We propose a reaction-diffusion equation to model the bidimensional propagation of a wet oxidation front in thin aluminum-rich layers. The model includes adequately the anisotropies of the system. It can be used with any starting geometry, and it can be applied to other oxides. Our numerical simulations for simple and complex starting geometries are in excellent agreement with experimental observations
The present work is devoted to describing the Quasi-Stationary-States (QSS) involved in the d-HMF model (dipole Hamiltonian Mean Field Model) introduced by Atenas and Curilef 2017 [1], which was inspired in the dipole-dipole interactions, neglecting the distance dependence. The model is a variation of the Ising model, but it involves long-range interactions. Its Hamiltonian is similiar to the HMF model, which is a toy model widely used by several authos for study the the dynamics and thermodynamics in systems with long range interactions. Both models share several properties, except the symmetry, which makes the d-HMF model interesting. In the study of the stationary solutions, we found analytical solutions of the Vlasov equation, which are highly non-linear. We have obtained the analytical solution for the equilibrium by Boltzmann-Gibbs distribution and for the QSS out of the equilibrium by means of a Vlasov distribution type Tsallis. The results are relevant because they are found by means of optimization and variational methods, which are different ways to commonly used in the literature, where several authors try to find distributions using a fitting parameter q contrasting with simulations or experimental data. Additionally, we attempt to formally connect to thermodynamics and Tsallis statistics.
[1] Atenas B and Curilef S., Phys. Rev. E 95, 022110 (2017).
The synchronization of 3 coupled mobile oscillators is characterized by applying some of the techniques used for 2 [1]. The use of the logistic map is proposed as a fundamental dynamic for each oscillator and an interaction model, initially fixed and then another distance dependent, based on the restricted three body problem, i.e. two oscillators strongly coupled and the other one with a weak coupling with each others. It is intended to study the periodicities of the synchronization factor [2] as an analysis tool. We will seek to apply the results found.
[1] R. O. E. Bustos-Espinoza and G. M. Ramírez Ávila, Condiciones de sincronización de dos osciladores móviles, Revista Boliviana de Física 22,1-7, 2012.
[2] R. O. E. Bustos-Espinoza and G. M. Ramírez Ávila, Synchronization conditions of coupled maps using periodicities, The European Physical Journal Special Topics, 225, 2697-2705, 2016
Complexity appears naturally in many fields, and the importance of its analysis and development is crucial in promoting a better understanding of its reach and applications. For instance, in written language, which is the representation of spoken language by means of a symbolic or text system, we observe repetitive patterns in word formation, prefixes, and suffixes. Moreover, syntax and correctness of sentences have large consequences in transmitting the message properly and adequately. Yet, the content and meaningful information are remarkably hidden in pattern construct, simply put in terms of combination, frequency of repetition, and inherent meaning. We have constructed a simple discrimination scheme based on quantitative measures that arise naturally in most written pieces and from it calculated a frequency using words from books of most traditional Latin-American writers.
We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit.
While the characterization of the synchronous behavior of a system of two mutually coupled light-controlled oscillators (LCOs) has been extensively studied, few studies were done when coupling strength takes high values and none for the case in which a strong coupling enables the transition to an oscillation death (OD) regime. We used a model for light-controlled oscillators to establish the synchronization conditions and also the situation in which a tendency to produce oscillation quenching is due to a strong coupling between these oscillators [1]. According to the model, there is a critical value for which the oscillation death appears, and above this one, the oscillation death is manifested with distinctive features. We experimentally verified the model predictions concerning the transition from synchronization to oscillation death as the coupling strength increases. We studied the route to OD numerically considering the differences of natural periods of the LCOs as well as the coupling strength. As a result of the variations of these variables, we computed the quantities characterizing the oscillation of the coupled LCOs, i.e., the amplitude and the common period or synchronization period. The above-mentioned variables characterize the transition leading to OD.
This transition exhibits among others: (i) a trend to diminish the difference of amplitudes, (ii) a kind of bursting behaviour in the signals, i.e., an increasing number of peaks, (iii) the period falls down as coupling increases.
We point out that our model is based on experimental results on oscillation death, which has been carefully detailed in pulse-coupled oscillators.
[1] G. Conde-Saavedra, G.M. Ramírez-Ávila, Experimental oscillation death in two mutually coupled light-controlled oscillators, Chaos, 28 (2018) 043112.
This work performs a fractal analysis of the galaxy distribution and presents evidence that can be described as a fractal system within the redshift range of the FORS Deep Field (FDF) galaxy survey data. The fractal dimension D was derived by means of the galaxy number densities calculated by Iribarrem et al. (2012) using the FDF luminosity function parameters and absolute magnitudes obtained by Gabasch et al. (2004, 2006) in the spatially homogeneous standard cosmological model with Ωm_0 = 0.3, ΩΛ_0 = 0.7 and H0 = 70 km s−1 Mpc−1. Under the supposition that the galaxy distribution forms a fractal system, the ratio between the differential and integral number densities γ and γ* obtained from the red and blue FDF galaxies provides a direct method to estimate D and implies that γ and γ* vary as power-laws with the cosmological distances, feature which provides a second method for calculating D. The luminosity distance dL , galaxy area distance dG and redshift distance dz were plotted against their respective number densities to calculate D by linear fitting. It was found that the FDF galaxy distribution is better characterized by two single fractal dimensions at successive distance ranges, that is, two scaling ranges in the fractal dimension. Two straight lines were fitted to the data, whose slopes change at z \(\approx\) 1.3 or z \(\approx\) 1.9 depending on the chosen cosmological distance. The average fractal dimension calculated using γ* changes from ⟨D⟩ = 1.4+0.7 −0.6 to ⟨D⟩ = 0.5+1.2−0.4 for all galaxies. Besides, D evolves with z , decreasing as the redshift increases. Small values of D at high z mean that in the past galaxies and galaxy clusters were distributed much more sparsely and the large-scale structure of the universe was then possibly dominated by voids.
Chagas disease is a vector-borne disease due to the parasite Trypanosoma cruzi that is mainly transmitted by triatomine insects (Triatominae). These bugs live in the human neighbourhood; they aggregated in wall or roof cracks during the day and go out to feed on animal or human blood at night. Understanding the group dynamics is essential for discerning the insects and parasite dispersion. Experiments where adults of Triatoma infestans were dropped at the base of an artificial wall (vertical surface) were carried out to analyse the aggregation behaviour and how the sex and the infection by T. cruzi affect the aggregation behaviour. Insects presented a high negative geotaxis and aggregative behaviour. Males showed a higher geotaxis and clustering than females, and infected insects than potentially weakly infected ones. An analysis of the networks formed by the clusters showed that females tend to cluster in a looser network when compared with males. Simulations demonstrated that the social part is essential in the clustering in triatomines.
We shall present explicitly the construction of the mKdV hierarchy in terms of graded affine algebras and show that it decomposes into positive and negative graded sub-hierarchies.
Moreover we shall extend the construction of the Backlund transformation for the sinh-Gordon model to all other positive and negative odd graded equations of motion generated by the same affine algebraic structure.
As an application we shall discuss the structure of integrable defects in which two solutions can be interpolated by Backlund transformation .
We explain the chaotic and hyperchaotic characteristics of the Tarka and show how the geometry of the instrument creates the conditions for this nonlinear behaviour. The generation of multiphonic sounds is analysed using spectral techniques. We confirm two particular musical behaviours and by increasing the blow pressure on different fingerings, peculiar changes from linear to nonlinear patterns are produced, leading ultimately to oscillation death.
Coexistence of states is an essential feature in the observation of domain walls, interfaces, shock waves, or fronts in macroscopic systems. The propagation of these nonlinear waves depends on the relative stability of the connected equilibria. In particular, one presumes a stable equilibrium to invade an unstable one, such as occur in combustion, in the spread of permanent contagious diseases, or the freezing of supercooled water. In this work, we show that an unstable state generically can invade a locally stable one in pattern-forming systems. We associate this effect to the lower free energy unstable state invading the locally stable but higher free energy state. Based on a one-dimensional model, we reveal the features required to observe this phenomenon. The scenario fulfills in the case of a first-order spatial instability. We show that in the photoisomerization transition on a dye-doped liquid crystal cell allow us to witness the front propagation from an unstable state.
We propose a thermodynamically consistent minimal model to study synchronization which is made of driven and globally interacting three-state units. This system exhibits at the mean-field level two bifurcations separating three dynamical phases: a single stable fixed point, a stable limit cycle indicative of synchronization, and multiple stable fixed points. These complex emergent dynamical behaviors are understood at the level of the underlying linear Markovian dynamics in terms of metastability, i.e. the appearance of gaps in the upper real part of the spectrum of the Markov generator. Stochastic thermodynamics is used to study the dissipated work across dynamical phases as well as across scales. This dissipated work is found to be reduced by the attractive interactions between the units and to nontrivially depend on the system size. When operating as a work-to-work converter, we find that the maximum power output is achieved far-from-equilibrium in the synchronization regime and that the efficiency at maximum power is surprisingly close to the linear regime prediction. [PRX 8, 031056 (2018)]
We furthermore find that the phenomenology of the three-state-model can also be observed for a whole class of driven Potts models with spin states q. It follows from thermodynamic consistency that the low- and high-temperature phase are universal for any q. We derive the critical point that destabilizes the symmetric fixed point and generically show that all models exhibit a Hopf bifurcation. Supported by numerical studies, we claim that there are two classes of universal (thermo)dynamical properties exhibited by the Potts model depending on q: If q is even the Hopf bifurcation occurs subcricital and there are only the two high- and low-temperature phases. Conversely, if q is odd, the Hopf bifurcation occurs supercritical, that is there is an additional intermediate phase characterized by stable oscillations. [PRE 99, 022135 (2019)]
Dispersion is the mechanism by which populations are distributed in a territory. Its study is an essential aspect in evolutionary ecology. The habitats themselves are heterogeneous in space and time. The dispersion may reflect a purely random movement or may be conditioned for the environment or other organisms. Understand what ways of dispersion confer some sort of selective advantage is an issue of growing interest in space ecology. In this talk, we will concentrate in diffusive models of competition type Lotka-Volterra in a focal plot of territory or habitats whose limits are subject to the restriction that individuals who leave the plot do not return more (they are absorbed by the surrounding territory, and it may be the case that the border is lethal). We will study different situations that could arise, such as the dispersion processes are the same, but the demography is potentially different. The main tool will be the theory of dynamic systems, specifically that of semi-dynamic systems over complete metric spaces. Theorems of existence and persistence of global attractors will allow us to establish predictions of long-term coexistence among competitors, under certain conditions of invasiveness that will be assumed.
It is established that for a special case of the oscillator with time dependent parameters leads to the problem of damped oscillator. The exact analytical solutions of a generalized classical harmonic oscillator with time dependent mass, frequency, two-photon parameter and external forces are obtained. By using the invariance property of the scaled Wronskian, these solutions are used to obtain the solutions of the quantum mechanical counterpart of the oscillator under Heisenberg picture. In order to discuss the applications of these solutions of the quantum mechanical oscillator, we calculate the exact analytical expressions for the second order variances of both the canonically conjugate quadratures. It is found that these variances do not depend on the time dependent driven terms. The second and higher variances of the field operators will be of use to investigate the squeezing, higher ordered squeezing, antibunching of photons of the input radiation field coupled to the said oscillator. We discuss few situations of physical interest where the mass is varying in time.
We report the triggering of localized and confined chaos described by a general cubic order damped nonlinear Schrodinger amplitude equation containing a conjugate amplitude term, representing the time-periodic parametric driving, and a spatially periodic term, representing the external potential that cuts and confines the chaotic patterns promoted by the former, leading to trapped chaotic space-localized structures that are supported against the damping. Numerical simulations in 1+1, 1+2, and 1+3 dimensions, Lagrangian and Hamiltonian theories for continuous fields, moments method, largest Lyapunov exponents, spectral distributions, and bifurcations diagrams are used to characterize and analyze these chaotic solitons.
Based on the work of the authors in [1], we review the mathematical framework that describes the dynamics of charged massless particles. The dynamics of such particles is characterized by the presence of a constraint equation that enforces the modulus of the velocity of the particle to be equal to the speed of light at any time of the evolution. Analytically, if we start with a set of initial conditions compatible with the constraints, the dynamical equations evolve the system by preserving the solution within the constrained surface. Numerically, this is not the case; the system seems to exhibit high sensitivity to the integration details, despite their refinement. A suitable modification of the original system that enlarges the field vector by incorporating the constraint as a dynamical variable proved to be more appropriate for numerical work. A similar numerical treatment performed for the massive case in the limit of zero mass and velocity closer to the speed of light shows that this case is strongly time-consuming from a computational viewpoint because of the singularity of such a limit. This latter result motivates further developments of our approach as an alternative to study the dynamics of charged ultrarelativistic particles.
[1] Ivan Morales, Bruno Neves, Zui Oporto, Olivier Piguet, “Behaviour of Charged Spinning Massless Particles”, Symmetry 10 (2017) no.1, 2 (2017-12-22) DOI: 10.3390/sym10010002
Synchronous flashing in fireflies is perhaps the first observed natural phenomenon displaying synchronization of a large ensemble. During a long time, this collective behavior was not recognized and validated as synchronous, but nowadays, it constitutes a paradigmatic example of synchronization. Despite this fact, there are not many efforts to model this astounding phenomenon realistically. One of the essential features of fireflies’ synchronization is the cooperative behavior of many fireflies giving rise to the emergence of synchronization without any leader, a fact that took a long time to be recognized. A review of the main attempts to build models allowing the explanation of how and why fireflies synchronize is done. The starting point is qualitative models based on simple observations. The latter served to formulate original mathematical models enabling not only to explain fireflies’ synchronization but also some other collective phenomena. Integrate-and-fire oscillators (IFOs) constitute a typical model to describe the fireflies’ synchronous behavior, and they have also inspired ones to build electronic circuits with similar features and adapted to fireflies in the sense that they communicate with each other by means of light pulses. The above-mentioned electronic circuits received the name of electronic fireflies or more technically, light-controlled oscillators (LCOs). These engines allowed a systematic study of synchronization from experimental, theoretical, and numerical viewpoints. They have also been used in a wide variety of situations ranging from simple cases of identical oscillators to scenarios where populations of dissimilar oscillators whose interaction does explain synchronization as well as the response to synchronization, a widespread phenomenon occurring in fireflies. For further details, see [1, 2].
[1] G.M. Ramírez-Ávila, J. Kurths, J.L. Deneubourg, Fireflies: A Paradigm in Synchronization, in: M. Edelman, E.E.N. Macau, M.A.F. Sanjuan (Eds.) Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives, Springer International Publishing, Cham, 2018, pp. 35-64.
[2] G.M. Ramírez-Ávila, J. Kurths, S. Depickère, J.-L. Deneubourg, Modeling Fireflies Synchronization, in: E.E.N. Macau (Ed.) A Mathematical Modeling Approach from Nonlinear Dynamics to Complex Systems, Springer International Publishing, Cham, 2019, pp. 131-156.
The Schwarzian derivative appears in many areas of mathematics such as projective differential geometry and differential equations. In the dynamical systems theory it appeared for the first time independently in Singer's Theorem (1978) and Michael Herman's Doctoral Thesis (1976). Although the Schwarzian derivative now appears frequently in one-dimensional dynamics, it wasn't know if there is a way to motivate its definition in purely dynamical systems concepts. In this lecture we will show that Singer's Theorem is sufficient to motivate the definition of the Schwarzian derivative only from the point of view of dynamical systems.
This is a joint work with Bernardo San Martin of the Universidad Católica del Norte, Chile.
We perform the opinion evolution analysis of a group of individuals who are under a decision making situation. This analysis is made using a mathematical model of opinion evolution which evolve in discrete steps of time and carried out through numerical simulations, where some of the parameters of the model are obtained from computational random draws. Firstly, we consider the case of a constant global external source's action on the group whose individuals do not interact; considering different connectivity types between the source and the group's members, namely, binary or continuous with constant or time dependent intensities. In this case we characterized the evolution of the opinion of the group under different values of the parameters which characterized the network. Here we made too, a comparation between the case when just the global external source interact over the individuals and the case when there is the global external source and the network of interaction between the individuals. Secondly, we consider the case of a complex network featured interaction among the individuals in addition to the action of the global source. Afterwards, we study the case in which are present in the group the so-called intransigents, whose main characteristic is that their opinion is always opposite and do not change; thus, hindering consensus situations. Finally, we address the situation in which there are only interactions between individuals, without considering the external source, finding a sensitivity to the initial conditions in individual opinions for the evolution of the opinion state. In all cases, we analyze and compare the effects of the model variants on the opinion of the group, where the achievement or not of consensus is an essential aspect of the study.
In this work we report the existence of tricorn-like structures of stable periodic orbits in the parameter plane of an optically injected semiconductor laser model (a continuous time dynamical system). These tricorns appear inside tongue-like structures that born through simple Shi’lnikov bifurcations. As we increase the linewidth enhancement factor-α of the laser, these tongues invade the locking zone of the laser and extends over the zone of stable orbits of period-1. This invation provocates a rich dynamic of overlap of parameter planes that produce an abundant multiestability.
As we increase α the tricorn also exhibits a phenomenon of codimension-3 rotating in clock and anti-clockwise in the plane of the injected field rate K vs. its detunning ω. We hope that the numerical evidence of the existence of tricorns that we present here motivates the study of mathematical conditions for their genesis. And also we encourage the experimental verification of these tricorn-like structures beacause our results also open new possibilities for optical switching between several different outputs of the laser in the neighbourhood of these structures.
Let G be a connected Lie group with Lie algebra g and Σ = (G, D) a controllable invariant control system.
A subset A ⊂ G is said to be isochronous if there exists a uniform time TA > 0 such that any two arbitrary elements in A can be connected by a positive orbit of Σ at exact time TA. In this paper, we search for classes of Lie groups G such that any Σ has the following property: there exists an increasing sequence of open neighborhoods (Vn)n≥0 of the identity in G such that the group can be decomposed in isochronous rings
Wn = Vn+1 − Vn. We characterize this property in algebraic terms and we show that three classes of Lie groups satisfy this property: completely solvable simply connected Lie groups, semisimple Lie groups and reductive Lie groups.
Chagas disease American trypanosomiasis is caused by a flagellated parasite: trypanosoma cruzi, transmitted by an insect of the genus Triatoma and also by blood transfusions. In Latin America, the number of infected people is approximately 6 million, with a population exposed to the risk of infection of 550000. It is our interest to develop a non-invasive, low-cost methodology, capable of detecting any early cardiac alteration. by T. cruzi.
We analyzed 24-hour RR records in patients with abnormal ECG (CH2), patients without ECG abnormalities (CH1) who had positive serological findings for Chagas disease and healthy (Control) matched by sex and age. We found significant differences between Control and CH2 that show the dysautonomy and enervation of the autonomic nervous system.
The matter in thermodynamic equilibrium, in quasi-two spatial dimensions, as temperature decrease, it can exhibit exotic states of matter, it corresponds to topological transitions associated to the emergency of pairs vortexes (Berezinskii–Kosterlitz–Thouless transition). This type of phenomena has been observed in different contexts such as Josephson junctions, superconductor thin films, among others, which allowed that Kosterlitz–Thouless obtaining the Nobel prize in the year 2016. We are interested in the study of this phenomena in out equilibrium systems— systems with temporarily modulated parameters, for this, we have considered a thin film of nematic liquid crystal with negative electric constant and with homeotropic anchoring. An oscillatory voltage is applied to nematic liquid crystal film, the balance between electric force and the elastic force of the molecules allow the creations of topological defects called vortexes. In a range of frequency between 500[μHz] and 0.5[Hz], the system described previously exhibit different transitions of topological phases in which pairs and networks of vortexes emerge, among other topological phenomena.