Fractal Dynamics Research Articles

Revealing fractal dynamics by dispersion tuning in a mode-locked fiber laser

Fractal dynamics was previously observed by tuning nonlinearity through pump currents in a mode-locked laser. Here we show that the fascinating fractal dynamics (the devil’s staircase) can also be accessed by tuning dispersion only. In this case, the fractal dimension is also the same as that of the devil’s staircase. Standard dispersion control via the cut-back methods is not precise enough to observe the fractal phenomena. A precise and readily dispersion tuning method through software control is thus proposed and confirmed by the numerical simulations. Third-order dispersion is later considered and does not destroy the phenomena, confirming the robustness of the devil’s staircase. Our work confirms the universality of the fractal phenomena in mode-locked lasers, and that the numerical simulations could stimulate future experimental investigations.

Fractal phototherapy: impact on the structure and function of the retina of rabbits with modelled retinal pigment epithelium atrophy

It is believed that in degenerative diseases of the retina, photostimulation by fractal dynamics signals activates neuroplasticity, thereby increasing the efficiency of visual rehabilitation. Previously, we showed a positive effect of fractal phototherapy (FF) on the electroretinogram (ERG) of healthy rabbits and demonstrated the safety of long-term photostimulation courses for the retina. The purpose of this work is to study the effect of FF on the functional activity and morphology of the retina in rabbits with a model of retinal pathology. Material and methods. We modelled an atrophy of retinal pigment epithelium (RPE) on both eyes of 50 rabbits. 30 days after the administration of bevacizumab, the animals were divided into two groups of 25 animals each. In the main group, photostimulation was performed using a device for FF, while in the control group incandescent lamps were used that create radiation of constant intensity. In both groups, 20-minute binocular light stimulation sessions were performed daily, five times a week. ERG and optical coherence tomography of the retina were performed before and after courses of treatment which lasted 1 week, 1 and 3 months. Results. Long-term courses of FF were shown to be safe for the morphology of the retina of animals with the RPE atrophy model. In all periods of observation, biochemical studies revealed no statistically significant changes in the content of norepinephrine and dopamine in the tear as compared with baseline values. In the main group, a slight positive effect of FF on rod and cone ERG was found after 5 FF sessions, while a significant increase in the amplitude of the transient and steady-state pattern-ERG (PERG), most pronounced after a 1-month FF course, was observed. Conclusions. A positive effect of FF on the functional activity of retinal ganglion cells (RGC) may suggest that prescribing a course of FF lasting up to 1 month (20 sessions) in diseases accompanied by a pathology of RGC is advisable, whereas for patients with a pathology of the macular region, such as AMD, an effective improvement in the activity of photoreceptors and bipolar cells could probably be achieved through a 1-week course of FF, conducted under the control of electroretinography.

Predictive analysis of stochastic stock pattern utilizing fractional order dynamics and heteroscedastic with a radial neural network framework

Modelling of high-dimension chaotic, noisy, and non-stationary time series of complex fractal dynamics is a big challenge. In this research work, a novel approach of Leverage Convolution LC ARFIMA– GARCH model is presented for sequential learning of irregular, fractal dynamic patterns of stochastic time series dynamics. Different classical methods including the auto-regressive approach indecently are unable to capture fragile patterns and consequently convert some weak signals into random errors by creating white noise patterns. The Convolution Leverage provides an additional degree of freedom for critical noisy points and asymmetrical distribution dynamics to regulate the frequency of imbalance and skewed observation. The designed transformation in the ARFIMA model preserves the loss of information by shifting the skewed data distribution toward a normal pattern. The pattern of population parameters in the proposed leverage paradigm provides out of box approach to track and extract additional information on population variance patterns in the form of an additional degree of freedom to stabilize imbalance signals. The model can provide reliable performance for long-range dependence, particularly for mean reversion chaotic phase variation at critical non-differentiable points. The performance of the dynamic model is verified on chaotic real data of the Ireland Stock Market (ISEQ). The result statistics confirmed the optimized outcome with the addition of the GARCH heteroscedastic multimodal and radial basis neural network. The novel technique can help to address inherited challenges in the imbalance learning and extreme events of the time series modeling by monitoring the chaotic trajectory of fractal physical phenomena, particularly in, finance healthcare, climate, and intelligent sustainability.

Gait variability, fractal dynamics, and statistical regularity of treadmill and overground walking recorded with a smartphone

BackgroundThe nonlinear variability present during steady-state gait may provide a signature of health and showcase one’s walking adaptability. Although treadmills can capture vast amounts of walking data required for estimating variability within a small space, gait patterns may be misrepresented compared to an overground setting. Smartphones may provide a low-cost and user-friendly estimate of gait patterns among a variety of walking settings. However, no study has investigated differences in gait patterns derived from a smartphone between treadmill walking (TW) and overground walking (OW). Research questionThis study implemented a smartphone accelerometer to compare differences in temporal gait variability and gait dynamics between TW and OW. MethodsSixteen healthy adults (8F; 24.7 ± 3.8 years) visited the laboratory on three separate days and completed three 8-minute OW and three TW trials, at their preferred speed, during each visit. The inter-stride interval was calculated as the time difference between right heel contact events located within the vertical accelerometery signals recorded from a smartphone while placed in participants front right pant pocket during walking trials. The inter-stride interval series was used to calculate stride time standard deviation (SD) and coefficient of variation (COV), statistical persistence (fractal scaling index), and statistical regularity (sample entropy). Two-way analysis of variance compared walking condition and laboratory visits for each measure. ResultsCompared to TW, OW displayed significantly (p < 0.01) greater stride time SD (0.014 s, 0.020 s), COV (1.26 %, 1.82 %), fractal scaling index (0.70, 0.79) and sample entropy (1.43, 1.63). No differences were found between visits for all measures. SignificanceSmartphone-based assessment of gait provides the ability to distinguish between OW and TW conditions, similar to previously established methodologies. Furthermore, smartphones may be a low-cost and user-friendly tool to estimate gait patterns outside the laboratory to improve ecological validity, with implications for free-living monitoring of gait among various populations.

Trade Off between Hydrodynamic and Thermodynamic Forces at the Liquid-Liquid Interface.

Viscous fingering (VF) instability has been investigated in the case of a partially miscible binary system by nonlinear numerical simulations. Partially miscible fluid systems offer the possibility of phase separation coupled with VF instability. The thermodynamics of such systems are governed by the Margules parameter (interaction parameter) as well as the fluid concentrations. Kinetics of the decomposition is also influenced by dynamical parameters such as the viscosity of the fluid, which incidentally also affects the hydrodynamic forces. Here, we explore the effects of concentration and Margules parameter in order to ascertain the trade-offs incurred between hydrodynamic and thermodynamic effects at the interface as well as the thermodynamics of the bulk. Based on the Gibb's free energy versus concentration curve, we select concentrations (i) outside spinodal and binodal regions, (ii) within binodal but outside the spinodal, and (iii) within the spinodal curve. We solve the modified Cahn-Hilliard-Hele-Shaw equation employing the COMSOL Multiphysics software. Applying high-resolution numerical simulations, we show a strong dependence of the thermodynamic forces on the concentration of the mixtures. Rapid phase separation and hence a faster rate of droplet formation have been found when the concentration lies inside the spinodal region. Further, we have investigated the correlation between the fractal dimension and dynamics of the system. The spatiotemporal studies presented in this work clearly illustrate the competition between hydrodynamic and thermodynamic forces and provide insights on the kinetics of decomposition and growth of interfacial instabilities.

River networks evolution under multiple stresses: A geometric and structural fractal perspective

Accurately quantifying changes in river networks is essential for studying and assessing the environmental effects of such changes. The fractal method has become a popular and effective tool for describing and capturing river network changes. However, it is usually limited to the geometric fractal dimension (Dg), whereas the structural fractal dimension (Ds) is rarely used. Despite this, the Ds could capture crucial structural information, which the Dg does not, such as interconnections among rivers and the overall basin connectivity. In this study, we examined the evolution of Dg and Ds in the Qinhuai River Basin from the 1960s to 2016, revealing the changes in river networks under natural and anthropogenic stresses from both the traditional geometric and novel structural fractal analysis perspective. We also determined the key factors causing fractal variation and their contributions. The results showed that river networks possess both structural and geometric fractality, with values ranging from 1.484 to 1.287 and 1.615–1.514, respectively. The values of Dg and Ds both declined, with Ds declining at a higher rate, suggesting that the river network weakened and deteriorated. The variances of Dg and Ds attributed to urbanization were 0.3139 and 0.3103, respectively, more than thirty times and almost twice as much as the variance attributed to precipitation features. Urbanization was identified as the most significant factor influencing fractal dynamics. These findings advance the understanding of river network evolution from a novel fractal perspective and provide support for integrating river restoration with urban planning practices.

Statistical analysis of self-affinity of energy dispatch in power generation in the Northeast region of Brazil

In this article, we study the fractal dynamics in time series of energy generation in the Northeast region of Brazil, considering a database used by the National Electric System Operator referring to semi-hourly information on the generation of various energy sources. Applying the DFA technique between the years 2020 and 2022, it was found that the dispatch of solar energy fluctuated in all years from superdiffusive behavior in the short term to persistent in the long term, whereas wind generation in all years varied from superdiffusive short-term to long-term subdiffusive. In all observation periods, there was a repeatability of the dispatch behavior of wind and solar sources, which can be explained by the fact that these sources always maintain a standard of availability for the energy matrix. When analyzing the behavior of solar generation, a greater predictability of energy availability was found, which can therefore be considered as a more viable source to make up for the lack of hydraulic energy. This analysis is based on historical time series numbers, not considering the costs of each energy source, environmental license situations, among other factors.

Thermodynamics and fractal dynamics of a nematic spin ice: A doubly frustrated pyrochlore Ising magnet

The Ising antiferromagnets on the triangular and on the pyrochlore lattices are two of the most iconic examples of magnetic frustration, paradigmatically illustrating many exotic properties such as emergent gauge fields, fractionalization, and topological order. In this paper, we show that the two instances of frustration can, remarkably, be combined in a single system, where they coexist without inducing conventional long-range ordering. Our results indicate that the system undergoes a first-order phase transition upon lowering the temperature, into a yet different frustrated phase that we characterize to exhibit nematic order. We argue that an extensive degeneracy survives down to zero temperature, at odds with a customary Pauling estimate. Dynamically, we find evidence of anomalous noise in the power spectral density, arising from an effectively fractal anisotropic motion of monopoles at low temperature. Published by the American Physical Society 2024

The Risk Function of Breast and Ovarian Cancers in the Avrami-Dobrzyński Cellular Phase-Transition Model.

Specifying the role of genetic mutations in cancer development is crucial for effective screening or targeted treatments for people with hereditary cancer predispositions. Our goal here is to find the relationship between a number of cancerogenic mutations and the probability of cancer induction over the lifetime of cancer patients. We believe that the Avrami-Dobrzyński biophysical model can be used to describe this mechanism. Therefore, clinical data from breast and ovarian cancer patients were used to validate this model of cancer induction, which is based on a purely physical concept of the phase-transition process with an analogy to the neoplastic transformation. The obtained values of model parameters established using clinical data confirm the hypothesis that the carcinogenic process strongly follows fractal dynamics. We found that the model's theoretical prediction and population clinical data slightly differed for patients with the age below 30 years old, and that might point to the existence of an ancillary protection mechanism against cancer development. Additionally, we reveal that the existing clinical data predict breast or ovarian cancers onset two years earlier for patients with BRCA1/2 mutations.

Neuronal Fractal Dynamics.

Synapse formation is a unique biological phenomenon. The molecular biological perspective of this phenomenon is different from the fractal geometrical one. However, these perspectives are not mutually exclusive and supplement each other. The cornerstone of the first one is a chain of biochemical reactions with the Markov property, that is, a deterministic, conditional, memoryless process ordered in time and in space, in which the consecutive stages are determined by the expression of some regulatory proteins. The coordination of molecular and cellular events leading to synapse formation occurs in fractal time space, that is, the space that is not only the arena of events but also actively influences those events. This time space emerges owing to coupling of time and space through nonlinear dynamics. The process of synapse formation possesses fractal dynamics with non-Gaussian distribution of probability and a reduced number of molecular Markov chains ready for transfer of biologically relevant information.

FAST AND ACCURATE POPULATION FORECASTING WITH TWO-SCALE FRACTAL POPULATION DYNAMICS AND ITS APPLICATION TO POPULATION ECONOMICS

One of the major challenges in population economics is accurately predicting population size. Incorrect predictions can lead to ineffective population control policies. Traditional differential models assume a smooth change in population, but this assumption is invalid when measuring population on a small-time scale. To address this change, we developed two-scale fractal population dynamics that can accurately predict population size with minimal experimental data. The Taylor series method is used to reveal the population’s dynamical properties, and the Padé technology is adopted to accelerate the convergence rate.

ON SELF-SIMILARITY OF QUADRANGLE

Wen [Some problems in fractal geometry, in Report in Chinese Conference on Fractal Geometry and Dynamical Systems, 2023] posed the following questions: when a quadrangle is a self-similar set with the open set condition (OSC). It is well known that any convex quadrangle is a self-similar set. For the concave quadrangles, this case is more complicated. Loosely speaking, some concave polygons may be not self-similar. In this paper, we prove two results. First, we give a necessary and sufficient condition such that a concave quadrangle is a self-similar set. Moreover, we also give a necessary condition for a concave quadrangle to be a self-similar set with the OSC. For the convex quadrangle, we investigate the trapezoid. We show under some condition that the trapezoid is a self-similar set with the OSC. In particular, for any trapezoid, if the ratio of the lengths of two bases is rational, then the trapezoid is a self-similar set with the OSC.

Fractal Phototherapy in Maximizing Retina and Brain Plasticity.

The neuroplasticity potential is reduced with aging and impairs during neurodegenerative diseases and brain and visual system injuries. This limits the brain's capacity to repair the structure and dynamics of its activity after lesions. Maximization of neuroplasticity is necessary to provide the maximal CNS response to therapeutic intervention and adaptive reorganization of neuronal networks in patients with degenerative pathology and traumatic injury to restore the functional activity of the brain and retina.Considering the fractal geometry and dynamics of the healthy brain and the loss of fractality in neurodegenerative pathology, we suggest that the application of self-similar visual signals with a fractal temporal structure in the stimulation therapy can reactivate the adaptive neuroplasticity and enhance the effectiveness of neurorehabilitation. This proposition was tested in the recent studies. Patients with glaucoma had a statistically significant positive effect of fractal photic therapy on light sensitivity and the perimetric MD index, which shows that methods of fractal stimulation can be a novel nonpharmacological approach to neuroprotective therapy and neurorehabilitation. In healthy rabbits, it was demonstrated that a long-term course of photostimulation with fractal signals does not harm the electroretinogram (ERG) and retina structure. Rabbits with modeled retinal atrophy showed better dynamics of the ERG restoration during daily stimulation therapy for a week in comparison with the controls. Positive changes in the retinal function can indirectly suggest the activation of its adaptive plasticity and the high potential of stimulation therapy with fractal visual stimuli in a nonpharmacological neurorehabilitation, which requires further study.

Fingerprints of decreased cognitive performance on fractal connectivity dynamics in healthy aging

Analysis of brain functional connectivity (FC) could provide insight in how and why cognitive functions decline even in healthy aging (HA). Despite FC being established as fluctuating over time even in the resting state (RS), dynamic functional connectivity (DFC) studies involving healthy elderly individuals and assessing how these patterns relate to cognitive performance are yet scarce. In our recent study we showed that fractal temporal scaling of functional connections in RS is not only reduced in HA, but also predicts increased response latency and reduced task solving accuracy. However, in that work we did not address changes in the dynamics of fractal connectivity (FrC) strength itself and its plausible relationship with mental capabilities. Therefore, here we analyzed RS electroencephalography recordings of the same subject cohort as previously, consisting of 24 young and 19 healthy elderly individuals, who also completed 7 different cognitive tasks after data collection. Dynamic fractal connectivity (dFrC) analysis was carried out via sliding-window detrended cross-correlation analysis (DCCA). A machine learning method based on recursive feature elimination was employed to select the subset of connections most discriminative between the two age groups, identifying 56 connections that allowed for classifying participants with an accuracy surpassing 92%. Mean of DCCA was found generally increased, while temporal variability of FrC decreased in the elderly when compared to the young group. Finally, dFrC indices expressed an elaborate pattern of associations—assessed via Spearman correlation—with cognitive performance scores in both groups, linking fractal connectivity strength and variance to increased response latency and reduced accuracy in the elderly population. Our results provide further support for the relevance of FrC dynamics in understanding age-related cognitive decline and might help to identify potential targets for future intervention strategies.

Sex differences in the fractal dynamics of force control during maximal handgrip

This work examines the temporal structure of force fluctuations during maximal handgrip with detrended fluctuation analysis (DFA α). Here, we assess the influence of fatigue and sex on force complexity during unimanual handgrip for the fatigued and the contralateral, non-fatigued hand. Participants randomly completed experimental sessions requiring fatiguing handgrip contractions or control measurements only. Maximal unimanual forces of both hands were measured before and after the fatigue trial or a time-matched control visit. DFA revealed substantially lower alpha values for females (PRE = 1.15, POST = 1.25) compared to males (PRE = 1.30, POST = 1.33) regardless of fatigue (p < 0.01, d = 0.738) for the dominant hand with a similar pattern observed for the contralateral, non-fatigued hand (p = 0.045, d = 0.561). Females also showed greater alpha changes (Δ = 0.09) versus males (Δ = 0.01) following fatigue (p = 0.028, ηp2 = 0.151). The data provide evidence of reduced force complexity during successive maximal handgrip contractions for females, but not males. Our findings highlight task-specific factors involving force control and demonstrate the utility of complexity analyses to provide insights regarding the influence of sex on motor control strategies.

Fractal dynamics and computational analysis of local fractional Poisson equations arising in electrostatics

In this paper, the local fractional natural decomposition method (LFNDM) is used for solving a local fractional Poisson equation. The local fractional Poisson equation plays a significant role in the study of a potential field due to a fixed electric charge or mass density distribution. Numerical examples with computer simulations are presented in this paper. The obtained results show that LFNDM is effective and convenient for application.

Local electrical characteristic of memristor structure in a high-resistance state obtained using electrostatic force microscopy: Fractal and multifractal dynamics of surface

A heterostructure BiFeO3/TiO2(Nt)Ti (BFOT) was obtained by the atomic layer deposition (ALD) method. After thermal treatment, the redistribution of Fe/Ti atoms forms an Aurivillius intermediate layered phase, and local charge capture centers are formed in the sample. Due to cationic non-stoichiometry, the BFO film exhibits p-type conductivity, while the nanotubes exhibit n-type conductivity due to oxygen vacancies. It was observed that lateral displacement of the sample can lead to ferroelectric switching, which can, in turn, affect the transition of the memristive structure from high-resistance (HRS) to low-resistance states (LRS). The hysteresis suppression tends to transition to an ohmic character and depends on the amplitude, frequency, and duration of the periodic signal. It has been found that compensation of static charge during resistive switching can affect the transport properties of the material. Fractal dimension analysis showed an acceleration of structure restructuring with increasing voltage, possibly contributing to the transition from an insulator to a metal in certain areas of the film volume. The joint analysis of piezoresponse force microscopy (PFM), electrostatic force microscopy (EFM), and fractal/multifractal dynamics showed a correlation between surface static charge and piezopotential. The new methodology described in this work can help understand the resistive switching processes in ferroelectric/semiconductor memristive structures.

Dynamics, circuit design, feedback control of a new hyperchaotic system and its application in audio encryption

In this study, a 4D hyperchaotic system is constructed based on the foundation of a 3D Lü chaotic system. The newly devised hyperchaotic system possesses a sole equilibrium point, showcasing a simplified system structure that reduces complexity. This simplification offers a clearer opportunity for in-depth analysis of dynamic behaviors in the realm of scientific research. The proposed hyperchaotic system undergoes an in-depth examination of its dynamical characteristics, including chaotic attractors, equilibrium point stability, Lyapunov exponents’ spectrum, and bifurcation diagram. Numerical analysis results reveal that the attractor of this hyperchaotic system exhibits highly complex, non-periodic, and fractal structural dynamics. Its motion demonstrates extreme sensitivity and randomness, even within a wide range of variations in parameter d, affirming its hyperchaotic properties with two positive Lyapunov exponents. Hyperchaotic bifurcation diagrams typically exhibit highly intricate structures, such as fractals, branches, and period doubling characteristics, signifying that even minor parameter adjustments can lead to significant changes in system behavior, presenting diversity and unpredictability. Subsequently, to further investigate the practical utility of this hyperchaotic system, a linear feedback control strategy is implemented. Through linear feedback control, the hyperchaotic system is stabilized at its unique equilibrium point. Experimental validation is conducted using both computer software simulation Matlab, electronic circuit simulation Multisim, and embedded hardware STM32. The results of these experiments consistently align, providing theoretical support for the application of this hyperchaotic system in practical domains. Finally, leveraging the hyperchaotic keys generated by this hyperchaotic system, audio encryption is achieved using a cross-XOR algorithm, which is then realized on the embedded hardware platform STM32. The results show that the audio encryption scheme based on the hyperchaotic system is feasible, and the method is simple to implement, has nonlinear characteristics and certain algorithm complexity, which can be applied to audio encryption, image encryption, video encryption, and more.

Prediction of early postoperative pain using sleep quality and heart rate variability.

Accurate predictions of postoperative pain intensity are necessary for customizing analgesia plans. Insomnia is a risk factor for severe postoperative pain. Moreover, heart rate variability (HRV) can provide information on the sympathetic-parasympathetic balance in response to noxious stimuli. We developed a prediction model that uses the insomnia severity index (ISI), HRV, and other demographic factors to predict the odds of higher postoperative pain. We recruited gynecological surgery patients classified as American Society of Anesthesiologists class 1-3. An ISI questionnaire was completed 1 day before surgery. HRV was calculated offline using intraoperative electrocardiogram data. Pain severity at the postanesthesia care unit (PACU) was assessed with the 0-10 numerical rating scale (NRS). The primary outcome was the model's predictive ability for moderate-to-severe postoperative pain. The secondary outcome was the relationship between individual risk factors and opioid consumption in the PACU. Our study enrolled 169 women. Higher ISI scores (p = 0.001), higher parasympathetic activity (rMSSD, pNN50, HF; p < 0.001, p < 0.001, p < 0.001), loss of fractal dynamics (SD2, alpha 1; p = 0.012, p = 0.039) in HRV analysis before the end of surgery were associated with higher NRS scores, while laparoscopic surgery (p = 0.031) was associated with lower NRS scores. We constructed a multiple logistic model (area under the curve = 0.852) to predict higher NRS scores at PACU arrival. The five selected predictors were age (OR: 0.94; p = 0.020), ISI score (OR: 1.14; p = 0.002), surgery type (laparoscopic or open; OR: 0.12; p < 0.001), total power (OR: 2.02; p < 0.001), and alpha 1 (OR: 0.03; p < 0.001). We employed a multiple logistic regression model to determine the likelihood of moderate-to-severe postoperative pain upon arrival at the PACU. Physicians could personalize analgesic regimens based on a deeper comprehension of the factors that contribute to postoperative pain.

On the universality of fractal breathers in mode-locked fibre lasers

Mode-locked fibre lasers can show fractal dynamics when they work on breathing soliton regime. Here, we investigate the existence range of this intriguing phenomena by varying the laser parameters including the modulation depth of saturable absorbers and the net cavity dispersion by extensive numerical simulations. It is found that this phenomena can exist in a relatively large range of laser parameters, indicating that fractal dynamics could be observed in many mode-locked lasers. Thus, mode-locked lasers provide an excellent platform to investigate fascinating dynamics of fractals.