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We consider expansive homeomorphisms with the specification property. We give a new simple proof of a large deviation principle for Gibbs measures corresponding to a regular potential and we establish a general symmetry of the rate function for the large deviations of the antisymmetric part, under time-reversal, of th...
Divergence in antigen response of the immune network is discussed, based on shape-space modelling. The present model extends the shape-space model by introducing the evolution of specificity of idiotypes.
We study the closed Hamiltonian dynamics of a free particle moving on a ring, over one section of which it interacts linearly with a single harmonic oscillator. On the basis of numerical and analytical evidence, we conjecture that at small positive energies the phase space of our model is completely chaotic except for...
The gravity model is a mathematical model that applies Newton's universal law of gravitation to socio-economic transport phenomena and has been widely used to describe world trade, intercity traffic flows, and business transactions for more than several decades. However, its strong nonlinearity and diverse network...
The statistical geometry of dispersing Lagrangian clusters of four particles (tetrahedra) is studied by means of high-resolution direct numerical simulations of three-dimensional homogeneous isotropic turbulence. We give the first evidence of a self-similar regime of shape dynamics characterized by almost two-dimensio...
Mushroom billiards have the remarkable property to show one or more clear cut integrable islands in one or several chaotic seas, without any fractal boundaries. The islands correspond to orbits confined to the hats of the mushrooms, which they share with the chaotic orbits.
The quantum resonances (QRs) of the kicked particle are studied in a most general framework by also considering {\em arbitrary} periodic kicking potentials. It is shown that QR can arise, in general, for {\em any rational} value of the Bloch quasimomentum.
We have studied the quantum and classical solutions of a particle constrained to move inside a sector circular billiard with angle $\theta_w$ and its pacman complement with angle $2\pi-\theta_w$. In these billiards rotational invariance is broken and angular momentum is no longer a conserved quantum number.
We investigate the spreading of passive tracers in closed basins. If the characteristic length scale of the Eulerian velocities is not very small compared with the size of the basin the usual diffusion coefficient does not give any relevant information about the mechanism of spreading.
Perturbation theory, the quasiclassical approximation and the quantum surface of section method are combined for the first time. This solves the long standing problem of quantizing the resonances and chaotic regions generically appearing in classical perturbation theory.
Josephson junctions (JJs) are by nature neuromorphic hardware devices capable of mimicking excitability and spiking dynamics. When coupled together or combined with other superconducting elements, they can emulate additional behaviors found in biological neurons.
Hidden Markov chains are widely applied statistical models of stochastic processes, from fundamental physics and chemistry to finance, health, and artificial intelligence. The hidden Markov processes they generate are notoriously complicated, however, even if the chain is finite state: no finite expression for their S...
We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos.
We consider the two-dimensional (2D) flow in a flat free-slip surface that bounds a three-dimensional (3D) volume in which the flow is turbulent. The equations of motion for the two-dimensional flow in the surface are neither compressible nor incompressible but strongly influenced by the 3D flow underneath the surface...
In this paper, two kinds of the exact singular solutions are obtained by the improved homogeneous balance (HB) method and a nonlinear transformation. The two exact solutions show that special singular wave patterns exists in the classical model of some nonlinear wave problems.
We consider transport properties of a double delta-kicked system, in a regime where all the symmetries (spatial and temporal) that could prevent directed transport are removed. We analytically investigate the (non trivial) behavior of the classical current and diffusion properties and show that the results are in good...
High frequency stochastic resonance (SR) phenomena, associated with fluctuational transitions between coexisting periodic attractors, have been investigated experimentally in an electronic model of a single-well Duffing oscillator bistable in a nearly resonant field of frequency $\omega_F$. It is shown that, with incr...
In this paper, a novel uncertain fractional-orders and parameters' inversion mechanism via the differential evolution algorithms (DE) with a general mathematical model is proposed for non-commensurate and hyper fractional chaotic systems. The problems of fractional-order chaos' inversion estimation are convert...
The statistical properties of a Hamiltonian $H_0$ perturbed by a localized scatterer are considered. We prove that when $H_0$ describes a bounded chaotic motion, the universal part of the spectral statistics are not changed by the perturbation.
The three-disk scatterer has served as a paradigm for semiclassical periodic orbit quantization of classical chaotic systems using Gutzwiller's trace formula. It represents an open quantum system, thus leading to spectra of complex eigenenergies.
Limit possibilities of observer-based synchronization systems under information constraints (limited information capacity of the coupling channel) are evaluated. We give theoretical analysis for multi-dimensional drive-response systems represented in the Lurie form (linear part plus nonlinearity depending only on meas...
The Kuramoto model, a paradigmatic framework for studying synchronization, exhibits a transition to collective oscillations only above a critical coupling strength in the thermodynamic limit. However, real-world systems are finite, and their dynamics can deviate significantly from mean-field predictions.
The extended Gross-Pitaevskii equation for the Bose-Einstein condensation of gases with attractive 1/r-interaction has a second solution which is born together with the ground state in a tangent bifurcation. At the bifurcation point both states coalesce, i.e., the energies and the wave functions are identical.
Populations of oscillators can display a variety of synchronization patterns depending on the oscillators' intrinsic coupling and the coupling between them. We consider two coupled, symmetric (sub)populations with unimodal frequency distributions and show that the resulting synchronization patterns may resemble th...
It was recently shown (Keating & Prado, {\it Proc. R. Soc.
In this paper we consider a system of strongly coupled logistic maps involving two parameters. We classify and investigate the stability of its fixed points.
In this paper we consider a simple general form of a deterministic system with power-law memory whose state can be described by one variable and evolution by a generating function. A new value of the system's variable is a total (a convolution) of the generating functions of all previous values of the variable wit...
We investigate the dependence of Poincaré recurrence-times statistics on the choice of recurrence-set, by sampling the dynamics of two- and four-dimensional Hamiltonian maps. We derive a method that allows us to visualize the direct relation between the shape of a recurrence-set and the values of its return probabilit...
The manuscript addresses the problem of finding all solutions of power flow equations or other similar nonlinear system of algebraic equations. This problem arises naturally in a number of power systems contexts, most importantly in the context of direct methods for transient stability analysis and voltage stability a...
The advection and diffusion of a passive scalar is investigated for a map of the 2-torus. The map is chaotic, and the limit of almost-uniform stretching is considered.
The statistical properties of velocity and acceleration fields along the trajectories of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. We present results for Lagrangian velocity structure functions, the acceleration probabilit...
Some phase space transport properties for a conservative bouncer model are studied. The dynamics of the model is described by using a two-dimensional measure preserving mapping for the variables velocity and time.
We present a methodology for the study of the dispersion of trajectories of stochastic processes in reconstructed phase spaces from observed data. The methodology allows to find ensembles of analog states, i.e. states that are infinitesimally close in the phase space.
We study a model of coupled oscillators with bidirectional first nearest neighbours coupling with periodic boundary conditions. We show that a stable phase-locked solution is decided by the oscillators at the borders between the major clusters, which merge to form a larger one of all oscillators at the stage of comple...
Lagrangian chaos is experimentally investigated in a convective flow by means of Particle Tracking Velocimetry. The Fnite Size Lyapunov Exponent analysis is applied to quantify dispersion properties at different scales.
Hyperchaos is a qualitatively stronger form of chaos, in which several degrees of freedom contribute simultaneously to exponential divergence of small changes. A hyperchaotic dynamical system is therefore even more unpredictable than a chaotic one, and has a higher fractal dimension.
We generalize earlier studies on the Laplacian for a bounded open domain $\Omega\in \real^2$ with connected complement and piecewise smooth boundary. We compare it with the quantum mechanical scattering operator for the exterior of this same domain.
Large systems of coupled oscillators subjected to a periodic external drive occur in many situations in physics and biology. Here the simple, paradigmatic case of equal-strength, all-to-all sine-coupling of phase oscillators subject to a sinusoidal external drive is considered.
We study the long time behaviour of a nonlinear oscillator subject to a random multiplicative noise with a spectral density (or power-spectrum) that decays as a power law at high frequencies. When the dissipation is negligible, physical observables, such as the amplitude, the velocity and the energy of the oscillator ...
The linear stability of pipe flow implies that only perturbations of sufficient strength will trigger the transition to turbulence. In order to determine this threshold in perturbation amplitude we study the \emph{edge of chaos} which separates perturbations that decay towards the laminar profile and perturbations tha...
In this communication, the incorrectness of phenomenological approach to the logistic growth equation, proposed by Castorina et al. is presented in detail. The correct phenomenological approach to logistic growth equation is also proposed here.
We investigate coupled circle maps in presence of feedback and explore various dynamical phases observed in this system of coupled high dimensional maps. We observe an interesting transition from localized chaos to spatiotemporal chaos.
We consider two models of deterministic active particles in an external potential. In the limit where the speed of a particle is fixed, both models coincide and can be formulated as a Hamiltonian system, but only if the potential is time-independent.
In this paper, we dealt the dynamic behavior of a magnetic structure, considering a shape memory spring and a non-ideal motor as excitation. It has fractal behavior of tits basins of attraction, for a set of parameters for energy harvesting.
This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed.
We propose to study echo dynamics in a random matrix framework, where we assume that the perturbation is time independent, random and orthogonally invariant. This allows to use a basis in which the unperturbed Hamiltonian is diagonal and its properties are thus largely determined by its spectral statistics.
The collinear hydrogen exchange reaction is a paradigm system for understanding chemical reactions. It is the simplest imaginable atomic system with $2$ degrees of freedom modeling a chemical reaction, yet it exhibits behaviour that is still not well understood - the reaction rate decreases as a function of energy bey...
We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the energy regimes is made.
A new simple method to measure the spatial distribution of the electric field in the plasma sheath is proposed. The method is based on the experimental investigation of vertical oscillations of a single particle in the sheath of a low-pressure radio-frequency discharge.
An exact non-perturbative calculation of the fourth-order anomalous correction to the scaling behaviour of a random shell-model for passive scalars is presented. Importance of ultraviolet (UV) and infrared (IR) boundary conditions on the inertial scaling properties are determined.
Extreme events such as rogue wave in optics and fluids are often associated with the merging dynamics of coherent structures. We present experimental and numerical results on the physics of extreme events appearance in a spatially extended semiconductor microcavity laser with intracavity saturable absorber.
In this paper, we explore the dynamics of a Hamiltonian system after a double van der Waals potential energy surface degenerates into a single well. The energy of the system is increased from the bottom of the potential well up to the dissociation energy, which occurs when the system becomes open.
We investigate explicit functions that can produce truly random numbers. We use the analytical properties of the explicit functions to show that certain class of autonomous dynamical systems can generate random dynamics.
We present large scale simulations for a one-dimensional chain of hard-point particles with alternating masses. We correct several claims in the recent literature based on much smaller simulations.
We report the identification of global phase synchronization (GPS) in a linear array of unidirectionally coupled Mackey-Glass time-delay systems exhibiting highly non-phase-coherent chaotic attractors with complex topological structure. In particular, we show that the dynamical organization of all the coupled time-del...
Dynamical systems driven by strong external signals are ubiquituous in nature and engineering. Here we study "echo state networks", networks of a large number of randomly connected nodes, which represent a simple model of a neural network, and have important applications in machine learning.
Low dimensional dynamics of large networks is the focus of many theoretical works, but controlled laboratory experiments are comparatively very few. Here, we discuss experimental observations on a mean-field coupled network of hundreds of semiconductor lasers, which collectively display effectively low-dimensional mix...
We study the dynamics of a collection of nonlinearly coupled limit cycle oscillators, relevant to systems ranging from neuronal populations to electrical circuits, under coupling topologies varying from a regular ring to a random network. We find that the trajectories of this system escape to infinity under regular co...
The inertia of particles driven by the turbulent flow of the surrounding fluid makes them prefer certain regions of the flow. The heavy particles lag behind the flow and tend to accumulate in the regions with less vorticity, while the light particles do the opposite.
We developed a powerful computational approach to elaborate on onset mechanisms of deterministic chaos due to complex homoclinic bifurcations in diverse systems. Its core is the reduction of phase space dynamics to symbolic binary representations that lets one detect regions of simple and complex dynamics as well as f...
Many biological processes and objects can be described by fractals. The paper uses a new type of objects - blinking fractals - that are not covered by traditional theories considering dynamics of self-similarity processes.
The correlation between level velocities and eigenfunction intensities provides a new way of exploring phase space localization in quantized non-integrable systems. It can also serve as a measure of deviations from ergodicity due to quantum effects for typical observables.
We study the fluctuations that are predicted in the autocorrelation function of an energy eigenstate of a chaotic, two-dimensional billiard by the conjecture (due to Berry) that the eigenfunction is a gaussian random variable. We find an explicit formula for the root-mean-square amplitude of the expected fluctuations ...
Kraichnan's model of passive scalar advection is studied as a case model for understanding the anomalous scaling in the anisotropic sectors. We show here that the solutions of the Kraichnan equation for the $n$ order correlations foliate into sectors that are classified by the irreducible representations of the SO...
Methods of dynamical system's theory are used for numerical study of transport and mixing of passive particles (water masses, temperature, salinity, pollutants, etc.) in simple kinematic ocean models composed with the main Eulerian coherent structures in a randomly fluctuating ocean -- a jet-like current and an edd...
In this communication we will re-examine the widely studied technique of phase space projection. By imposing a time domain constraint (TDC) on the residual noise, we deduce a more general version of the optimal projector, which includes those appearing in previous literature as subcases but does not assume the indepen...
We formulate, for continuous-time dynamical systems, a sufficient condition to be a gradient-like system, i.e. that all bounded trajectories approach stationary points and therefore that periodic orbits, chaotic attractors, etc. do not exist. This condition is based upon the existence of an auxiliary function defined ...
It has been proved that the spanning tree from a given network has the optimal synchronizability, which means the index $R=\lambda_{N}/\lambda_{2}$ reaches the minimum 1. Although the optimal synchronizability is corresponding to the minimal critical overall coupling strength to reach synchronization, it does not guar...
Employing some recent results in dynamics of systems with invariant subspaces we find evidence in both truncated and full axisymmetric mean-field dynamo models of a recently discovered type of intermittency, referred to as in-out intermittency. This is a generalised form of on-off intermittency that can occur in syste...
It was demonstrated in [Phys. Rev. E 86, 021104, (2012)], that the ground-state wave functions for a large variety of one-dimensional spin-1/2 models are multifractals in the natural spin-z basis. We present here the details of analytical derivations and numerical confirmations of these results.
We show that the number of KLD (Karhunen-Lo`eve decomposition) modes D_KLD(f) needed to capture a fraction f of the total variance of an extensively chaotic state scales extensively with subsystem volume V. This allows a correlation length xi_KLD(f) to be defined that is easily calculated from spatially localized data....
We present a numerical study of charged particle motion in a time-independent magnetic field in cylindrical geometry. The magnetic field model consists of an unperturbed reversed-shear helical part and a perturbation consisting of a superposition of modes.
We present a dissipative system with unstable dynamics called unstable dissipative system which are capable of generating a multi-stable behavior, i.e., depending on its initial condition the trajectory of the system converge to a specific basin of attraction. A piecewise linear (PWL) systems is generated based on uns...
Finding the distribution of vibro-acoustic energy in complex built-up structures in the mid-to-high frequency regime is a difficult task. In particular, structures with large variation of local wavelengths and/or characteristic scales pose a challenge referred to as the mid-frequency problem.
We study the dynamics of $N$-dimensional lattices of nonchaotic Rulkov neurons coupled with a flow of electrical current. We consider both nearest-neighbor and next-nearest-neighbor couplings, homogeneous and heterogeneous neurons, and small and large lattices over a wide range of electrical coupling strengths.
A $G$-strand is a map $g(t,{s}):\,\mathbb{R}\times\mathbb{R}\to G$ for a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. The SO(3)-strand is the $G$-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain.
An improved $\eps$ expansion in the $d$-dimensional ($d > 2$) stochastic theory of turbulence is constructed by taking into account pole singularities at $d \to 2$ in coefficients of the $\eps$ expansion of universal quantities. Effectiveness of the method is illustrated by a two-loop calculation of the Kolmogorov ...
We study smooth, caustic-free, chaotic semiclassical dynamics on two-dimensional phase space and find that the dynamics can be approached by an iterative procedure which constructs an approximation to the exact long-time semiclassical propagator. Semiclassical propagation all the way to the Heisenberg time, where indi...
We show that introducing quenched disorder into a circle map leads to the suppression of quasiperiodic behavior in the limit of large system sizes. Specifically, for most parameters the fraction of disorder realizations showing quasiperiodicity decreases with the system size and eventually vanishes in the limit of inf...
Scaling analysis of nuclear giant resonance transition probabilities with increasing level of complexity in the background states is performed. It is found that the background characteristics, typical for chaotic systems lead to nontrivial multifractal scaling properties.
Our work presents a study on the nonlinear dynamical behavior for a microcavity semiconductor containing a quantum well. Using an external periodic perturbation in energy level we observe the periodic-doubling, quasiperiodic, and direct route to chaos as forcing strength is changed.
We study the effects of additive noise on the lifetimes of chaotic repellors. Using first order perturbation theory we argue that noise will increase the lifetime if the escape holes lie in regions where the unperturbed density is higher than in the immediate vicinity and that it decreases if the density is lower.
An algorithm is presented to update the multi-fractal spectrum of a time series in constant time when new data arrives. The discrete wavelet transform (DWT) of the time series is first updated for the new data value.
A numerical study has been done of collisions between protons and hydrogen atoms, treated as classical particles, at low impact velocities. The presence of chaos has been looked for by investigating the processes with standard techniques of the chaotic--scattering theory.
The periodic motion of a classical point particle in a one-dimensional double-well potential acquires a surprising degree of complexity if friction is added. Finite uncertainty in the initial state can make it impossible to predict in which of the two wells the particle will finally settle.
We consider escape from chaotic maps through a subset of phase space, the hole. Escape rates are known to be locally constant functions of the hole position and size.
The field theoretic renormalization group (RG) is applied to the problem of a passive scalar advected by the Gaussian self-similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient. The energy spectrum in the inertial range has the form $E(k)\propto k^{1-\epsilon}$, and...
We present a method for the analysis of electroencephalograms (EEG). <br>In particular, small signals due to stimulation, so called evoked potentials, have to be detected in the background EEG.
This paper develops the geometry and analysis of the averaged Euler equations for ideal incompressible flow in domains in Euclidean space and on Riemannian manifolds, possibly with boundary. The averaged Euler equations involve a parameter $\alpha$; one interpretation is that they are obtained by ensemble averaging th...
We devote our studies to the subject of weakly nonintegrable dynamics of systems with a macroscopic number of degrees of freedom. Our main points of interest are the relations between the timescales of thermalization and the timescales of chaotization; the choice of appropriate observables and the structure of equatio...
We investigate exceptional points, which are branch point singularities of two resonance eigenstates, in spectra of the hydrogen atom in crossed external electric and magnetic fields. A procedure to systematically search for exceptional points is presented, and their existence is proven.
A conjecture due to Bohigas, Giannoni and Schmit (BGS), stating that the energy level correlations of quantum chaotic systems generically obey the laws of random matrix theory, is given a precise formulation for quantized symplectic maps. No statement is made about any individual quantum map.
We present a dynamical system that naturally exhibits two unstable attractors that are completely enclosed by each others basin volume. This counter-intuitive phenomenon occurs in networks of pulse-coupled oscillators with delayed interactions.
Nonlinear dynamics of a bouncing ball moving in gravitational field and colliding with a moving limiter is considered. Displacement of the limiter is a quadratic function of time.
A cellular automaton model is used to describe the dynamics of the catalytic oxidation of $CO$ on a $Pt(100)$ surface. The cellular automaton rules account for the structural phase transformations of the $Pt$ substrate, the reaction kinetics of the adsorbed phase and diffusion of adsorbed species.
The true dynamical randomness is obtained as a natural fundamental property of deterministic quantum systems. It provides quantum chaos passing to the classical dynamical chaos under the ordinary semiclassical transition, which extends the correspondence principle to chaotic systems.
Homoclinic and heteroclinic motions in dynamics equations on time scales is investigated. The utilized time scale is a specific one such that it is a union of disjoint compact intervals.
Simple dynamical systems -- with a small number of degrees of freedom -- can behave in a complex manner due to the presence of chaos. Such systems are most often (idealized) limiting cases of more realistic situations.
Three kinds of fractal solutions of model of reactors are presented. The first kind concerns the structure of Feigenbaum_s diagram on the limit of chaos.
A class of dynamical models of turbulence living on a one-dimensional dyadic-tree structure is introduced and studied. The models are obtained as a natural generalization of the popular GOY shell model of turbulence.

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