Chaotic Regime Research Articles

A Nonlinear Quark–Gluon Cascade Converges and Transits to a Chaotic Regime

A Nonlinear Quark–Gluon Cascade Converges and Transits to a Chaotic Regime

Synchronization of chaos in semiconductor gas discharge model with local mean energy approximation

The delta synchronization is a useful method to analyze appearance of chaotic synchronization in gas discharge systems. In recent studies, the generalized synchronization method has been implemented in various gas discharge systems. However, synchronization is not detected with this conventional method. In our previous study, we introduced the delta synchronization method and applied it to the gas discharge-semiconductor system (GDSS) via the one-dimensional ‘simple’ fluid model approach. In the present study, we implement this method in the more detailed (in terms of plasma chemical reactions and treatment of the electron transport) fluid model, namely the ‘extended’ fluid model or ‘local mean energy approximation’ model. The description of the GDSS model is given, and a bifurcation diagram demonstrates the system’s transition to the chaotic regime. The unpredictable motion, which proves the existence of Poincaré chaos, and the delta synchronized motion are confirmed by the numerical simulations, and corresponding algorithms are given. The time sequences corresponding to the unpredictability and delta synchronization are presented in tables. For consistency, the absence of generalized synchronization is also shown via the auxiliary system approach. The numerical characteristics indicating the degrees of chaos and synchronization are described and implemented in the analysis. These features are also used to compare our results to the simpler model.

Probing chaos by magic monotones

There is a property of a quantum state called ``magic.'' As shown by the Gottesman-Knill theorem, so-called stabilizer states, which are composed of only Clifford gates, can be efficiently computed on a classical computer, and thus quantum computation gives no advantage. Nonstabilizer states are called magic states, which are necessary to achieve the universal quantum computation. Magic (monotone) is the measure of the amount of nonstabilizer resource, and it measures how difficult it is for a classical computer to simulate the state. We study magic of states in the integrable and chaotic regimes of the higher-spin generalization of the Ising model through two quantities: ``mana'' and ``robustness of magic'' (RoM). We find that in the chaotic regime, mana increases monotonically in time in the early-time region, and at late times these quantities oscillate around some nonzero value that increases linearly with respect to the system size. Our result also suggests that under chaotic dynamics, any state evolves to a state whose mana almost saturates the optimal upper bound; i.e., the state becomes ``maximally magical.'' We find that RoM also shows similar behaviors. On the other hand, in the integrable regime, mana and RoM behave periodically in time in contrast to the chaotic case. In addition to mana and RoM, for the early-time behavior of magic, we study the stabilizer R\'enyi entropy, which can be numerically computed for larger systems than mana and RoM. In the anti-de Sitter/conformal field theory correspondence, classical spacetime emerges from the chaotic nature of the dual quantum system. Our results suggest that magic of quantum states is strongly involved in the emergence of spacetime geometry.

Mixing performance of T-shape micromixers equipped with 3D printed Gyroid matrices: A numerical evaluation

This paper introduces innovative T-shape micromixers equipped with various 3D printed Gyroid matrices aiming to boost the mixing performance. A regular Gyroid matrix has been adapted into more complicated twisted, functionally graded, and stochastic Gyroid structures to promote a chaotic flow regime. A numerical model has been derived and validated to investigate the impact of various Gyroid matrices on the mixing index, pressure drop, and performance index of T-shape micromixers at various Reynolds and Schmidt numbers. The results indicate that, among all structures, the twisted Gyroid matrix provides the highest enhancement in the mixing performance by boosting the mixing index up to 172%. Moreover, the twisted micromixer provided a 135% higher mixing index with a mixing channel that is 41% shorter as compared to a full-length plain micromixer. The novel design, therefore, could be implemented in developing more compact and more efficient microfluidic systems in chemical and pharmaceutical integrated micro-operations.

Dissipative Kerr solitons, breathers, and chimera states in coherently driven passive cavities with parabolic potential.

We analyze the stability and dynamics of dissipative Kerr solitons (DKSs) in the presence of a parabolic potential. This potential stabilizes oscillatory and chaotic regimes, favoring the generation of static DKSs. Furthermore, the potential induces the emergence of new dissipative structures, such as asymmetric breathers and chimera-like states. Based on a mode decomposition of these states, we unveil the underlying modal interactions.

Purification and scrambling in a chaotic Hamiltonian dynamics with measurements

Chaotic transverse-field Ising model with measurements exhibits interesting purification dynamics. Ensemble of non-unitary dynamics of a chaotic many-body system with measurements exhibits a purification phase transition. We numerically find that the law of the increase dynamics of the purity changes by projective measurements in the model. In order to study this behavior in detail, we construct the formalism of the tripartite mutual information (TMI) for non-unitary time evolution operator by using the state-channel map. The numerical result of the saturation value of the TMI indicates the existence of a measurement-induced phase transition. This implies the existence of two distinct phases, mixed phase and purified phase. Furthermore, the real-space spread of the TMI is investigated to explore spatial patterns of information spreading. Even in the purified phase, the spatial pattern of the light cone spread of quantum information is not deformed, but its density of information propagation is reduced on average by the projective measurements. We also find that this spatial pattern of the TMI distinguishes the chaotic and integrable regimes of the system.

Continuous and discrete data assimilation with noisy observations for the Rayleigh-Bénard convection: a computational study

Obtaining accurate high-resolution representations of model outputs is essential to describe the system dynamics. In general, however, only spatially- and temporally-coarse observations of the system states are available. These observations can also be corrupted by noise. Downscaling is a process/scheme in which one uses coarse scale observations to reconstruct the high-resolution solution of the system states. Continuous Data Assimilation (CDA) is a recently introduced downscaling algorithm that constructs an increasingly accurate representation of the system states by continuously nudging the large scales using the coarse observations. We introduce a Discrete Data Assimilation (DDA) algorithm as a downscaling algorithm based on CDA with discrete-in-time nudging. We then investigate the performance of the CDA and DDA algorithms for downscaling noisy observations of the Rayleigh-Bénard convection system in the chaotic regime. In this computational study, a set of noisy observations was generated by perturbing a reference solution with Gaussian noise before downscaling them. The downscaled fields are then assessed using various error- and ensemble-based skill scores. The CDA solution was shown to converge towards the reference solution faster than that of DDA but at the cost of a higher asymptotic error. The numerical results also suggest a quadratic relationship between the ℓ2 error and the noise level for both CDA and DDA. Cubic and quadratic dependences of the DDA and CDA expected errors on the spatial resolution of the observations were obtained, respectively.

TARDYS Quantifiers: Extracting Temporal and Reversible DYnamical Symmetries

One of the great challenges in complex and chaotic dynamics is to reveal the details of its underlying determinism. This can be manifest in the form of temporal correlations or structured patterns in the dynamics of a measurable variable. These temporal dynamical structures are sometimes a consequence of hidden global symmetries. Here, we identify the temporal (approximate) symmetries of a semiconductor laser with external optical feedback, based on which we define the Temporal And Reversible DYnamical Symmetry (TARDYS) quantifiers to evaluate the relevance of specific temporal correlations in a time series. We show that these symmetries are also present in other complex dynamical systems, letting us extrapolate one system’s symmetries to characterize and distinguish chaotic regimes in other dynamical systems. These symmetries, natural of the dynamics of the laser with feedback, can also be used as indicators in forecasting regular-to-chaos transitions in mathematical iterative maps. We envision that this can be a useful tool in experimental data, as it can extract key features of the deterministic laws that govern the dynamics of a system despite the lack of knowledge of those specific quantitative descriptions.

Simulation for flow across heated square cylinders

AbstractThe thermal lattice Boltzmann method is used to examine forced convection heat transfer from six inline heated square cylinders for Re = 100 at 0.5 ≤ s/d ≤ 4.0, where s is the distance between the surfaces of two cylinders and d is the cylinder size. Such a heat transfer is transient in nature for which the present work reports heat transfer regimes such as synchronous, quasiperiodic and chaotic. For 0.5 ≤ s/d ≤ 1.5 the heat transfer is synchronous, for 1.5 ≤ s/d ≤ 3.0 it is quasiperiodic and for 3.0 ≤ s/d ≤ 4.0 it is chaotic in nature at Re = 100. These regimes are confirmed through cylinder Nusselt number signals, its power spectra, and heat wake interference. The appearance of heat transfer regimes for inline heated cylinders is similar to the appearance of flow regimes for inline unheated cylinders except for the fact that transition from synchronous to quasiperiodic regime occurs at s/d = 1.5 for heat transfer and at s/d = 1.1 for flow. The synchronous heat transfer regime is characterized by a single heat wake that envelopes all cylinders while quasiperiodic heat transfer regime is characterized by the formation of thermal blobs in the gap between cylinders. A chaotic heat transfer regime is characterized by the shedding of thermal blobs and interference of thermal blobs by downstream cylinders. Regardless of spacing, the average Nusselt number encountered by cylinders is smaller than that for the isolated cylinder. The novelty of the work is that transitions occurring in the flow of heat are considered for an understanding of heat flow from bluff bodies.

Dynamical Interpretation of Logistic Map using Euler’s Numerical Algorithm

In the last two decades, the dynamics of difference and differential equations have found a celebrated place in science and engineering such as weather forecasting, secure communication, transportation problems, biology, the population of species, etc. In this article, we deal with the dynamical behavior of the logistic map using Euler’s numerical algorithm. The dynamical properties of Euler’s type logistic system are derived analytically as well as experimentally. In the analytical section, the dynamical properties such as fixed point, period-doubling, and irregularity are examined followed by s few theorems. Further, in the experimental section, the dynamical properties of Euler’s type logistic system are studied using period-doubling bifurcation plots. Because the dynamics of the Euler’s map depend on the Euler’s control parameter h, therefore, three major cases are discussed for all the dynamical properties for h = 0.1, 0.4, and 0.7. The result shows that as the value of parameter h decreases from 1 to 0 the growth rate parameter r increases rapidly. Therefore, the improved chaotic regime in bifurcation plots may improve the chaos based applications in science and engineering such as secure communication.

Experimental observation of chaotic hysteresis in Chua's circuit driven by slow voltage forcing

The Chua's circuit has been considered as a paradigm for the investigation of chaos, but it still presents many unexplored dynamics. Chaotic hysteresis is an effect observed when the hysteresis phenomenon and chaotic dynamics phenomenon act simultaneously. It is an interesting phenomenon since it is observed in a variety of disciplines, and it has been utilized in explaining the mechanism of bursting oscillations, rich dynamics in thermal convection, and economic financial crisis, to mention a few. In this paper, we report the experimental observation of chaotic hysteresis in a Chua's electronic circuit driven by both a dc and a slow triangular voltage source. One and two hysteresis loops were observed in single and double scroll chaotic regimes, respectively. We present variations of the attractor that are induced by voltage changes, as well as the dependence of the hysteresis effect on both the driving frequency and the distance from the threshold passing from single to double-scroll chaotic regimes. Our results help in expanding our knowledge on unexplored dynamics of Chua's circuit as well as contribute to chaos control theory with a potential for applications.

How Does Noise Induce Order?

In this paper we present a general result with an easily checkable condition that ensures a transition from chaotic regime to regular regime in random dynamical systems with additive noise. We show how this result applies to a prototypical family of nonuniformly expanding one dimensional dynamical systems, showing the main mathematical phenomenon behind Noise Induced Order. Accepted at Journal of Statistical Physics

Static and Dynamic Bifurcations Analysis of a Fluid-Conveying Pipe with Axially Moving Supports Surrounded by an External Fluid

In this study, nonlinear vibration and stability of a pipe conveying fluid, with axially moving supports, immersed in an external fluid is studied. The equation of motion is derived with the aid of the Newtonian method via employing the Euler–Bernoulli beam theory and plug flow assumption. Considering the stretching effect, the nonlinear equation of motion is obtained, and it is discretized via the Galerkin method. Afterward, the stability of the system is investigated using two different approaches, and their results are compared. Numerical simulations show the system has complex dynamic behavior. To unveil the dynamics of the system, various analysis techniques such as bifurcation diagram of Poincaré map, phase portrait, power spectrum, and time trajectories are used. Limit cycle, period-n, quasi-periodic, and chaotic regime are observed. It is found that pipe can flutter around the trivial solution or post-buckling equilibrium points. Moreover, the influences of some system parameters on the chaotic behavior of the pipe are discussed.

Onset of universality in the dynamical mixing of a pure state

We study the time dynamics of random density matrices generated by evolving the same pure state using a Gaussian orthogonal ensemble (GOE) of Hamiltonians. We show that the spectral statistics of the resulting mixed state is well described by random matrix theory (RMT) and undergoes a crossover from the GOE to the Gaussian unitary ensemble (GUE) for short and large times respectively. Using a semi-analytical treatment relying on a power series of the density matrix as a function of time, we find that the crossover occurs in a characteristic time that scales as the inverse of the Hilbert space dimension. The RMT results are contrasted with a paradigmatic model of many-body localization in the chaotic regime, where the GUE statistics is reached at large times, while for short times the statistics strongly depends on the peculiarity of the considered subspace.

Organized periodic structures and coexistence of triple attractors in a predator–prey model with fear and refuge

We report the existence of periodic structures viz., Arnold tongues, and shrimp-shaped structures in the transitional and chaotic regimes in bi-parameter spaces of a predator–prey model. The model is constructed taking into consideration of two important effects: the prey refuge and fear of predation risk. The fixed points, their existence, and stability behaviors are analyzed. The occurrences of transcritical and Neimark–Sacker bifurcations are studied in both analytical and numerical approaches. The complex dynamical behaviors of the predator–prey model are explored in the bi-parameter space with the help of the largest Lyapunov exponent and isoperiodic diagrams. A completely new kind of U-shaped periodic structure has emerged by merging two Arnold tongues of the same period in the parameter plane. The transition to chaos also occurs through period-bubbling process. The coexistence of triple heterogeneous attractors indicates the unpredictability of the present deterministic model. The basin sets of triple coexisting attractors are drawn, and we observe the presence of Wada basin boundaries. Our studies reveal that the oscillations of the predator–prey populations in certain control parameter regions are highly dependent upon the initial densities of the populations.

Self-generation of Möbius solitons and chaotic waveforms in magnonic-optoelectronic oscillators under simultaneous action of optic and magnonic nonlinearities

Self-generation of microwave nonlinear waveforms in the magnonic-optoelectronic oscillator (MOEO) was investigated. Nonlinear dynamics of the MOEO was due to both optical and magnonic paths of the oscillator circuit. Four-magnon parametric interactions in the magnonic path and cosine transfer function of the electro-optical modulator caused double nonlinearity of the MOEO. Gain coefficient was used as a control parameter. We found that during a route from regular to chaotic dynamics, the oscillator generates two unusual waveforms: symmetry-breaking soliton-like modes of Möbius type and periodic pulses with chaotic amplitude modulation. Nonlinear waveforms were characterized using a time series analysis. Peculiarities of the signals and their spectra in regular and chaotic regimes of self-generation are discussed. We expect that the multiple nonlinearity of the MOEO may be useful for investigation of various fundamental effects in complex time-delayed systems and for development of novel circuits for neuromorphic computing.

Analyticity constraints bound the decay of the spectral form factor

Quantum chaos cannot develop faster than λ≤2π/(ℏβ) for systems in thermal equilibrium [Maldacena, Shenker & Stanford, JHEP (2016)]. This `MSS bound' on the Lyapunov exponent λ is set by the width of the strip on which the regularized out-of-time-order correlator is analytic. We show that similar constraints also bound the decay of the spectral form factor (SFF), that measures spectral correlation and is defined from the Fourier transform of the two-level correlation function. Specifically, the inflection exponentη, that we introduce to characterize the early-time decay of the SFF, is bounded as η≤π/(2ℏβ). This bound is universal and exists outside of the chaotic regime. The results are illustrated in systems with regular, chaotic, and tunable dynamics, namely the single-particle harmonic oscillator, the many-particle Calogero-Sutherland model, an ensemble from random matrix theory, and the quantum kicked top. The relation of the derived bound with other known bounds, including quantum speed limits, is discussed.

Coupling Techniques for Nonlinear Ensemble Filtering

We consider filtering in high-dimensional non-Gaussian state-space models with intractable transition kernels, nonlinear and possibly chaotic dynamics, and sparse observations in space and time. We propose a novel filtering methodology that harnesses transportation of measures, convex optimization, and ideas from probabilistic graphical models to yield robust ensemble approximations of the filtering distribution in high dimensions. Our approach can be understood as the natural generalization of the ensemble Kalman filter (EnKF) to nonlinear updates, using stochastic or deterministic couplings. The use of nonlinear updates can reduce the intrinsic bias of the EnKF at a marginal increase in computational cost. We avoid any form of importance sampling and introduce non-Gaussian localization approaches for dimension scalability. Our framework achieves state-of-the-art tracking performance on challenging configurations of the Lorenz-96 model in the chaotic regime.

Magnetized plasma pressure filaments: Analysis of chaotic and intermittent transport events driven by drift-Alfvén modes

The origin of intermittent fluctuations in an experiment involving several interacting electron plasma pressure filaments in close proximity, embedded in a large linear magnetized plasma device, is investigated. The probability density functions of the fluctuations on the inner and outer gradient of the filament bundle are non-Gaussian and the time series contain uncorrelated Lorentzian pulses that give the frequency power spectral densities an exponential shape. A cross-conditionally averaged spatial reconstruction of a temporal event reveals that the intermittent character is caused by radially and azimuthally propagating turbulent structures with transverse spatial scales on the order of the electron skin depth. These eruption events originate from interacting pressure gradient-driven drift-Alfvén instabilities on the outer gradient and edge of the filament bundle. The temporal Lorentzian shape of the intermittent structures and exponential spectra are suggestive of deterministic chaos in the underlying dynamics; this conclusion is supported by the complexity–entropy analysis (CH-plane) that shows the experimental time series are located in the chaotic regime.

Geometric integration and dynamic properties

This paper reviews the Lie Group’s usage and applications in numerical methods for ordinary differential equations. We present comparisons between the midpoint symplectic and the usual Runge–Kutta methods for distinct dynamical behaviors ranging from integrable to chaotic regimes. Simulation results show that the first has better precision in the regular region of the phase space, according to a statistical indicator defined in this work. Close to the homoclinic crossing, this performance degrades sharply.