Alexandre Wagemakers

Basin entropy: a new tool to analyze uncertainty in dynamical systems

In nonlinear dynamics, basins of attraction link a given set of initial conditions to its corresponding final states. This notion appears in a broad range of applications where several outcomes are possible, which is a common situation in neuroscience, economy, astronomy, ecology and many other disciplines. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. From this respect, a proper classification of this unpredictability is clearly required. To address this issue, we introduce the basin entropy, a measure to quantify this uncertainty. Its application is illustrated with several paradigmatic examples that allow us to identify the ingredients that hinder the prediction of the final state. The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied. Additionally, we provide a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than log2, the basin is fractal.

Isochronous synchronization in mutually coupled chaotic circuits

This paper examines the robustness of isochronous synchronization in simple arrays of bidirectionally coupled systems. First, the achronal synchronization of two mutually chaotic circuits, which are coupled with delay, is analyzed. Next, a third chaotic circuit acting as a relay between the previous two circuits is introduced. We observe that, despite the delay in the coupling path, the outer dynamical systems show isochronous synchronization of their outputs, i.e., display the same dynamics at exactly the same moment. Finally, we give here the first experimental evidence that the central relaying system is not required to be of the same kind of its outer counterparts.

Experimental demonstration of bidirectional chaotic communication by means of isochronal synchronization

We give the first experimental demonstration of simultaneous bidirectional communication through chaotic carriers thanks to the phenomenon of isochronal synchronization. Two Mackey-Glass electronic circuits with chaotic behaviour exchange their signals through a coupling line with delay. When the internal feedback of the circuits and the coupling are accurately matched, isochronal synchronization arises. Under this dynamical regime, we introduce a binary message at both outputs and recover it at the opposite circuit. Finally, we discuss the security of this kind of communication system by analyzing the message recovered by a potential eavesdropper.

Synchronization of electronic genetic networks

We describe a simple analog electronic circuit that mimics the behavior of a well-known synthetic gene oscillator, the repressilator, which represents a set of three genes repressing one another. Synchronization of a population of such units is thoroughly studied, with the aim to compare the role of global coupling with that of global forcing on the population. Our results show that coupling is much more efficient than forcing in leading the gene population to synchronized oscillations. Furthermore, a modification of the proposed analog circuit leads to a simple electronic version of a genetic toggle switch, which is a simple network of two mutual repressor genes, where control by external forcing is also analyzed.

Design of a New Differential Chaos-Shift-Keying System for Continuous Mobility

Conventional differential chaos-shift-keying systems (DCSK) are not the most suitable for supporting continuous-mobility scenarios. Therefore, in this paper an improved continuous-mobility differential chaos-shift-keying system (CM-DCSK) is presented that provides greater agility and improved performance in fast fading channels without accurate channel estimation while still being simple compared to a conventional DCSK system. A new DCSK frame signal is designed to reach this goal. In our new frame design, each reference sample is followed by a data carrier sample. This modification of the system design reduces the hardware complexity of DCSK because it requires a shorter wideband delay line and significantly improves the performance over fast fading channels while keeping the non-coherent nature of the transmission system. Once the design is explained, the bit error rate performance is computed over a multipath fast fading channel and compared to the conventional DCSK system. Simulation results confirm the advantages of this new noncoherent spread-spectrum design that can support mobility.

Chaos-Based Turbo Systems in Fading Channels

The growing demand for ubiquitous wireless communication services requires powerful coding and modulation schemes to counteract the signal degradation in the air interface, preferably without resorting to costly hardware. Previous work has demonstrated that chaos-based coded modulations could be robust in the flat fading channel. This paper illustrates that, as the flat fading channel quality degrades, parallel concatenated chaos-based coded modulations can provide better error performance than non-chaotic counterparts. Therefore, this joint coding and modulation approach may contribute to power saving in common adverse channel conditions. Moreover, it is shown that it is possible to manage its spectral efficiency while keeping its robustness under fading.

BUILDING ELECTRONIC BURSTERS WITH THE MORRIS–LECAR NEURON MODEL

We propose a method for the design of electronic bursting neurons, based on a simple conductance neuron model. A burster is a particular class of neuron that displays fast spiking regimes alternating with resting periods. Our method is based on the use of an electronic circuit that implements the well-known Morris–Lecar neuron model. We use this circuit as a tool of analysis to explore some regions of the parameter space and to contruct several bifurcation diagrams displaying the basic dynamical features of that system. These bifurcation diagrams provide the initial point for the design and implementation of electronic bursting neurons. By extending the phase space with the introduction of a slow driving current, our method allows to exploit the bistabilities which are present in the Morris–Lecar system to the building of different bursting models.

Testing for Basins of Wada.

Nonlinear systems often give rise to fractal boundaries in phase space, hindering predictability. When a single boundary separates three or more different basins of attraction, we say that the set of basins has theWada property and initial conditions near that boundary are even more unpredictable. Many physical systems of interest with this topological property appear in the literature. However, so far the only approach to study Wada basins has been restricted to two-dimensional phase spaces. Here we report a simple algorithm whose purpose is to look for the Wada property in a given dynamical system. Another benefit of this procedure is the possibility to classify and study intermediate situations known as partially Wada boundaries.

PARTIAL CONTROL OF TRANSIENT CHAOS IN ELECTRONIC CIRCUITS

We present an analog circuit implementation of the novel partial control method, that is able to sustain chaotic transient dynamics. The electronic circuit simulates the dynamics of the one-dimensional slope-three tent map, for which the trajectories diverge to infinity for nearly all the initial conditions after behaving chaotically for a while. This is due to the existence of a nonattractive chaotic set: a chaotic saddle. The partial control allows one to keep the trajectories close to the chaotic saddle, even if the control applied is smaller than the effect of the applied noise, introduced into the system. Furthermore, we also show here that similar results can be implemented on a circuit that simulates a horseshoe-like map, which is a simple extension of the previous one. This encouraging result validates the theory and opens new perspectives for the application of this technique to systems with higher dimensions and continuous time dynamics.

Chaotic dynamics and fractal structures in experiments with cold atoms

We use tools from nonlinear dynamics to the detailed analysis of cold atom experiments. A powerful example is provided by the recent concept of basin entropy which allows to quantify the final state unpredictability that results from the complexity of the phase space geometry. We show here that this enables one to reliably infer the presence of fractal structures in phase space from direct measurements. We illustrate the method with numerical simulations in an experimental configuration made of two crossing laser guides that can be used as a matter wave splitter.