Fractal Phase Research Articles

Matryoshka multistability: Coexistence of an infinite number of exactly self-similar nested attractors in a fractal phase space

Multistability, and its special types such as megastability and extreme multistability, is an important phenomenon in modern nonlinear science that provides several possible practical applications. In this paper, we propose a new special type of multistability when the infinite number of exactly self-similar attractors nested inside each other coexist in a system. We called it matryoshka multistability due to its resemblance to the famous Russian wooden doll. We theoretically explain and experimentally confirm the properties of a new type of multistable behavior using two representative examples based on the Chua and Sprott Case J chaotic systems. In addition, we construct an adaptive controller for synchronizing two Chua-type matryoshka multistable systems when the amplitude of the master system is of arbitrary scale. The proposed type of multistability can find several applications in chaotic communication, cryptography, and data compression.

Machine learning wave functions to identify fractal phases

Machine learning wave functions to identify fractal phases

Changes in oscillatory and nonlinear dynamic processes of microcirculation in patients with obliterating atherosclerosis of lower limb arteries after revascularization

Aim: To study the nature of changes in oscillatory and nonlinear dynamic processes in the skin microcirculatory system by laser Doppler flowmetry in patients with obliterating atherosclerosis of the lower limbs arteries (OALLA) after limb revascularization.Material and Methods. 27 male patients with OALLA before and after endovascular revascularization of the affected limb (median age 63.0 [60.0; 69.0] years) were studied. Microcirculation (MC) of the foot skin with assessment of nonlinear dynamic processes and spectral wavelet analysis of blood flow fluctuations was studied by laser Doppler flowmetry. The normalized amplitude indices of blood flow fluctuations were determined in frequency ranges reflecting: endothelial, neurogenic, myogenic, respiratory, pulse factors of hemocirculation. Bypass parameters and nutritive blood flow were assessed. An occlusion test was performed to determine capillary blood flow reserve. The study of nonlinear dynamic processes included assessment of fractal dimension, entropy determination and phase portrait analysis.Results. Limb revascularization in patients with OALLA resulted in improvement of the clinical picture accompanied by statistically significant increase in nutritive blood flow (+9.7%) and reserved dilatation potential of microvessels (+43.2%), decrease in arteriolo-venular blood shunting (–5.0%) and venous plethora (–14.3%). The analysis of nonlinear dynamic processes of the MC showed that after angioplasty, along with the remaining deficit of oscillatory processes energy, there was a decrease in the fractal dimension index (–14.3%), indicating the limitation of lability of the functional system of the microvascular bed. At the same time, an increase in the chaotization of the regulatory mechanisms of the peripheral blood flow was established.Conclusions. The results showed positive functional changes of the MC system associated with the improved clinical picture in patients with OALLA after limb revascularization. At the same time, changes in the nonlinear dynamics parameters indicate a compensatory increase in the chaotization of the system together with the remaining limitation of its functional lability and the energy deficit of oscillatory processes.

Entanglement phases in large- N hybrid Brownian circuits with long-range couplings

We develop solvable models of large-$N$ hybrid quantum circuits on qubits and fermions with long-range power-law interactions and continuous local monitoring, which provide analytical access to the entanglement phase diagram and error-correcting properties of many-body entangled non-equilibrium states generated by such dynamics. In one dimension, the long-range coupling is irrelevant for $\alpha>3/2$, where $\alpha$ is the power-law exponent, and the models exhibit a conventional measurement-induced phase transition between volume- and area-law entangled phases. For $1/2<\alpha<3/2$ the long-range coupling becomes relevant, leading to a nontrivial dynamical exponent at the measurement-induced phase transition. More interestingly, for $\alpha<1$ the entanglement pattern receives a sub-volume correction for both area-law and volume-law phases, indicating that the phase realizes a quantum error correcting code whose code distance scales as $L^{2-2\alpha}$. While the entanglement phase diagram is the same for both the interacting qubit and fermionic hybrid Brownian circuits, we find that long-range free-fermionic circuits exhibit a distinct phase diagram with two different fractal entangled phases.

Localization and fractality in disordered Russian Doll model

Motivated by the interplay of Bethe-Ansatz integrability and localization in the Richardson model of superconductivity, we consider a time-reversal symmetry breaking deformation of this model, known as the Russian Doll Model (RDM), and implement diagonal on-site disorder. The localization and ergodicity-breaking properties of the single-particle spectrum are analyzed using a large-energy renormalization group (RG) over the momentum-space spectrum. Based on the above RG, we derive an effective Hamiltonian of the model, discover a fractal phase of non-ergodic delocalized states – with the fractal dimension different from the paradigmatic Rosenzweig-Porter model – and explain it in terms of the developed RG equations and the matrix-inversion trick.

Non-linear dynamic processes and their correlation with indicators of microcirculation in patients with obliterating atherosclerosis of the lower extremities arteries according to laser doppler flowmetry

Objective: To study features of nonlinear dynamic and oscillatory processes in the skin microcirculatory flow in patients with obliterating atherosclerosis in lower extremities arteries (OALEA).Material and methods. 56 male patients with obliterating atherosclerosis in the arteries of lower extremities and 15 practically healthy individuals were taken into the study. Microcirculation in the skin on feet with the assessment of nonlinear dynamic processes and spectral wavelet analysis of blood fl ow fl uctuations was studied by laser Doppler flowmetry, device “LACK-M” (firm Lazma, Russia). Normalized amplitude parameters of blood flow fluctuations in frequency ranges of hemocirculation tonus-forming factors were analyzed: endothelial, neurogenic, myogenic. Myogenic microvascular tone and capillary blood flow indices were calculated. To study nonlinear dynamic processes, the following parameters were assessed: fractal dimension, entropy and phase portrait analysis.Results. Spastic-atonic microcirculation disorders were revealed in patients with OALEA. They were characterized with metarteriole constriction and with restriction of capillary blood flow, with an increased myogenic tone and dilated arterioles. Statistically significant shifts in parameters of nonlinear dynamics in this case- in particular, decrease in entropy (–10.6 %) and in dimensions of phase portrait (–9.3%) under energy deficit in oscillatory microcirculation processes (–20.8 %) – can be interpreted as a decrease in complexity of laser Doppler flowmetric signal and better ordering in regulatory mechanisms of the peripheral blood flow.Conclusion. In patients with obliterating atherosclerosis in the arteries of lower extremities, the obtained results indicate significant functional microcirculation disorders of the spastic-atonic nature leading to the restriction of nutritive blood flow and accompanied by the lack of energy in oscillatory processes. These results have also shown the decrease in the system chaos severity which indicates limitation of connections of microcirculation control factors, simplification of the mechanisms of functioning of the microvascular bed and, as a consequence, decrease in the compensatory adaptive potential.

Non-Hermitian Rosenzweig-Porter random-matrix ensemble: Obstruction to the fractal phase

We study the stability of nonergodic but extended (NEE) phases in non-Hermitian systems. For this purpose, we generalize the so-called Rosenzweig-Porter random-matrix ensemble, known to carry a NEE phase along with the Anderson localized and ergodic ones, to the non-Hermitian case. We analyze, both analytically and numerically, the spectral and multifractal properties of the non-Hermitian case. We show that the ergodic and localized phases are stable against the non-Hermitian nature of matrix entries. However, the stability of the fractal phase depends on the choice of the diagonal elements. For purely real or imaginary diagonal potential, the fractal phase is intact, while for a generic complex diagonal potential the fractal phase disappears, giving way to a localized one.

Research on Fractal Evolution Characteristics and Safe Mining Technology of Overburden Fissures under Gully Water Body

A fractal realizes the quantitative characterization of complex and disordered mining fracture networks, and it is of great significance to grasp the fractal characteristics of rock movement law to guide mine production. To prevent the water-conducting fracture (WF) under the gullies from conducting the surface water body, and to realize the purpose of safe production and surface water body protection. The evolution of overburden fissures in the working face with shallow buried gulley landform and thick bedrock conditions is studied. The development height of water-conducting fracture (DHWF) is theoretically analyzed. The evolution characteristics of overlying fissures with different mining heights were observed by similarity simulation, and the observation results were analyzed by fractal theory. The results show that the main factor that determines the height of WF is mining height. The working face is mined at different mining heights, and the corresponding indexes such as the height of the WF, the area of the caving zone and the fractal dimension are related to engineering phenomena. In particular, the appearance and disappearance of the separation space correspond to the fractal dimension fluctuation phase. The safe mining technology under a gully water body, which mainly reduces mining height, is adopted, and the fissures of the working face are not connected to the surface water body after mining.

Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian

We study the response of an isolated quantum system governed by the Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to a perturbation controlled by a small parameter. We focus on the density of states, local density of states and the eigenfunction amplitude overlap correlation functions which are calculated exactly using the mapping to the supersymmetric nonlinear sigma model. We show that the susceptibility of eigenfunction fidelity to the parameter of perturbation can be expressed in terms of these correlation functions and is strongly peaked at the localization transition: It is independent of the effective disorder strength in the ergodic phase, grows exponentially with increasing disorder in the fractal phase and decreases exponentially in the localized phase. As a function of the matrix size, the fidelity susceptibility remains constant in the ergodic phase and increases in the fractal and in the localized phases at modestly strong disorder. We show that there is a critical disorder strength inside the insulating phase such that for disorder stronger than the critical the fidelity susceptibility decreases with increasing the system size. The overall behavior is very similar to the one observed numerically in a recent work by Sels and Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity susceptibility in a disordered XXZ spin chain.

Universal Entanglement Transitions of Free Fermions with Long-range Non-unitary Dynamics

Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand the effect of long-range hopping that decays withr&amp;#x2212;&amp;#x03B1;in non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large-NSYK2chain and a single-flavor fermion chain and we show that they share the same phase diagram. When&amp;#x03B1;&amp;#x003E;0.5, we observe two critical phases with subvolume entanglement scaling: (i)&amp;#x03B1;&amp;#x003E;1.5, a logarithmic phase with dynamical exponentz=1and logarithmic subsystem entanglement, and (ii)0.5&amp;#x003C;&amp;#x03B1;&amp;#x003C;1.5, a fractal phase withz=2&amp;#x03B1;&amp;#x2212;12and subsystem entanglementSA&amp;#x221D;LA1&amp;#x2212;z, whereLAis the length of the subsystemA. These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays asL/T. We then confirm that the results are also valid for the static SYK2chain, indicating the phase diagram is universal for general free-fermion systems. We also discuss phase diagrams in higher dimensions and the implication in measurement-induced phase transitions.

Construction of Fractal Order and Phase Transition with Rydberg Atoms.

We propose the construction of a many-body phase of matter with fractal structure using arrays of Rydberg atoms. The degenerate low energy excited states of this phase form a self-similar fractal structure. This phase is analogous to the so-called "type-II fracton topological states." The main challenge in realizing fractonlike models in standard condensed matter platforms is the creation of multispin interactions, since realistic systems are typically dominated by two-body interactions. In this work, we demonstrate that the van der Waals interaction and experimental tunability of Rydberg-based platforms enable the simulation of exotic phases of matter with fractal structures, and the study of a quantum phase transition involving a fractal ordered phase.

Emergent fractal phase in energy stratified random models

We study the effects of partial correlations in kinetic hopping terms of long-range disordered random matrix models on their localization properties. We consider a set of models interpolating between fully-localized Richardson’s model and the celebrated Rosenzweig-Porter model (with implemented translation-invariant symmetry). In order to do this, we propose the energy-stratified spectral structure of the hopping term allowing one to decrease the range of correlations gradually. We show both analytically and numerically that any deviation from the completely correlated case leads to the emergent non-ergodic delocalization in the system unlike the predictions of localization of cooperative shielding. In order to describe the models with correlated kinetic terms, we develop the generalization of the Dyson Brownian motion and cavity approaches basing on stochastic matrix process with independent rank-one matrix increments and examine its applicability to the above set of models.

Entropy variation in a fractal phase space

In this paper we recall for physicists how it is possible, using the principle of maximization of the Boltzmann-Shannon entropy, to derive the Burr-Singh-Maddala (BurrXII) double power law probability distribution function and its approximations (Pareto, loglogistic ..) and extension (GB2…) first used in econometrics. This is possible using a deformation of the power function, as this has been done in complex systems for the exponential function. We give to that distribution a deep stochastic interpretation using the theory of Weron et al. Applied to thermodynamics the entropy nonextensivity can be accounted for by assuming that the asymptotic exponents are scale dependent. Therefore functions which describe phenomena presenting power-law asymptotic behaviour can be obtained without introducing exotic forms of the entropy.

Pluto’s Haze Abundance and Size Distribution from Limb Scatter Observations by MVIC

Abstract The New Horizons spacecraft observed Pluto and Charon at solar-phase angles between 16° and 169°. In this work, we use the Multispectral Visible Imaging Camera (MVIC) observations to construct multiwavelength phase curves of Pluto’s atmosphere, using the limb scatter technique. Observational artifacts and biases were removed using Charon as a representative airless body. The size and distribution of the haze particles were constrained using a Titan fractal aggregate phase function. We find that monodispersed and log-normal populations cannot simultaneously describe the observed steep forward scattering, indicative of wavelength-scale particles, and the non-negligible backscattering indicative of particles much smaller than the wavelength. Instead, we find it necessary to use bimodal or power-law distributions, especially below ∼200 km, to properly describe the MVIC observations. Above 200 km, where the atmosphere is isotropically scattering, a monodisperse, log-normal, or a bimodal/power law approximating a monodispersed population is able to fit the phase curves well. As compared to the results of previously published articles, we find that Pluto’s atmosphere must contain haze particle number densities an order of magnitude greater for small (∼10 nm) and large (∼1 μm) radii, and relatively fewer intermediate sizes (∼100 nm). These conclusions support a lower aggregate aerosol growth rate than that found by Gao et al., indicating a higher charge-to-radius ratio, upwards of 60e − μm−1. In order to generate large particles with a lower growth rate, the atmosphere must also have a lower sedimentation velocity (&lt;∼0.01 m s−1 at 200 km), which is possible with a fractal dimension of less than 2.

Unwrapping the phase portrait features of adventitious crackle for auscultation and classification: a machine learning approach.

The paper delves into the plausibility of applying fractal, spectral, and nonlinear time series analyses for lung auscultation. The thirty-five sound signals of bronchial (BB) and pulmonary crackle (PC) analysed by fast Fourier transform and wavelet not only give the details of number, nature, and time of occurrence of the frequency components but also throw light onto the embedded air flow during breathing. Fractal dimension, phase portrait, and sample entropy help in divulging the greater randomness, antipersistent nature, and complexity of airflow dynamics in BB than PC. The potential of principal component analysis through the spectral feature extraction categorises BB, fine crackles, and coarse crackles. The phase portrait feature-based supervised classification proves to be better compared to the unsupervised machine learning technique. The present work elucidates phase portrait features as a better choice of classification, as it takes into consideration the temporal correlation between the data points of the time series signal, and thereby suggesting a novel surrogate method for the diagnosis in pulmonology. The study suggests the possible application of the techniques in the auscultation of coronavirus disease 2019 seriously affecting the respiratory system.

Fractalizing quantum codes

We introduce "fractalization", a procedure by which spin models are extended to higher-dimensional "fractal" spin models. This allows us to interpret type-II fracton phases, fractal symmetry-protected topological phases, and more, in terms of well understood lower-dimensional spin models. Fractalization is also useful for deriving new spin models and quantum codes from known ones. We construct higher dimensional generalizations of fracton models that host extended fractal excitations. Finally, by applying fractalization to a 2D subsystem code, we produce a family of locally generated 3D subsystem codes that are conjectured to saturate a quantum information storage tradeoff bound.

Bifurcating subsystem symmetric entanglement renormalization in two dimensions

We introduce the subsystem symmetry-preserving real-space entanglement renormalization group and apply it to study bifurcating flows generated by linear and fractal subsystem symmetry-protected topological phases in two spatial dimensions. We classify all bifurcating fixed points that are given by subsystem symmetric cluster states with two qubits per unit cell. In particular, we find that the square lattice cluster state is a quotient-bifurcating fixed point, while the cluster states derived from Yoshida's first order fractal spin liquid models are self-bifurcating fixed points. We discuss the relevance of bifurcating subsystem symmetry-preserving renormalization group fixed points for the classification and equivalence of subsystem symmetry-protected topological phases.

Fractal analysis for estimating electricity demand through the application of Hurst’s re-staging exponent

This article presents a methodology for fractal analysis using the Hurst re-staging exponent for the generation prediction of a hydroelectric power plant. Which is an innovative technique to address this problem since literature usually performs it through classical and lately Bayesian statistical processes. The importance of the work implies amalgamation of concepts from fractal theory, chaos, phase space, delay time and Hurst exponent, necessary in the analysis that will determine an estimate of the future behavior of a chaotic series. Thus, the problem is addressed from the qualitative analysis through the theory of chaos.

Wave Beams with a Fractal Structure, Their Properties and Applications: A Literature Review

Concepts of formation and propagation of radiation with a fractal amplitude–phase profile are analyzed and systematized on a single methodological basis using main points of fractal physics. Motion of a plane wave through fractal phase and amplitude screens is considered. Characteristics of beams with a stochastic and a regular self-similar structure are discussed. A range of questions on direct formation of waves with a fractal amplitude and phase distribution in laser systems is examined. The analysis shows that scaling characteristics complement the space–time parameters traditionally used for describing light beams. In particular, mutually complementing are statistical and fractal data on propagation of beams in the turbulent atmosphere. To date, the mathematical and physical apparatus has been developed that allows self-similar and scaling properties of radiation to be adequately described. With this apparatus, it is possible to develop efficient methods for optical diagnostics of fractal structures using probing light beams. A high degree of stability of Fourier images of fractals is an important factor for increasing stability and noise immunity of optical information transmission channels based on fractal beams.

Time series analysis of duty cycle induced randomness in thermal lens system

The present work employs time series analysis, a proven powerful mathematical tool, for investigating the complex molecular dynamics of the thermal lens (TL) system induced by the duty cycle (C) variation. For intensity modulation, TL spectroscopy commonly uses optical choppers. The TL formation involves complex molecular dynamics that vary with the input photothermal energy, which is implemented by varying the duty cycle of the chopper. The molecular dynamics is studied from the fractal dimension (D), phase portrait, sample entropy (S), and Hurst exponent (H) for different duty cycles. The increasing value of C is found to increase D and S, indicating that the system is becoming complex and less deterministic, as evidenced by the phase portrait analysis. The value of H less than 0.5 conforms the evolution of the TL system to more antipersistent nature with C. The increasing value of C increases the enthalpy of the system that appears as an increase in full width at half maximum of the refractive index profile. Thus the study establishes that the sample entropy and thermodynamic entropy are directly related.