Narrow Results

The location and basins of attraction of attractors involved in nonlinear smooth dynamical systems play a crucial role in understanding the dynamics of these systems across a wide range of parameter values. Nonlinear dynamical systems, described by autonomous nonlinear coupled ordinary differential equations with three or more variables, often exhibit chaotic dynamics within specific parameter ranges. In the literature, three distinct routes to chaos are well known: period doubling, crisis, and intermittency. Recent research in the field of nonlinear dynamical systems focuses on the characterization and identification of hidden attractors. The primary objective of this study is to investigate the dynamics and the successive local and global bifurcations that give rise to chaotic behavior in a Modified Lorenz System (MLS). Our findings reveal that chaos emerges via the type II intermittency route. Furthermore, we identify the chaotic attractor as a hidden attractor, with its existence corroborated by the dissipative nature of the model beyond the subcritical Hopf bifurcation. Key contributions of this research include the derivation of analytical criteria for a degenerate pitchfork bifurcation, the numerical calculation of the first Lyapunov exponent, and a semi-analytic proof of the homoclinic bifurcation. These results enhance our understanding of the complex dynamics within the MLS and contribute to the broader field of nonlinear dynamics and chaos theory.

The distribution of a single increment of the limit lognormal process of Mandelbrot, several representations of its Mellin transform, and an explicit analytic continuation of the Selberg integral are reviewed. The intermittency invariance of the limit lognormal construction is used to establish a functional Feynman–Kac equation that captures the entire stochastic dependence structure of the limit process. This equation is a general rule of intermittency differentiation that quantifies how the joint distribution of an arbitrary number of increments of the limit process evolves as a function of intermittency. The solution is represented by means of a formal intermittency expansion and shown to be an exactly renormalized expansion in the joint centered moments of the limit process. The expansion coefficients are related to a novel extension of the Selberg integral.

We show that the intermittent collective behavior shown by several CA has the statistical properties of the on–off intermittency. The average transient time for trajectories to fall in a ε neighborhood of the off state scales with ε^-1. The laminar and burst phases are symmetric and the scaling exponent is -3/2 for the distribution of the phases. Also, we show that the size distribution of clusters follows a power law with a scaling exponent of -2.

In this paper, a search for power-law fluctuations with fractality and intermittency analysis to explore the QCD phase diagram and the critical point is summarized. Experimental data on self-similar correlations and fluctuations with respect to the size of phase–space volume in various high energy heavy-ion collisions are presented, with special emphasis on background subtraction and efficiency correction of the measurement. Phenomenological modeling and theoretical work on the subject are discussed. Finally, we highlight possible directions for future research.

We investigate the classical evolution of a ϕ⁴ scalar field theory, using in the initial state random field configurations possessing a fractal measure expressed by a noninteger mass dimension. These configurations resemble the equilibrium state of a critical scalar condensate. The measures of the initial fractal behavior vary in time following the mean field motion. We show that the remnants of the original fractal geometry survive and leave an imprint in the system time averaged observables, even for large times compared to the approximate oscillation period of the mean field, determined by the model parameters. This behavior becomes more transparent in the evolution of a deterministic Cantor-like scalar field configuration. We extend our study to the case of two interacting scalar fields, and we find qualitatively similar results. Therefore, our analysis indicates that the geometrical properties of a critical system initially at equilibrium could sustain for several periods of the field oscillations in the phase of nonequilibrium evolution.

Most signatures of new physics have been studied on the transverse plane with respect to the beam direction at the LHC where background is much reduced. In this paper we propose the analysis of inclusive longitudinal (pseudo)rapidity correlations among final-state (charged) particles in order to search for (un)particles belonging to a hidden sector beyond the Standard Model, using a selected sample of p–p minimum bias events (applying appropriate off-line cuts on events based on, e.g. minijets, high-multiplicity, event shape variables, high-p_⊥ leptons and photons, etc.) collected at the early running of the LHC. To this aim, we examine inclusive and semi-inclusive two-particle correlation functions, forward–backward correlations, and factorial moments of the multiplicity distribution, without resorting to any particular model but under very general (though simplifying) assumptions. Finally, motivated by some analysis techniques employed in the search for quark–gluon plasma in heavy-ion collisions, we investigate the impact of such intermediate (un)particle stuff on the (multi)fractality of parton cascades in p–p collisions, by means of a Lévy stable law description and a Ginzburg–Landau model of phase transitions. Results from our preliminary study seem encouraging for possible dedicated analyses at LHC and Tevatron experiments.

Modeling statistical properties of motion of a Lagrangian particle advected by a high-Reynolds-number flow is of much practical interest and complement traditional studies of turbulence made in Eulerian framework. The strong and nonlocal character of Lagrangian particle coupling due to pressure effects makes the main obstacle to derive turbulence statistics from the three-dimensional Navier–Stokes equation; motion of a single fluid-particle is strongly correlated to that of the other particles. Recent breakthrough Lagrangian experiments with high resolution of Kolmogorov scale have motivated growing interest to acceleration of a fluid particle. Experimental stationary statistics of Lagrangian acceleration conditioned on Lagrangian velocity reveals essential dependence of the acceleration variance upon the velocity. This is confirmed by direct numerical simulations. Lagrangian intermittency is considerably stronger than the Eulerian one. Statistics of Lagrangian acceleration depends on Reynolds number. In this review we present description of new simple models of Lagrangian acceleration that enable data analysis and some advance in phenomenological study of the Lagrangian single-particle dynamics. Simple Lagrangian stochastic modeling by Langevin-type dynamical equations is one the widely used tools. The models are aimed particularly to describe the observed highly non-Gaussian conditional and unconditional acceleration distributions. Stochastic one-dimensional toy models capture main features of the observed stationary statistics of acceleration. We review various models and focus in a more detail on the model which has some deductive support from the Navier–Stokes equation. Comparative analysis on the basis of the experimental data and direct numerical simulations is made.

On the basis of the lattice Boltzmann method for the Navier–Stokes equation, we have done a numerical experiment of a forced turbulence in real space and time. Our new findings are summarized into two points. Firstly, in the analysis of the mean-field behavior of the velocity field using the exit-time statistics, we have verified Kolmogorov's scaling and Taylor's hypothesis at the same time. Secondly, in the analysis of the intermittent velocity fluctuations using a non-equilibrium probability distribution function and the wavelet denoising, we have clarified that the coherent vortices sustain the power-law velocity correlation in the non-equilibrium state.

In this paper, the two-dimensional (2D) turbulence perturbed by arrays of cylinders placed both horizontally and vertically is investigated by Immersed Boundary Lattice Boltzmann Method (IB-LBM). The energy spectrum reveals the coexistence of the inverse and direct cascades in 2D grid turbulence. By observing at the distribution of fluxes in space, the energy and enstrophy fluxes have explained the physical mechanism of the double cascades where the two Kolmogorov laws for structure functions are simultaneously observed. The results of vortex statistics by the conditional analysis, which are based on a new and accurate vortex identification criteria called Liutex, show that the algebraic number density $n(A)\sim A^{-2}$, where $A$ is vortex area. The time-evolving vortex number density distribution constructs a theoretical framework involving a three-part: $n(A,t)=A^{-0.5}(A_{min}\le A<A_{-})$; $t^{-1}{A}^{-1}(A_{-}<A<A_{+})$; $t^7{A}^{-8}(A_{+}<A\le A_{max})$, which is satisfied with the prediction well. The relationship between the vortex circulation $\left|\mathrm{\Gamma }\right|(A)$ and vortex area $A$ is $\left|\mathrm{\Gamma }\right|(A)\sim A$ and the one between the kinetic energy of vortex $Ee(A)$ and $A$ is $Ee(A)\sim A$ in the range where $n(A)\sim A^{-2}$. Moreover, it has been found that vortices contain about 30% of the total energy of the flow by studying the energy ratio of all vortices to the entire flow field. What is more, it is an interesting phenomenon is that there is only a range where $E(k)\sim k^{-5/3}$ in the energy spectrum for the coherent structure field which is obtained by using Liutex as the extraction of vortices. The probability density function (PDF) of the fluctuations of longitudinal velocity shows that an indication of small intermittency in the direct cascade and the absence of intermittency in the inverse cascade range. On the other hand, the scaling exponents $\zeta (p)$ of the structure function for the inverse cascade are consistent with Kr67 model, which shows the absence of intermittency. While the measured intermittency parameters are $J=0.006$ and $H=0.921$, which explains that there is a very weak intermittent correction in the direct cascade, and ESS has verified the existence of intermittency in our 2D turbulence.

We report our results on nonperiodic experimental time series of pressure in a spark ignition engine. The experiments were performed for a low rotational velocity of a crankshaft and a relatively large spark advance angle. We show that the combustion process has many chaotic features. Surprisingly, the reconstructed attractor has a characteristic butterfly shape similar to a chaotic attractor of Lorentz type. The suitable recurrence plot shows that the dynamics of the combustion is a nonlinear multidimensional process mediated by stochastic noise.

In this paper, coupling properties of regular and chaotic calcium oscillations are examined. Synchronized calcium signals among coupled cells in tissue, where calcium ions were found to be one of the most important second messengers, have proven indispensable for proper and reliable functioning of living organisms. When modeling such systems, it is of particular interest to determine, which internal system properties guarantee best coupling abilities and herewith physiologically the most efficient signal transduction between cells. We found that local contractive properties of attractors in phase space, quantified by the local divergence, represent one of the crucial system properties that determine synchronization abilities of coupled regular and chaotic oscillators. In particular, parts of attractors with close to zero local divergence largely facilitate synchronization of initially unsynchronized oscillators. For bursting oscillations, this is fully in agreement with previous studies showing that synchronization abilities of bursters are closely related with the slow passage effect. We extended this concept with the help of local divergence and succeeded to apply our theory also to other oscillatory regimes, like regular spiking and complex chaotic oscillations.

We investigate the bifurcation structures in a two-dimensional parameter space (PS) of a parametrically excited system with two degrees of freedom both analytically and numerically. By means of the Rényi entropy of second order K₂, which is estimated from recurrence plots, we uncover that regions of chaotic behavior are intermingled with many complex periodic windows, such as shrimp structures in the PS. A detailed numerical analysis shows that the stable solutions lose stability either via period doubling, or via intermittency when the parameters leave these shrimps in different directions, indicating different bifurcation properties of the boundaries. The shrimps of different sizes offer promising ways to control the dynamics of such a complex system.

This paper introduces a type of one-dimensional nonsmooth nonlinear discrete dynamic system. We find a direct route to chaos from stable period-two point, and this is called Sudden Occurrence of Chaos. It is completely different from the three routes from regular motion to chaos — period-doubling bifurcation chaos, intermittency and quasi-periodicity chaos. Furthermore, we present some examples of sudden occurrence of chaos from m-period directly to chaos.

Intermittent instability is commonly observed in switching power supplies during the design and development phase. It manifests as symmetrical period-doubling bifurcation in the time domain with long intermittent periods. Such intermittent operation is considered undesirable in practice and is usually avoided by appropriate adjustments of circuit parameters. This paper explores the mechanism and conditions for the emergence of intermittency in a common voltage-mode controlled buck converter. It is found that interference at frequencies near the switching frequency or its rational multiples will induce intermittent operation. The strengths and frequencies of the interfering signals determine the type and period of intermittency. The problem is analyzed by transforming the conventional parameter-bifurcation analysis to a time-bifurcation analysis. Analytical results are verified by simulations and experimental measurements.

We discuss a general mechanism that induces periodic switching between different dynamic phenomena in rings of cells. Such intermittent behavior is normally modeled by asymptotically stable heteroclinic cycles. Our mechanism does not require a heteroclinic cycle in the ring of cells, instead a Van der Pol oscillator forces our system so that it periodically changes state.

We investigate this mechanism numerically in a number of different examples. These simulations demonstrate a variety of interesting forms of intermittency.

Multistability is an interesting phenomenon of nonlinear dynamical systems. To gain insights into the effects of noise on multistability, we consider the parameter region of the Lorenz equations that admits two stable fixed point attractors, two unstable periodic solutions, and a metastable chaotic "attractor". Depending on the values of the parameters, we observe and characterize three interesting dynamical behaviors: (i) noise induces oscillatory motions with a well-defined period, a phenomenon similar to stochastic resonance but without a weak periodic forcing; (ii) noise annihilates the two stable fixed point solutions, leaving the originally transient metastable chaos the only observable; and (iii) noise induces hopping between one of the fixed point solutions and the metastable chaos, a three-state intermittency phenomenon.

We discuss a system with a discontinuity moments generator to obtain the solutions. It is proven that the system is chaotic and has shadowing property [Anosov, 1967; Bowen, 1975; Hammel et al., 1987; Robinson, 1995] if the generator possesses these characteristics. Illustrative examples are provided.

Intermittent search processes alternate between two different stochastic motions in order to reach a given target. If the faster motion has a lower probability to detect the target, a question arises concerning the efficiency of both processes, and it may be possible to minimize the search time by a convenient choice of the parameters. This argument has been used to interpret observations in molecular biology or to explain the behavior of animals when searching for food. It can also have interesting consequences for the kinetics of reactions in heterogeneous media. In particular, the reaction kinetics in a biological cell can be enhanced when the active molecules occasionally bind to molecular motors that inactivate their reactivity and carry them far away. Here, we present a synthesis of the recent results obtained on these topics, with new perspectives and possible applications of intermittent behavior in reaction kinetics to be soon developed.

We study analytically and numerically the reinjection probability density for type-II intermittency. We find a new one-parameter class of reinjection probability density where the classical uniform reinjection is a particular case. We derive a new duration probability density of the laminar phase. New characteristic relations e^β(-1 < β < 0) appear where the exponet β deepens on the reinjection probability distributions. Analytical results are in agreement with the numerical simulations.

In this paper, we consider compact, invariant sets in Hamiltonian systems in order to extend the concept of crisis to such systems. We focus on crisis-induced intermittency in several systems where two invariant sets merge, obtaining scaling laws for the residence times and for the probability distribution decay as a function of a critical parameter. The connection to hitherto known crisis-induced intermittency in dissipative systems is discussed.