiking behaviors in chaotic or periodic regimes. The simple high pass filter (washout filter) is compared with an all pass filter which only affects the phase characteristics of the input signal. Experimental measurements performed on the electronic implementation of the FHN are in good agreement with numerical simulations. Apart a small difference in their amplitude corrections that remain in both cases within a few percent with respect to the unperturbed spiking signal, we prove the key role of the phase characteristics of the two filters in achieving control.

We report a detailed experimental study of the complex behavior of a dc low-pressure plasma discharge tube of the type commonly used in commercial illuminated signs, in a microfluidic chip recently proposed for visible analog computing and other practical devices. Our experiments reveal a clear quasiperiodicity route to chaos, the two competing frequencies being the relaxation frequency and the plasma eigenfrequency. Based on an experimental volt-ampere characterization of the discharge, we propose a macroscopic model of the current flowing in the plasma. The model, governed by four autonomous ordinary differential equations, is used to compute stability diagrams for periodic oscillations of arbitrary period in the control parameter space of the discharge. Such diagrams show self-pulsations to emerge remarkably organized into intricate mosaics of stability phases with extended regions of multistability (overlap). Specific mosaics are predicted for the four dynamical variables of the discharge. Their experimental observation is an open challenge.

The control of quantum entanglement in systems in contact with environment plays an important role in information processing, cryptography and quantum computing. However, interactions with the environment, even when very weak, entail decoherence in the system with consequent loss of entanglement. Here we consider a system of two coupled oscillators in contact with a common heat bath and with a time dependent oscillation frequency. The possibility to control the entanglement of the oscillators by means of an external sinusoidal perturbation applied to the oscillation frequency has been theoretically explored. We demonstrate that the oscillators become entangled exactly in the region where the classical counterpart is unstable, otherwise when the classical system is stable, entanglement is not possible. Therefore, we can control the entanglement swapping from stable to unstable regions by adjusting amplitude and phase of our external controller. We also show that the entanglement rate is approximately proportional to the real part of the Floquet coefficient of the classical counterpart of the oscillators. Our results have the intriguing peculiarity of manipulating quantum information operating on a classical system.

A system to assess timber deformations has been developed using an array of optical distance sensors (ODS), each of which controls a strip of the timber determining how it deviates from flatness. The main characteristics of the ODS system are compactness, modularity and real-time elaboration software. The shape of the timber is reconstructed by sampling the distance between the timber and the ODS while the timber is running in the wood plant. The working principle of the ODS is based on the triangulation of a light spot. A low-power laser diode projects a light spot on the timber surface. The image of the spot moves on a position-sensitive detector (PSD) according to the distance between the optical system and the timber. Suitable software for computer-aided operation was developed at an industrial plant producing wood cutting systems. A prototype of the ODS system has been constructed and tested under real operating conditions.

When FitzHugh-Nagumo (FHN) driven oscillators are coupled, their dynamics tend to be synchronized. We show that the chaotically spiking neurons change their internal dynamics to subthreshold oscillations, the phenomenon referred to as firing death. These dynamical changes are observed below the critical coupling strength at which the transition to full chaotic synchronization occurs. Moreover, we find various dynamical regimes in the subthreshold oscillations, namely, regular, quasiperiodic, and chaotic states. We show numerically that these dynamical states may coexist with large-amplitude spiking regimes and that this coexistence is characterized by riddled basins of attraction. The reported results are obtained for neurons implemented in the electronic circuits as well as for the model equations. Finally, we comment on the possible scenarios where the coupling-induced firing death could play an important role in biological systems.

The FitzHugh-Nagumo neurons driven by a periodic forcing undergo a period-doubling route to chaos and a transition to mixed-mode oscillations. When coupled, their dynamics tend to be synchronized. We show that the chaotically spiking neurons change their internal dynamics to subthreshold oscillations, the phenomenon referred to as firing death. These dynamical changes are observed below the critical coupling strength at which the transition to full chaotic synchronization occurs. Moreover, we find various dynamical regimes in the subthreshold oscillations, namely, regular, quasiperiodic, and chaotic states. We show numerically that these dynamical states may coexist with large-amplitude spiking regimes and that this coexistence is characterized by riddled basins of attraction. The reported results are obtained for neurons implemented in the electronic circuits as well as for the model equations. Finally, we comment on the possible scenarios where the coupling-induced firing death could play an important role in biological systems. DOI: 10.1103/PhysRevE.87.022919

We show that the chaotically spiking neurons coupled in a ring configuration changes their internal dynamics to subthreshold oscillations, the phenomenon referred to as firing death. These dynamical changes are observed below the critical coupling strength at which the transition to full chaotic synchronization occurs. We find various dynamical regimes in the subthreshold oscillations, namely, regular, quasi-periodic and chaotic states. We show numerically that these dynamical states may coexist with large amplitude spiking regimes and that this coexistence is characterized by riddled basins of attraction. Moreover, we show that under a particular coupling configuration, the neural network exhibits bistability between two configurations of clusters. Each cluster composed of two neurons undergoes independent chaotic spiking dynamics. As an appropriate external perturbation is applied to the system, the network undergoes changes in the clusters configuration, involving different neurons at each time. We hypothesize that the winning cluster of neurons, responsible for perception, is that exhibiting higher mean frequency. The clusters features may contribute to an increase of local field potential in the neural network. The reported results are obtained for neurons implemented in the electronic circuits as well as for the model equations.

When an ambiguous stimulus is observed, our percep- tion undergoes dynamical changes between two states, a situation extensively explored in association with the Necker cube. Such phenomenon refers to bistable per- ception. Here, we present a model neural network composed of forced FitzHugh-Nagumo neurons, im- plemented also experimentally in an electronic circuit. We show, that under a particular coupling configu- ration, the neural network exhibit bistability between two configurations of clusters. Each cluster composed of two neurons undergoes independent chaotic spiking dynamics. As an appropriate external perturbation is applied to the system, the network undergoes changes in the clusters configuration, involving different neu- rons at each time. We hypothesize that the winning cluster of neurons, responsible for perception, is that exhibiting higher mean frequency. The clusters fea- tures may contribute to an increase of local field po- tential in the neural network.

We report on the feedback control of a periodically driven FitzHugh-Nagumo system displaying various dynamical regimes, regular and chaotic, including chaotic spiking. The feedback term is composed of a high-pass or an all-pass filter. We provide a characterization of the efficiency of the filters in controlling the dynamical state of the system, both experimentally (electronic circuit) and numerically (model equations). We demonstrate the better efficiency of the all-pass filter as we change the coupling strength of the feedback. Moreover, we show that both filters change the appearance of bifurcations, that is the main result of the controlling effect. However, all-pass filter revealed to have a stronger efficiency in the bifurcation control. Finally, we discuss the relation of the feedback method based on filters with the delayed feedback control scheme.

With a modulated CO2 laser as a standard model of periodically driven multistable systems, we experimentally demonstrate that a small-amplitude optoelectronic feedback perturbation can efficiently transform a bursting chaotic system to a nonchaotic one. Numerical simulations are in excellent agreement with the experimental results. The control has also been equally effective in the case of a driven FitzHugh–Nagumo model of Neuroscience. 2011 Elsevier B.V. All rights reserved.

We experimentally demonstrate that a suitable opto-electronic feedback perturbation may efficiently control multistability related nonlinear phenomena (“bursting” or crisis-induced intermittency) in a modulated CO2 laser, even in presence of noise (stochastic multistability). We believe these investigations are relevant for their implications in neural networks, coding and telecommunication purposes.

We study, both numerically and experimentally, the synchronization of uncoupled excitable systems due to a common noise. We consider two identical FitzHugh-Nagumo systems, which display both spiking and non-spiking behaviours in chaotic or periodic regimes. An electronic circuit provides a laboratory implementation of these dynamics. Synchronization is tested with both white and coloured noise, showing that coloured noise is more effective in inducing synchronization of the systems. We also study the effects on the synchronization of parameter mismatch and of the presence of intrinsic (not common) noise, and we conclude that the best performance of coloured noise is robust under these distortions.

Stochastic disturbances and spikes (sudden sharp fluctuations of any system parameter), commonly observed among natural and laboratory-scale systems, can perturb the multistable dynamics significantly and become a serious impediment when the device is designed for a certain dynamical behavior. We experimentally demonstrate that suitable periodic modulation of any system parameter may efficiently control such stochastic multistability related problems. The control mechanism is verified individually with two standard models (namely, an analog circuit of Lorenz equations and a cavity-loss modulated CO 2 laser), against three externally introduced disturbing signals, (namely, white Gaussian noise, pink noise, and train of spikes). Indeed, with both the systems, it has been observed that the modulation is capable to significantly control untoward jumps to coexisting attractors that otherwise would have occurred due to either of the disturbances. These results establish the robustness and wide applicability of this control mechanism in resolving stochastic multistability related problems.

A photosedimentation technique is used to analyze the size composition of soil samples. The number and size of the particles are determined, respectively, by the Stokes formula and the Beer–Lambert law, measuring time-of-flight and laser light attenuation simultaneously and hence evaluating solution turbidity. A simple software procedure has been developed to obtain fractional volume size distribution, taking into account the particle’s optical properties depending mainly on its size and refractive index. Laboratory measurements on calibrated particulates, showing their reproducibility and validation as well as a classification of ground samples, are presented. Size distribution data can then be utilized to obtain a textural classification of the soil samples for agricultural applications.

In this paper we study how to avoid escapes in open dynamical systems in the presence of dissipation and forcing, as it occurs in realistic physical situations. We use as a prototype model the Helmholtz oscillator, which is the simplest nonlinear oscillator with escapes. For some parameter values, this oscillator presents a critical value of the forcing for which all particles escape from its single well. By using the phase control technique, weakly changing the shape of the potential via a periodic perturbation of suitable phase $\ensuremath{\phi}$, we avoid the escapes in different regions of the phase space. We provide numerical evidence, heuristic arguments, and an experimental implementation in an electronic circuit of this phenomenon. Finally, we expect that this method might be useful for avoiding escapes in more complicated physical situations.

Recurrence networks are a novel tool of nonlinear time series analysis allowing the characterisation of higher-order geometric properties of complex dynamical systems based on recurrences in phase space, which are a fundamental concept in classical mechanics. In this Letter, we demonstrate that recurrence networks obtained from various deterministic model systems as well as experimental data naturally display power-law degree distributions with scaling exponents $\gamma$ that can be derived exclusively from the systems' invariant densities. For one-dimensional maps, we show analytically that $\gamma$ is not related to the fractal dimension. For continuous systems, we find two distinct types of behaviour: power-laws with an exponent $\gamma$ depending on a suitable notion of local dimension, and such with fixed $\gamma=1$., Comment: 6 pages, 7 figures

Here we study how to control the dynamics of excitable systems by using the phase control technique. Excitable systems are relevant in neuronal dynamics and therefore this method might have important applications. We use the periodically driven FitzHugh-Nagumo (FHN) model, which displays both spiking and non-spiking behaviours in chaotic or periodic regimes. The phase control technique consists of applying a harmonic perturbation with a suitable phase phi that we adjust in search of different behaviours of the FHN dynamics. We compare our numerical results with experimental measurements performed on an electronic circuit and find good agreement between them. This method might be useful for a better understanding of excitable systems and different phenomena in neuronal dynamics.