arXiv subjects
Publications and source records attributed to Alexandre Wagemakers.
We present a review of the different techniques available to study a special kind of fractal basins of attraction known as Wada basins, which have the intriguing property of having a single boundary separating three or more basins. We expose several approaches to identify this topological property that rely on different, but not exclusive, definitions of the Wada property.
In this work we expand the possibilities of a known class of chaos-based coded modulation (CCM) systems to address the problem of high-power amplifier (HPA) nonlinearity. The hypothesis is that the CCM nonlinear nature can be exploited to counteract its effects on the system performance, avoiding the need of a predistorter. We propose an optimization method for the design of the CCM to prove this hypothesis. The results show that, for a given back-off level, a nonlinear mapping function for the chaos-based encoder and decoder can be designed to compensate the HPA effect on the error rate.
In this article, we propose an index modulation system suitable for optical communications, where the intensity of the light is modulated along the time, space and frequency dimensions, building a Frequency-Hopping Spatial Multi-Pulse Position Modulation (FH-SMPPM). We analyze its performance from the point of view of its efficiency in power and spectrum, and its implementation complexity and required receiver latency. We analyze its error probability in the case of the non-turbulent free-space optical (FSO) channel. We derive formulas for the average symbol and bit error probabilities, and the simulation results show that they are tight enough for a wide range of signal-to-noise ratios and system parameters. We compare FH-SMPPM with other proposed index modulated systems of the same nature, and we highlight its distinctive advantages, like the flexibility to build an appropriate waveform under different constraints and objectives. Moreover, FH-SMPPM shows to be better performing in BER/SER and/or offer advantages in efficiency with respect to those alternatives, thus offering the possibility to be adopted for a variety of contexts.
The massive deployment of light-emitting diode (LED) lightning infrastructure has opened the opportunity to reuse it as visible light communication (VLC) to leverage the current RF spectrum crisis in indoor scenarios. One of the main problems in VLC is the limited dynamic range of LEDs and their nonlinear response, which may lead to a severe degradation in the communication, and more specifically in the bit error rate (BER). This is aggravated by the extensive usage of multicarrier multiplexing, based on optical orthogonal frequency division multiplexing (O-OFDM), characterized by a high peak-to-average power ratio. Here, we present a chaos-based coded modulation (CCM) setup specifically adapted to the LED nonlinearities. It replaces the usual modulation, while keeping the multicarrier O-OFDM structure unchanged. First, we obtain a semi-analytical bound for the bit error probability, taking into account the LED nonlinear response. The bound results particularly tight for the range of signal-to-noise ratio of interest. Then, we propose a method to design the modulator based on optimization techniques. The objective function is the semi-analytical bound, and the optimization is applied to a parameterization of the CCM conjugation function. This appropriately shapes the chaotic waveform and leads to BER improvements that outperform classical counterparts under ideal predistortion.
First conceived as a topological construction, Wada basins abound in dynamical systems. Basins of attraction showing the Wada property possess the particular feature that any small perturbation of an initial condition lying on the boundary can lead the system to any of its possible outcomes. The saddle-straddle method, described here, is a new method to identify the Wada property in a dynamical system based on the computation of its chaotic saddle in the fractalized phase space. It consists of finding the chaotic saddle embedded in the boundary between the basin of one attractor and the remaining basins of attraction by using the saddle-straddle algorithm. The simple observation that the chaotic saddle is the same for all the combinations of basins is sufficient to prove that the boundary has the Wada property.
In this article, we propose an index modulation system suitable for optical communications, based on jointly driving the time and frequency of the signal: an index-time frequency hopping (I-TFH) system. We analyze its performance from the point of view of its efficiency in power and spectrum, and its behavior in terms of error probability for the non-turbulent free-space optical (FSO) channel. We compare I-TFH with already proposed index modulated systems of the same nature, but where the amplitude or the number of transmitters are driven instead of the signal frequency. We derive and compare approximations for the average symbol and bit error probabilities of all these systems. The simulation results show that said approximations are tight enough for a wide range of signal-to-noise ratios and system parameters. Moreover, I-TFH shows to be better performing in BER and/or power efficiency than the comparative alternatives, and may offer interesting properties in a variety of contexts.
When several dynamical systems interact, the transmission of the information between them necessarily implies a time delay. When the time delay is not negligible, the study of the dynamics of these interactions deserve a special treatment. We will show here that under certain assumptions, it is possible to set to zero a significant amount of time-delayed connections without altering the global dynamics. We will focus on graphs of interactions with identical time delays and bidirectional connections. With these premises, it is possible to find a configuration where a number $n_z$ of time delays have been removed with $n_v-1 \leq n_z \leq n_v^2/4$, where $n_v$ is the number of dynamical systems on a connected graph.
Trying to imagine three regions separated by a unique boundary seems a difficult task. However, this is exactly what happens in many dynamical systems showing Wada basins. Here, we present a new perspective on the Wada property: A Wada boundary is the only one that remains unaltered under the action of merging the basins. This observation allows to develop a new method to test the Wada property, which is much faster than the previous ones. Furthermore, another major advantage of the merging method is that a detailed knowledge of the dynamical system is not required.
Time delays may cause dramatic changes to the dynamics of interacting oscillators. Coupled networks of interacting dynamical systems can behave unexpectedly when the signal between the vertices are time delayed. It has been shown for a very general class of systems that the time delays can be rearranged as long as the total time delay over the constitutive loops of the network is conserved. This fact allows to reduce the number of time delays of the problem without loss of information. There is a theoretical lower bound for this number, but in many cases we can find a numerical solution that beats this limit. Here we propose a formulation of the problem and a numerical method to even further reduce the number of time delays in a network.
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.
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 $\log 2 $, the basin is fractal.
Delay differential equations take into account the transmission time of the information. These delayed signals may turn a predictable system into chaotic, with the usual fractalization of the phase space. In this work, we study the connection between delay and unpredictability, in particular we focus on the Wada property in systems with delay. This topological property gives rise to dramatical changes in the final state for small changes in the history functions.