Self-organized Criticality Research Articles

Space–Time Structures in the Auroral Oval: Approaches to Modeling

The interaction of the magnetospheric–ionospheric (MI) system surrounding the Earth with the environment (solar wind) occurs in the form of a series of transient processes at different scales. The largest of them, magnetic storms, are obviously triggered by disturbances in the solar wind (direct driving). The role of the internal dynamics of the MI system, which is caused to a large extent by the nonlinearity and temporal delays of the loading–unloading processes of energy and particle from the solar wind into the magnetosphere, becomes more significant at smaller scales (substorms, pseudobreakups, injections, and activations). A typical dynamic state of the MI system is characterized as self-organized criticality or turbulence, which are characterized by statistical scale invariance (scaling) in the fluctuation distributions of many characteristics. The dynamics of the MI system is projected into the region of the auroral oval, the very existence of which is due to this dynamics. The space–time structure of auroral disturbances largely reflects the structure of processes in the MI plasma. The description of this structure is important both for studying the fundamental study of plasma processes and for many topical applied problems related to the propagation of radio waves in the ionosphere and vital activity at high latitudes. The paper discusses approaches and developments for constructing a model of the space–time structure of the auroral oval, based on fractal and multifractal characteristics.

Sandpile Universality in Social Inequality: Gini and Kolkata Measures.

Social inequalities are ubiquitous and evolve towards a universal limit. Herein, we extensively review the values of inequality measures, namely the Gini (g) index and the Kolkata (k) index, two standard measures of inequality used in the analysis of various social sectors through data analysis. The Kolkata index, denoted as k, indicates the proportion of the 'wealth' owned by (1-k) fraction of the 'people'. Our findings suggest that both the Gini index and the Kolkata index tend to converge to similar values (around g=k≈0.87, starting from the point of perfect equality, where g=0 and k=0.5) as competition increases in different social institutions, such as markets, movies, elections, universities, prize winning, battle fields, sports (Olympics), etc., under conditions of unrestricted competition (no social welfare or support mechanism). In this review, we present the concept of a generalized form of Pareto's 80/20 law (k=0.80), where the coincidence of inequality indices is observed. The observation of this coincidence is consistent with the precursor values of the g and k indices for the self-organized critical (SOC) state in self-tuned physical systems such as sand piles. These results provide quantitative support for the view that interacting socioeconomic systems can be understood within the framework of SOC, which has been hypothesized for many years. These findings suggest that the SOC model can be extended to capture the dynamics of complex socioeconomic systems and help us better understand their behavior.

Response of cross-correlations between high PM2.5 and O3 with increasing time scales to the COVID-19: different trends in BTH and PRD.

The air pollution in China currently is characterized by high fine particulate matter (PM2.5) and ozone (O3) concentrations. Compared with single high pollution events, such double high pollution (DHP) events (both PM2.5 and O3 are above the National Ambient Air Quality Standards (NAAQS)) pose a greater threat to public health and environment. In 2020, the outbreak of COVID-19 provided a special time window to further understand the cross-correlation between PM2.5 and O3. Based on this background, a novel detrended cross-correlation analysis (DCCA) based on maximum time series of variable time scales (VM-DCCA) method is established in this paper to compare the cross-correlation between high PM2.5 and O3 in Beijing-Tianjin-Heibei (BTH) and Pearl River Delta (PRD). At first, the results show that PM2.5 decreased while O3 increased in most cities due to the effect of COVID-19, and the increase in O3 is more significant in PRD than in BTH. Secondly, through DCCA, the results show that the PM2.5-O3 DCCA exponents α decrease by an average of 4.40% and 2.35% in BTH and PRD respectively during COVID-19 period compared with non-COVID-19 period. Further, through VM-DCCA, the results show that the PM2.5-O3 VM-DCCA exponents [Formula: see text] in PRD weaken rapidly with the increase of time scales, with decline range of about 23.53% and 22.90% during the non-COVID-19 period and COVID-19 period respectively at 28-h time scale. BTH is completely different. Without significant tendency, its [Formula: see text] is always higher than that in PRD at different time scales. Finally, we explain the above results with the self-organized criticality (SOC) theory. The impact of meteorological conditions and atmospheric oxidation capacity (AOC) variation during the COVID-19 period on SOC state are further discussed. The results show that the characteristics of cross-correlation between high PM2.5 and O3 are the manifestation of the SOC theory of atmospheric system. Relevant conclusions are important for the establishment of regionally targeted PM2.5-O3 DHP coordinated control strategies.

Nonstationary but quasisteady states in self-organized criticality.

The notion of self-organized criticality (SOC) was conceived to interpret the spontaneous emergence of long-range correlations in nature. Since then many different models have been introduced to study SOC. All of them have a few common features: externally driven dynamical systems self-organize themselves to nonequilibrium stationary states exhibiting fluctuations of all length scales as the signatures of criticality. In contrast, we have studied here in the framework of the sandpile model a system that has mass inflow but no outflow. There is no boundary, and particles cannot escape from the system by any means. Therefore, there is no current balance, and consequently it is not expected that the system would arrive at a stationary state. In spite of that, it is observed that the bulk of the system self-organizes to a quasisteady state where the grain density is maintained at a nearly constant value. Power law distributed fluctuations of all lengths and time scales have been observed, which are the signatures of criticality. Our detailed computer simulation study gives the set of critical exponents whose values are very close to their counterparts in the original sandpile model. This study indicates that (i) a physical boundary and (ii) the stationary state, though sufficient, may not be the necessary criteria for achieving SOC.

Compressibility, Metallization, and Relaxation in Nonstoichiometric Chalcogenide Glass g-As3Te2 at High Hydrostatic Pressure versus “Classic” g-As2Te3 Glass

The volume and conductivity of nonstoichiometric chalcogenide glass g-As3Te2 have been investigated at high hydrostatic pressures (up to 8.5 GPa), and results have been compared with earlier data for stoichiometric chalcogenide glass g-As2Te3. Structural and Raman studies of g-As3Te2 glass have revealed a greater significance of As–As pair correlations in the range of medium-range order compared with “classic” chalcogenide glass g-As2Te3. Even at such a large excess of arsenic, a high concentration of “improper” Te–Te neighbors has been observed because of chemical disorder. Under normal conditions, the thermal gap (0.43–0.48 eV) and resistivity (104 Ω cm) of glass g-As3Te2 are greater than those of g-As2Te3. The elastic behavior of g-As3Te2 glass, as well as of g-As2Te3, under compression has been observed at pressures up to 1 GPa, the initial values of bulk moduli for these glasses being nearly coincident. Polyamorphic transformation in g-As3Te2 (with softening of relaxing bulk modulus) is more diffuse and extends to higher pressures (from 1.5 to 4.0 GPa). The metallization process in g-As3Te2 is also more diffuse: metallic conductivity is reached at pressures of 5.5–6.0 GPa. As in the case of the stoichiometric glass, the baric dependences of the bulk modulus exhibit a kink in the pressure range 4–5 GPa. Up to maximal pressures, the volume and resistivity relax logarithmically in time with roughly the same rate as in the case of g-As2Te3. The residual densification of g-As3Te2 after pressure release is roughly twice as high as for g-As2Te3 and equals 3.5%, the conductivity of the compacted glass is about three orders of magnitude higher than that of the as-prepared sample. Under normal conditions, a considerable relaxation of the volume and resistivity has been observed. As for densified g-GeS2 glass, the logarithmic kinetics of this relaxation has been successfully described in terms of our earlier model based on the concept of relaxation self-organized criticality with the activation energy (1.3 eV) remaining unchanged up to 5 × 106 s.

Probabilistic Wildfire Risk Assessment and Modernization Transitions: The Case of Greece

Wildfire is the primary cause of deforestation in fire-prone environments, disrupting the forest transition process generated by multiple social-ecological drivers of modernization. Given the positive feedback between climate change and wildfire-driven deforestation, it seems necessary to abstract the primary- or micro-characteristics of wildfire event(s) and focus on the general behavior of the phenomenon across time and space. This paper intends to couple wildfire self-organizing criticality theory (SOC) and modernization statistics to propose a verisimilar explanation of the phenomenon’s evolution in the past decades and a prediction of its trends in Greece. We use power law distributions of the fire frequency–magnitude relationship to estimate the basic SOC parameters and the Weibull reliability method to calculate large-size wildfires’ conditional probability as a time function. We use automatic linear modeling to search for the most accurate relationship between wildfire metrics and the best subset of modernization predictors. The discussion concentrates on reframing the political debate on fire prevention vs. suppression, its flaws and limitations, and the core challenges for adopting more efficient wildfire management policies in Greece.

TOTEs and trauma 2: a rat maternal care and stress epigenetic schematic

PurposeThis conceptual article presents a schematic of rat maternal behavior and niche stress epigenetic effects as a case study that is then aligned with current evolutionary concepts, which raises new questions regarding immigrant assimilation and niche dynamics.Design/methodology/approachThe necessary background material for rat maternal and niche(s) stress factors are incorporated into a recursive, test-operate-test (rTOT), information-only-transfer, schematic (Patterson, 2023), which is an extension and refinement of the test-operate-test-exit (TOTE) schematic of Miller et al. (1960).FindingsThe generated epigenetic rTOT demonstrates the fundamental evolutionary unit of the flexible organism within its niche(s). The rTOT also confirms that epigenetic processes, epigenetic inheritance and phenotype plasticity are significant conceptual tools for understanding evolution. The teleology of rat adaptations for niche fitness via maternal behavior has been demonstrated. Sterling's (2011) allostasis, or predictive homeostasis, is extended to include species-niche(s) interaction(s) that are governed by recursive information feedback loops that function via self-organized criticality (SOC) for species and niche(s). Use of a rat model for biosocial issues in humans is strengthened.Research limitations/implicationsEpigenetic rTOT only covers the species side of the evolutionary unit. Niche(s) require(s) a separate rTOT schematic. The information modeled does not include the entire system producing epigenetic effects but models a substantial portion of it.Practical implicationsEpigenetic rTOT demonstrates the utility of phenotypic plasticity, epigenetics and epigenetic inheritance as explanations for inheritable behavior patterns. rTOT is a useful computational model for evolutionary issues. The issues involved in niche modeling using an rTOT schematic are briefly reviewed.Social implicationsWhen the demonstrated epigenetic model of rat genetics and inherited behavior are applied to the issues of immigrant enclaves, epigenetic complications for the difficulties of assimilation into the culture within which the enclaves are embedded become apparent. However, the questions raised must be addressed with extreme care to avoid cultural imperialism. Such cultural issues must be modeled with an rTOT application that covers the materials involved. The limitations of human Learning III restrictions when attempting to model Learning IV issues are addressed. Research into the means by which abuse and trauma are maintained by epigenetic means is urgently needed.Originality/valueThe rTOT schematic visualizes rat maternal behavior and stress epigenetic effects that produce inheritable behavior patterns, which answers Jablonka's (2017) request for new computational modeling representations. The concept of allostasis, or predictive homeostasis, (Sterling, 2011) is extended to the niche(s) of the organism under study so that allostasis becomes a fully cybernetic concept governed by SOC for both the organism and its niche(s). This new case study confirmed evolutionary effects of epigenetics, epigenetic inheritance and phenotypic plasticity. Niche control of organism evolution is presented. Epigenetic applications for immigrant assimilation issues have been suggested and niche dynamic questions have been raised.

Evidence for the Self-organized Criticality Phenomenon in the Prompt Phase of Short Gamma-Ray Bursts

The prompt phase of gamma-ray bursts (GRBs) contains essential information regarding their physical nature and central engine, which are as yet unknown. In this paper, we investigate the self-organized criticality phenomenon in GRB prompt phases as done in X-ray flares of GRBs. We obtain the differential and cumulative distributions of 243 short GRB pulses, such as peak flux, FWHM, rise time, decay time, and peak time in the fourth BATSE Time-Tagged Event Catalog with the Markov Chain Monte Carlo technique. It is found that these distributions can be well described by power-law models. In particular, comparisons are made with 182 short GRB pulses in the third Swift GRB Catalog from 2004 December to 2019 July. The results are essentially consistent with the BATSE ones. We notice that there is no obvious power-law index evolution across different energy bands for either BATSE or Swift short GRBs. The joint analysis suggests that the GRB prompt phase can be explained by a fractal-diffusive self-organized criticality system with the spatial dimension S = 3 and the classical diffusion β = 1. Our findings show that GRB prompt phases and X-ray flares possess the very same magnetically dominated stochastic process and mechanism.

Neuronal avalanches: Sandpiles of self-organized criticality or critical dynamics of brain waves?

Analytical expressions for scaling of brain wave spectra derived from the general nonlinear wave Hamiltonian form show excellent agreement with experimental "neuronal avalanche" data. The theory of the weakly evanescent nonlinear brain wave dynamics [Phys. Rev. Research 2, 023061 (2020); J. Cognitive Neurosci. 32, 2178 (2020)] reveals the underlying collective processes hidden behind the phenomenological statistical description of the neuronal avalanches and connects together the whole range of brain activity states, from oscillatory wave-like modes, to neuronal avalanches, to incoherent spiking, showing that the neuronal avalanches are just the manifestation of the different nonlinear side of wave processes abundant in cortical tissue. In a more broad way these results show that a system of wave modes interacting through all possible combinations of the third order nonlinear terms described by a general wave Hamiltonian necessarily produces anharmonic wave modes with temporal and spatial scaling properties that follow scale free power laws. To the best of our knowledge this has never been reported in the physical literature and may be applicable to many physical systems that involve wave processes and not just to neuronal avalanches.

The universality of power law slopes in the solar photosphere and transition region observed with HMI and IRIS

We compare the size distributions of self-organized criticality (SOC) systems in the solar photosphere and the transition region, using magnetogram data from Helioseismic and Magnetic Imager (HMI) and Interface Region Imaging Spectrograph (IRIS) data. For each dataset we fit a combination of a Gaussian and a power law size distribution function, which yields information on four different physical processes: (i) Gaussian random noise in IRIS data; (ii) spicular events in the plages of the transition region (described by power law size distribution in IRIS data); (iii) salt-and-pepper small-scale magnetic structures (described by the random noise in HMI magnetograms); and (iv) magnetic reconnection processes in flares and nanoflares (described by power law size distributions in HMI data). We find a high correlation (CCC = 0.90) between IRIS and HMI data. Datasets with magnetic flux balance are generally found to match the SOC-predicted power law slope αF = 1.80 (for mean fluxes F), but exceptions occur due to arbitrary choices of the HMI field-of-view. The matching cases confirm the universality of SOC-inferred flux size distributions, and agree with the results of Parnell et al. (ApJ, 2009, 698, 75–82), αF = 1.85 ± 0.14.

Self-organized criticality of magnetic avalanches in disordered ferrimagnetic material.

We observe multiple steplike jumps in a Dy-Fe-Ga-based ferrimagnetic alloy in its magnetic hysteresis curve at 2K. The observed jumps are found to have a stochastic character with respect to their magnitude and the field position, and the jumps do not correlate with the duration of the field. The distribution of jump size follows a power law variation indicating the scale invariance nature of the jumps. We have invoked a simple two-dimensional random bond Ising-type spin system to model the dynamics. Our computational model can qualitatively reproduce the jumps and their scale-invariant character. It also elucidates that the flipping of antiferromagnetically coupled Dy and Fe clusters is responsible for the observed jumps in the hysteresis loop. These features are described in terms of the self-organized criticality.

Two slip regimes in sheared granular fault

In the stick-slip cycles of faults, the evolving contact state inside the gouge is a major contribution to the evolution of fault friction. However, the microscopic contact changes within stick-slip cycles associated with the possible existence of multiple fault slip regimes are still obscure. By simulating a sheared fault containing a granular gouge, we examine the particle-level contact evolution inside the gouge and the corresponding gouge stress field heterogeneity at fault slip. We find two regimes of fault slip driven by the heterogeneity of the gouge stress field: in the case of weak gouge stress field heterogeneity, local contact rearrangements give rise to macroscopic friction coefficient drop and accumulated gouge stress field heterogeneity (dispersive regime); in the instance of strong stress field heterogeneity, catastrophic failure occurs via contact rearrangement sweeping the whole fault and homogenizes the stress field distribution in the gouge (pervasive regime). In the language of self-organized criticality, the accumulation of gouge stress field heterogeneity is an inevitable consequence of increased entropy in the preparation stage of a large earthquake, which eventually destroys the long-range stress field correlations in the gouge and resets at the beginning of a new cycle of a large earthquake.

Self-organized criticality of traffic flow: Implications for congestion management technologies

Self-organized criticality (SOC) is a celebrated paradigm from the 90’s for understanding dynamical systems naturally driven to its critical point, where the power-law dynamics taking place make predictions practically impossible, such as in stock prices, earthquakes, pandemics and many other problems in science related to phase transitions. Shortly thereafter, it was realized that traffic flow might be in the SOC category, implying that conventional traffic management strategies seeking to maximize the local flows can become detrimental. This paper shows that the Kinematic Wave model with triangular fundamental diagram, and many other related traffic models, indeed exhibit SOC, thanks in part to the fractal nature of traffic exposed here on the one hand, and our need to get to our destinations as soon as possible, on the other hand.Important implications for congestion management of traffic near the critical region are discussed, such as: (i) Jam sizes obey a power-law distribution with exponent 1/2, implying that both its mean and variance become ill-defined and therefore impossible to estimate. (ii) Traffic in the critical region is chaotic in the sense that predictions becomes extremely sensitive to initial conditions. (iii) However, aggregate measures of performance such as delays and average speeds are not heavy tailed, and can be characterized exactly by different scalings of the Airy distribution, (iv) Traffic state time–space “heat maps” are self-affine fractals where the basic unit is a triangle, in the shape of the fundamental diagram, containing 3 traffic states: voids, capacity and jams. This fractal nature of traffic flow calls for analysis methods currently not used in our field.

Similar properties between gamma-ray emission of 3C 454.3 and solar GeV flares

ABSTRACTBased on the survey data observed by Fermi-Large Area Telescope (LAT), we elaborate the statistical characteristics of gamma-ray flares from 3C 454.3 and solar GeV flares among flare parameters, such as isotropic energy (Eγ), peak luminosity (LP), and duration time (TDuration). We find two significant correlations as: $T_{\rm Duration} \propto E_{\gamma }^{0.31\pm 0.03}$ and $L_{\rm P} \propto E_{\gamma }^{0.61\pm 0.03}$ for 3C 454.3. The exponents are in a better agreement those of the Sun from the first Fermi-LAT solar flare catalogue, namely $T_{\rm Duration,\odot } \propto E_{\gamma ,\odot }^{0.38\pm 0.08}$ and $L_{\rm P,\odot } \propto E_{\gamma ,\odot }^{0.81\pm 0.08}$. The relationship of TDuration–Eγ and LP–Eγ could be interpreted naturally as due to magnetic dissipation through reconnection. On top of that the frequency distributions of isotropic energy, peak luminosity, and duration time of gamma-ray emission of 3C 454.3 show power-law forms, and the waiting time distribution can be described by a non-stationary Poisson process. These distribution behaviours are comparable to those of the Sun, Sagittarius A*, and M87, and follow the prediction of a self-organized criticality model. All statistical properties suggest that similar to the physical process accounting for solar GeV flares and X-ray flares in supermassive black hole systems, magnetic reconnection cloud govern the energy-release, and particle acceleration process for gamma-ray flares of 3C 454.3.

Retinal neovascularization as self-organized criticality on ultra-widefield fluorescein angiography imaging of diabetic retinopathy.

Do the distributions of surface area of non-perfusion (NP) and neovascularization (NV) on ultra-widefield fluorescein angiography (UWF FA) in patients with diabetic retinopathy (DR) differ significantly? Inclusion criteria were patients who had a UWF FA taken for DR at the Kellogg Eye Center from January 2009 to May 2018. Exclusion criteria included previous panretinal photocoagulation and significant media opacity (e.g., vitreous haemorrhage or significant cataract). UWF FAs were manually segmented for surface areas of NP and NV. The total areas per patient were organized in a histogram, and logarithmically binned to test against power law and exponential distributions. Then, a computational model was constructed in Python 3.7 to suggest a mechanistic explanation for the observed distributions. Analysis of areas of NV across 189 images demonstrated a superior fit by the least squares method to a power law distribution (p = 0.014) with an R2 fit of 0.9672. Areas of NP over 794 images demonstrated a superior fit with an exponential distribution instead (p = 0.011). When the far periphery was excluded, the R2 fit for the exponential distribution was 0.9618. A computational model following the principles of self-organized criticality (SOC), akin to earthquake and forest fire models, matched these datasets. These distributions inform what useful statistics may be applied to study of these imaging characteristics. Further, the difference in event distribution between NV and NP suggests that the two phenomena are mechanistically distinct. NV may follow SOC, propagating as a catastrophic event in an unpredictable manner.

Self-organized collective oscillations in networks of stochastic spiking neurons.

The theory of self-organized bistability (SOB) is the counterpart of self-organized criticality for systems tuning themselves to the edge of bistability of a discontinuous phase transition, rather than to the critical point of a continuous one. As far as we are concerned, there are currently few neural network models that display SOB or rather its non-conservative version, self-organized collective oscillations (SOCO). We show that by slightly modifying the firing function, a stochastic excitatory/inhibitory network model can display SOCO behaviors, thus providing some insights into how SOCO behaviors can be generated in neural network models.

A semigroup approach to a reaction–diffusion system with cross-diffusion

In this paper, we prove existence, uniqueness and special properties of the solution to a parabolic–elliptic reaction–diffusion system with cross-diffusion in an open bounded domain in Rd, as well as in Rd,d=1, 3, for various classes of the system parameters. In the case of a bounded domain and of Neumann boundary conditions we develop an approach based on a semigroup technique in the dual space (H1(Ω))′. This gives the possibility of considering singular initial data, distributions or functionals that may represent relevant physical data. The influence of the system parameters upon the qualitative behavior of the solution is investigated and some sufficient conditions for the existence of global or local solutions, and for global solutions in the case of small initial data are provided. In the case of the whole space, the problem is studied for a singular diffusion expressed by a multivalued operator. Existence is proved by passing to the limit in an approximating problem for which existence follows via a semigroup approach in H−1(Rd), d=1, 3. An argument for the uniqueness is developed in Beppo Levi spaces. Finally, the results are illustrated by a model exhibiting a self-organized criticality.

Scale invariance in x-ray flares of gamma-ray bursts

X-ray flares are generally believed to be produced by the reactivation of the central engine, and may have the same energy dissipation mechanism as the prompt emission of gamma-ray bursts (GRBs). X-ray flares can therefore provide important clues to understanding the nature of the central engines of GRBs. In this work, we study for the first time the physical connection between differential size and return distributions of X-ray flares of GRBs with known redshifts. We find that the differential distributions of duration, energy, and waiting time can be well fitted by a power-law function. In particular, the distributions for the differences of durations, energies, and waiting times at different times (i.e., the return distributions) well follow a $q$-Gaussian form. The $q$ values in the $q$-Gaussian distributions remain nearly steady for different temporal interval scales, implying a scale-invariant structure of GRB X-ray flares. Moreover, we verify that the $q$ parameters are related to the power-law indices $\alpha$ of the differential size distributions, characterized as $q=(\alpha+2)/\alpha$. These statistical features can be well explained within the physical framework of a self-organizing criticality system.

Power-law Distribution and Scale-invariant Structure from the First CHIME/FRB Fast Radio Burst Catalog

We study the statistical property of fast radio bursts (FRBs) based on a selected sample of 190 one-off FRBs in the first CHIME/FRB catalog. Three power law models are used in the analysis, and we find the cumulative distribution functions of energy can be well fitted by bent power law and thresholded power law models. The distribution functions of fluctuations of energy well follow the Tsallis q-Gaussian distribution. The q values in the Tsallis q-Gaussian distribution are constant with small fluctuations for different temporal scale intervals, indicating a scale-invariant structure of the bursts. The earthquakes and soft gamma repeaters show similar properties, which are consistent with the predictions of self-organized criticality systems.

Учет сложности в задачах транспортного спроса

The article describes the theoretical basis for taking into account the complexity in solving problems of transport demand, the analysis of the concept of “self-organized criticality” is carried out. The problems of describing complex conduct in transport systems are solved, the prerequisites or causes of complex behavior are indicated, the traditional definitions of a “complex transport system” are formalized. Examples of some models demonstrating complex behavior in transport systems are given. Objective: formulation of research directions in the theory of transport systems, which should be developed with the help of conceptual and mathematical devices of individual sciences, such as complexity theory, nonlinear dynamics, to solve problems of transport demand, accompanied by flows of various events and elements in transport macrosystems, to take into account the complexity of problems in the theory of transport systems in modeling the dynamics of transport demand. Approaches used: development of mathematical models of nonlinear transport systems demonstrating complex behavior, including models with power-law distribution of system characteristics within the framework of self-organized criticality, as well as models demonstrating bifurcations. Methods and (or) methodological apparatus of research: methods of the theory of self-organized criticality and synergetics, allowing to take into account the properties of a complex system. The scientific novelty lies in the use of methods and approaches of the theory of self-organized criticality and complexity theory for a wide class of models of transport systems used in problems of transport demand and based on traffic flow models. Directions for further research, recommendations: it is required to analyze real statistics of the behavior of transport systems that correspond to various states of the transport system, including complex behavior, as well as the formulation and solution of problems of the theory of transport macrosystems.