Complex systems and physics education

Abstract

Synergetics as a scientific area of research is in demand by society. The context of synergetics makes it possible for scientists of diferent specializations to interact fruitfully in the language of systematic understanding and search for new solutions. The presented work raises the question of how the theory of self-organization can help in the reformation of the higher education system, why this is relevant, and what can lead to the training of both teachers and students within the framework of an interdisciplinary approach. In the future, we will highlight the most important characteristics of complex systems and the simplest and at the same time conceptually simplest methods for analyzing complexity. As part of the complex systems modeling course, which will first be presented to students of physics and mathematics, and then, possibly, to students of other specialties, we present signals of seismic activity, gravitational waves and magnetic activity, and demonstrate how we can identify critical or crash phenomena in such systems. This kind of analysis can serve as a good basis for the formation of professional skills and universal competencies.

Keywords

• eol>synergetics
• complex systems
• non-equilibrium
• self-organization
• interdisciplinarity

1. Introduction

In 2021, Syukuro Manabe, Klaus Hasselmann, and Giorgio Parisi were awarded the Nobel Prize in Physics “for groundbreaking contributions to our understanding of complex physical systems” [ 1 ]. That is a sign that the study of complex systems is of paramount importance. Nevertheless, we need to deal with the problems of their implementation in the educational process.

The education system in the world today is in a state of crisis. This is evidenced by the following trends: a further increase in the number of illiterate people in the world; the widespread decline in the quality of education; the growing gap between education and culture, education and science; alienation of the student from the educational process.

This situation in the world at the present stage makes the problem of finding a new paradigm of education urgent, since the possibility of sustainable development of society, successful overcoming of global problems, regional and national conflicts characteristic of the present time of the development of civilization is closely related to the achieved level of education of all members of society [ 2 ]. But the education system is always based on a certain scientific understanding of the world and man, which determines the goals and objectives of education, its content, principles and methods.

The heyday of education in the XVII-XVIII centuries, which happened through the development and spread of classical mechanics of the New Time, led to the determination of the picture of the world, where the studied elements are unchangeable, and the laws of classical mechanics are universal and apply to all types of motion of matter.

In such a world there was no place for chance, and irreversibility and probability were usually associated with the incompleteness of knowledge. In this case, each phenomenon has a cause and at the same time there is a cause of other phenomena. Cause and efect form a chain that comes from the past, permeates the present and disappears in the future. This meant that all the processes taking place in the world were predetermined and led to the search for initial elements, having discovered which, it is possible to accurately predict the future.

Therefore, such ideological and methodological principles as rationalism, determinism, mechanismism and reductionism began to dominate in scientific knowledge, which also had a decisive influence on the education system: on the forms of knowledge acquisition, presentation of material, organizational principles of education.

The discovery by synergetics of the processes of self-organization in inanimate nature clearly shows that the transition from disorder to order, accompanied by the emergence of self-organization and stable structures, the replacement of old structures with new ones occurs according to specific internal laws inherent in certain forms of the movement of matter. Ultimately, it is the qualitative and quantitative criteria of self-organization that characterize the level of complexity and perfection of the corresponding forms of movement [ 3 ]. Based on these ideas, it is possible to develop a classification of types, forms, properties of matter according to their degree of complexity, perfection of organization, and thereby the degree of development. In this regard, development itself appears as a very complex, self-organizing process of movement from simple to complex, from less organized and perfect to more organized and perfect. In other words, development, in contrast to the movement that characterizes any changes in general, acts as a directed change associated with the emergence of a new one.

The post-non-classical stage of the development of science shows that rigid determinism and reductionism, which serve as the basis of the mechanistic view of the world, cannot be considered as universal principles of scientific knowledge, since an extensive class of phenomena and processes does not fit into the framework of linear, equilibrium and reversible schemes. In the world around us, a very real irreversibility plays an essential role, which is the basis of the majority of self-organization processes. Reversibility and rigid determinism in the world are applicable only in simple limiting cases, and irreversibility and randomness should be considered not as an exception, but as a general rule.

To integrate the synergetics approach into the educational process, it is important to instill in students ways of setting and solving problems of being and developing complex systems in various spheres: economic, social, natural, etc. It is equally important, at the beginning of studying the methods of studying complex systems, to instill in students at first or repeat with them the concepts of self-organization, chaos, destructive phenomena, to voice the diference between complex and complicated systems, etc.

Complex systems are a field of research that is now acquiring the characteristic features of a well-formed area of science with its own object, conceptual apparatus, and methods of analysis [ 4 ]. The concept of a complex system is gradually becoming one of the fundamental concepts of modern science, or, more broadly, it is increasingly appearing in a general cultural context. The expansion of the scope of application of this concept, as well as the identification and awareness of an increasing number of phenomena where it is applicable, causes dificulties in its exact definition. Although the science of complex systems covers a broad interdisciplinary field of research, the methods and concepts of physics (dynamical systems theory, quantum mechanics, statistical physics) are central to it.

So, the processes of self-organization in non-equilibrium conditions correspond to the dialectical interaction between chance and necessity, fluctuations and deterministic laws. Near bifurcations, the main role is played by chaos, randomness, while deterministic connections dominate in the intervals between bifurcations. The ways of development of self-organizing systems are not predetermined. Probability appears not as a product of our ignorance, but as an inevitable expression of chaos at the points of bifurcations. This means the end of the classical ideal of omniscience and creates the need to revise the principle of mechanical rationalism as the dominant scientific explanation of reality. The traditional education system, based on the principles of classical science, cannot efectively fulfill the role of a means of mastering the world by a person.

Hence, there is a need to integrate the principles and ideas of the complex systems paradigm in the sphere of education.

2. Analysis of previous studies

Analysis of scientific sources and publications shows that today there is an opinion that synergetics could provide significant assistance in the search for a new paradigm of education. A synergistic approach to understanding patterns operating in nature is associated with the names of Haken [ 5, 6, 7, 8 ], Haken and Schiepek [ 9 ], Nicolis and Prigogine [ 10 ], Prigogine and Stengers [ 11, 12 ], Prigogine [ 13, 14 ]. Some scientists believe that synergetics, as a theory of self-organization of complex systems, describes the general (common) that is in their development, education is a complex system, and therefore synergetics, which today is developed by various branches of scientific knowledge, necessarily becomes its new philosophy. However, despite the existence of a suficient number of works devoted to the application of synergetics in various spheres of human activity, the methodological and practical context of synergetics in the philosophy of education remains insuficiently developed. This is especially true for applying a synergistic approach to understanding the higher education system.

When forming interdisciplinary specialists, a significant component may be the humanitarization of education [ 15 ]. It is humanitarization that is designed to provide an educational synthesis of humanitarian, technical and natural science disciplines based on multi-level integration of the entire complex of knowledge. The growing trend of all scientific knowledge, which manifests itself in the transition from focusing on individual disciplines studied in the course of higher education, to strengthening interdisciplinary ties, requires qualitatively new approaches to the content of education and reorientation in teaching. Humanitarization and interdisciplinarity of education involve not only the intensive introduction of humanities disciplines into purely technical higher educational institutions, but also the enrichment of Natural Science and technical disciplines with material that reveals the struggle of scientific ideas, the human destinies of scientists-discoverers, the dependence of socio-economic and scientific progress on personal and moral qualities of a person, his creative abilities. In contrast to the traditional interdisciplinary approach in education, the goal is not only to provide knowledge, but also to teach to hear and understand colleagues working in diferent specialties, to develop skills of dialogue between specialists in diferent branches of scientific knowledge. The need for such a dialogue is becoming more and more palpable. Since the theoretical physicist Hermann Haken introduced this concept into scientific use [ 5 ], the world has been accumulating some experience in the use of synergetics and in the study of social and educational systems.

As Andrushchenko and Svyrydenko [16] notes, “... domestic scientific schools are not always ready to accept the success of other schools and directions, when traditionalism and conservatism enter the fight against innovation; threats of marginalization of carriers of foreign experience in the educational or scientific environment are actualized”. These processes in their content, scientists believe, contradict the logic of scientific cooperation, the spirit of partnership and exchange of views, in which the long-term experience of the closed educational system, its focus on ensuring the ideological function of education plays an important role. The nonlinear complexity of methodological renewal, thanks to the creativity of creative processes, creates new horizons for the emerging future [17]. This idea is supported by Kremen and Illyin [18], noting that the principles and ideas of synergetics have significant heuristic and methodological potential. In this regard, scientists believe that a synergistic approach to education and upbringing can be the basic one for solving many problems in the field of education.

When using and teaching approaches that are part of the synergetics paradigm, as already noted, chaos seems to be the engine of change. From the point of view of synergetics, personality development appears as a constant movement from one state of the system to another, in which chaos, chance, creation/destruction, passage of bifurcation points, etc.are natural states of the system, successively replacing each other, building a continuous chain of transformations [19]. Research conducted in schools and universities shows that interactive chaotic environments are very productive for developing creative thinking. The results of work in this area were presented by Davis-Seaver et al. [20], who analyzed the learning process at three levels – from a single point of balance, statement of fact, statement of a single point of view to learning on the verge of chaos, when there are many points of view, when reasoning develops in diferent directions, when students listen to the opinions of others and on this basis develop their own judgments. The role of the teacher is not to spread knowledge and evaluate the correctness of judgments, but to monitor the progress of reasoning and transfer the learning process from one level to another. As a result, the understanding becomes deeper, more versatile, and the incentives for learning are largely created by the energy of the group, and not by the diligence of the teacher. In the context of revealing a person’s creative abilities, a synergistic approach to education seeks not to eradicate chaos, but to find the relationship between order and disorder that would be most fruitful [21].

The above-mentioned concept of chaos from the point of view of synergetics loses its negative connotation. As Prigogine and Stengers [ 12 ] notes, instability can be a condition for stable and dynamic development. Only systems that are far from equilibrium are able to organize and evolve spontaneously. Thus, there is no development without instability. And if the system is strict against the implementation of new units, new units (‘innovators’) die”. In higher education, self-organizing systems are the Student, Teacher, their interrelation, etc. [22].

Jacobson and Wilensky [23], Wilensky and Jacobson [24] emphasize diferent research issues that need to be explored. They present such principles in studying complex phenomena as • experiencing complex systems phenomena; • making the complex systems conceptual framework explicit [25]; • encouraging collaboration, discussion, and reflection; the design of environments for learning about complex systems needs to take advantage of lessons learned from the extensive research on pedagogy that foster collaboration, discussion, and reflection [ 26]; • constructing theories, models, and experiments; • learning trajectories for deep understandings and explorations.

With a given appropriate conceptual and representational scafolding in the learning environment, students should be able to tap into their everyday experiences and channel and enhance these experiences to construct understandings of complex systems that are cognitively robust. Nowadays, students should have more possibilities to explore world through computational modeling which progressive scientists use almost everyday.

Jackson [27] and other, such as Pagels [28], have observed how the use of computational tools in science allows dramatically enhanced capabilities to investigate complex and dynamical systems that otherwise could not be systematically investigated by scientists. These computational modeling approaches include cellular automata, network and agent-based modeling, neural networks, genetic algorithms, Monte Carlo simulations, and so on that are generally used in conjunction with scientific visualization techniques. Examples of complex systems that have been investigated with advanced computational modeling techniques include climate change [29], urban transportation models [ 30, 31, 32 ], and economics [ 33, 34, 35, 36 ]. New communities of scientific practice have also emerged in which computational modeling techniques, in particular agent-based models and genetic algorithms, are being used to create synthetic worlds such as artificial life [ 37, 38 ] and societies [ 39 ] that allow tremendous flexibility to explore theoretical and research questions in the physical, biological, and social sciences that would be dificult or impossible in “real” or nonsynthetic settings.

Methodology of nonlinear synthesis based on scientific principles evolution and co-evolution of complex structures of the world, can form the basis of futurological research, designing various ways of human development into the future [ 40 ]. As the environmental, economic, and political problems of humanity have become global, complex and nonlinear, traditional ideas about individual responsibility are becoming questionable. We need to study and teach new models of collective behavior that take into account the diferent degrees of our individual abilities and understanding of what is happening.

We believe that the study of the apparatus of physics, graph theory, and computer science is now of paramount importance for the further development of both our society and the entire universe.

In further we need to understand how to grow an interest of students in constructing and revising computational models with multi-agent or qualitative modeling software, and how model building activities may enhance student conduct of real world experiments related to the phenomena under consideration [ 41, 42, 43 ].

3. The most important properties of complex systems to be studied

Based on the previously described characteristics and the direction in which we should move, it becomes clear that synergetics (the theory of complex systems) is the foundation of almost any system. Including pedagogical. Although the initial direction of research within this paradigm was physical systems, the latest objects of research on various manifestations of complexity also appear in the context of business organization and economics. For example, Wheatley [ 44 ] suggests that we view organizations as being more like living organisms than machines. As such, we need to modify traditional views on controlling organizations. Wheatley [ 44 ] argues that organizations are dynamic, nonlinear networks of relationships and cannot be separated into parts while maintaining their essential identity.

In complex systems, it is worth highlighting that they are • dynamic; • non-equilibrium and have the potential to change suddenly and may take one path out of an infinite number of others (bifurcate); • open systems, that is interchange energy (and information) with their surroundings; • depended. What happens next depends on what happened previously; • systems where the whole is more than the sum of its parts; • causal and yet indeterminate; • irreversible, since the interaction of parts together is transforming; • multi-agent. They composed of a diversity of agents that interact with each other, mutually afect each other, and in so doing generate novel, emergent behavior for the system as a whole. The system is constantly adapting to the conditions around it and over time it evolves; • co-evolving and move spontaneously towards the edge of chaos.

3.1. Time series data

In order to maintain students’ interest in studying complex systems and their corresponding data analysis tools, programming languages, etc. [ 45, 46 ], it is important to select truly interesting and complex systems (series). It is equally important that the studied systems are within the framework of the specialty that students are guided by. However, since we strive for an interdisciplinary approach, the study, for example, by biologists of the corresponding nonlinear methods on the example of the same socio-economic series or physical ones can also be beneficial for general development.

Complexity theory is subdivided into hard and soft complexity. Hard complexity theory stands for analytical analysis that concern with the nature of reality, while soft complexity aims to describe social and living systems. Davis and Sumara [ 47 ] proposes such term as “complexity thinking” which lies somewhere in between hard and soft skills. We support such idea and would like to promote it among ordinary citizens who are not specialists and, particularly, among universities and their student. Focusing on interdisciplinarity, both hard and soft skills, teachers and students will be more creative and productive in their further research. Knowing about interconnections across diferent disciplines, there are much more possibilities for collaborative research between diferent faculties and there is larger probability that people will be able to ifnd common topics for communication and will be engaged to cooperate.

The goal of this work is to present the basic characteristics of complex systems, which should be introduced to students during the course of studying complex systems, and the basic sets of methods that allow analyzing the varying randomness (complexity) of the system during the development of the studied signals.

In this paper, we present some of the most fundamental, applied, robust, and powerful methods on the example of three physical signals: seismic (SEI), gravitational wave (GW), and the distribution storm time (Dst) index.

SEI dataset constructed by Bladford [ 48 ]. Each event has 2048 points fixed at a seismic recording station in Scandinavia.

We used GW data GW150914 from Events of LIGO Open Science Center and select strain data (H1 and L1) after noise subtraction [49] (https://www.ligo.org/detections/GW150914.php).

The Dst index is an index of magnetic activity derived from a network of nearequatorial geomagnetic observatories that measures the intensity of the globally symmetrical equatorial electrojet (“ring current”). Dst is maintained at National Centers for Environmental Information [50] from 1957 to the present. Dst equivalent equatorial magnetic disturbance indices are derived from hourly scalings of low-latitude horizontal magnetic variation. They show the efect of the globally symmetrical westward flowing high altitude equatorial ring current, which causes the “main phase” depression worldwide in the H-component field during large magnetic storms.

In this paper, the time series of hourly values of the storm on March 13, 1989 is investigated. It is the strongest storm in the space age in several ways; the power system of the province of Quebec was out of order. The peak of the storm falls in the middle of the time series (point 1000).

In order to study changes of complexity dynamically, i.e., to get not only one value that will characterize the whole system, but an array of values, where each value will reflect the complexity of a signal in a specific period, we use sliding window approach [51].

In figure 1 is presented the dynamics of all physical signals that could be studied during physics classes. However, students of other faculties can also be interested.

3.2. Fat-tailed distribution

When studying complex systems, we inevitably encounter power distributions characterized by thick tails. A classic example is the power-law of dividing words by their frequency of use in a text, known as Zipf’s law [52].

In economics, this is the law of wealth distribution among individuals [53]; in demography, the distribution of cities by their size [54]; in biology, the distribution of the size of forest patches [55]; in scientometry, the distribution of citations [56]. In general, a wide class of phenomena is described in the framework of distributions with a degree dependence, but the researcher (student) will have to find out the nature of such a dependence, which can be caused by many factors: critical phenomena, processes with preference, self-organized criticality, multiplicative processes with connections, optimization and path-dependent nonergodic processes, the phase space of which decreases with evolution [57, 58, 59, 60, 61].

First of all, it will be important to build an empirical distribution for our data (figure 2). Having visualized the series we study in this paper, we can already be convinced of the non-Gaussian dynamics of the presented systems.

In the course of our research, we have determined that the Lévy -stable distribution most successfully covers the key statistical characteristics of both the economic [ 62, 63, 64 ] and those systems presented in this paper. Figures 3a to 3c show the window dynamics of the index derived from the Lévy distribution that characterizes the “heaviness” of tails.

3.3. Multifractality

When studying various types of systems, we often encounter both fractal (self-similar) structures and sets of diferent fractal dimensions [ 65 ]. In such problems, it is necessary to take into account the entire range of critical indicators that characterize diferent moments in the distribution of observed quantities. Such properties usually relate to the term “multifractality” [ 66 ].

There are several diferent algorithms that allow the obtention of multifractal spectra from time series. The most famous is the MF-DFA [ 67, 68, 69 ].

Based on the MF-DFA procedure, we select the maximum value of such a quantitative characteristic of multifractality as the singularity strength [ 70 ], although in the corresponding section of fractal (multifractal) analysis, it would be necessary to characterize and demonstrate the dynamics of all multifractality indicators. The following figure shows the window dynamics of the maximum value of the singularity strength.

3.4. Network analysis

Equally important is the network analysis of complex systems. Today, networks play a central role in modeling complex systems, as they ofer a way to describe diferent types of relationships between agents that act as endpoints in the network. Complex networks can characterize information, social, economic, biological, neural, and other systems [ 71, 72, 73, 74 ]. For example, a society can be represented as a network, where each individual (university, wealth, city) can be represented as nodes of a graph, and the connection between them through edges. For cities, edges can represent a road, where the possibilities of movement can vary, and therefore a diferent weight can be determined for each edge.

In general, the computer network model is a random graph, the law of mutual arrangement of edges and vertices for which is defined by the probability distribution.

The simplest of networked objects, so-called Erdös-Rényi, or random graphs. Such graphs can be characterized within the framework of the Poisson distribution, but most complex systems, as already noted, are characterized within the framework of distributions with heavy tails.

One of the most interesting characteristics of networks is the vertex degree. The vertex degree distribution for many real-world networks shows a power-law dependence. Such networks are called scale-independent. Scale-free networks are often characterized by very short average distances between randomly chosen pairs of nodes that may have a strong impact on the whole dynamics.

In addition to the topology of graphs, you can also study their quantitative characteristics. In our case, using the window procedure, we get a variable graph representation of our signal over time. For the presented work, we calculated the maximum vertex degree of the graph (), since this measure is one of the conceptually simplest measures, although many other measures can be represented. It is worth noting that there are also various algorithms for converting a time series to a graph. We would like to emphasize the visibility graph algorithms [ 64, 75, 76, 77, 78 ] (figure 5) and one based on recurrence analysis [79] (figure 6).

3.5. Recurrence analysis

Processes in nature are characterized by pronounced recurrent behavior, such as periodicity or irregular cyclicity.

Moreover, the recurrence (repeatability) of states in the sense of passing a further trajectory quite close to the previous one is a fundamental property of dissipative dynamical systems. This property was noted in the 1880s by the French mathematician Henri Poincaré and subsequently formulated in the form of the “recurrence theorem”, published in 1890 [80].

The essence of this fundamental property is that, despite the fact that even the smallest perturbation in a complex dynamical system can lead the system to an exponential deviation from its state, after a while the system tends to return to a state that is somewhat close to the previous one, and goes through similar stages of evolution.

In 1987, Eckmann et al. [81] proposed a method for mapping the recurrence of phase space trajectories to × matrix. The appearance of a recurrence diagram allows us to judge the nature of processes occurring in the system, the presence and influence of noise, states of repetition and fading (laminarity), and the implementation of sudden changes (extreme events) during the evolution of the system. If you look at recurrent diagrams in more detail, you can find small-scale structures (textures) consisting of simple points, diagonal, horizontal, and vertical lines, which in turn correspond to chaotic, repetitive, or laminar states.

Using combinations of these states, Zbilut and Webber [82], Webber and Zbilut [83] developed a tool for calculating a series of measures based on the distribution of recurrent points on a recurrence matrix. Later, the toolkit for quantitative recurrent analysis was supplemented by Marwan and Kurths [84]. The tools of quantitative recurrent analysis include the recurrence rate, determined by the ratio of recurrent points to the total number of points on the recurrence matrix under study. In addition to the recurrence measure, in the course of analyzing complex systems, it would be possible to present such measures as determinism, divergence, entropy, trend, and so on [ 78, 64, 51, 85, 86, 87, 88 ].

In this paper, we will focus on the recurrence rate and present it for the already specified series (figure 7).

3.6. Entropy and non-extensive statistics

The Boltzmann-Gibbs statistical entropy and the classical statistical mechanics associated with it are extremely useful tools for studying a wide range of simple systems that are characterized by a small range of space-time correlations (short memory), the additivity of noise, the presence of intense chaos, the ergodicity of dynamic processes, the Euclidean geometry of phase space, the locality of interaction between elements, the Gaussian probability distributions, etc.

The Boltzmann-Gibbs statistical entropy is a fundamental concept of the school section and the university course of thermodynamics and statistical physics.

In statistical mechanics, entropy denotes the number of possible configurations of a thermodynamic system. The notion of entropy can be associated with the uncertainty in the system [89, 90]. In 1948, Shannon [91] transformed classical statistical entropy to information entropy. Since then, a number of other types of information entropy have been developed [ 92, 93, 94, 95 ].

In order to study many real-world systems, it is necessary to go beyond the standard course of thermodynamics, statistical physics, and classical Shannon entropy. A whole range of natural, artificial and social systems, which, unlike those mentioned above, are characterized by a long range of spatio-temporal correlations and non-Gaussian processes.

Since the non-Gaussian and multifractal behavior of the studied systems was presented previously, we will depict the autocorrelation function in the figure 8a, as it should demonstrate an indicator decline. This fact will indicate the dependence of the following values on the previous ones.

It is also worth mentioning that such systems are characterized by multiplicative noise, the presence of weak chaos (vanishing maximum Lyapunov exponent), non-ergodicity of dynamic processes, hierarchy (usually multifractality) of the geometry of the phase space, the presence of asymptotically power-law statistical distributions. A fairly wide class of these complex systems (although not all) it is adequately described by non-additive statistics based on the Tsallis parametric entropy.

Figures 8b to 8d show the -Gaussian distribution from the Tsallis statistics for the considered series in comparison with the classical Gaussian one.

3.7. Reversibility and irreversibility

The last characteristic that we would like to mention is time-reversibility. Temporary irreversibility is a key property of non-equilibrium systems.

Again, such systems are characterized by the presence of memory, while reversibility increases with more noisy and unpredictable signals. Thus, by calculating the irreversibility, we determine the degree of nonlinearity and predictability. It is important to note that the significant time reversibility excludes linear Gaussian processes as a model of generating dynamics. Within the framework of the systems we are considering, we need to think about methods of nonlinear dynamics and non-Gaussian ones [ 96, 97 ].

Over the past decade, various methods have been proposed for calculating the degree of irreversibility in systems [ 98, 99, 100, 101, 102, 103, 104, 105 ] and we have presented how to use some of them for crises identification [ 77 ]. For pedagogical purposes, along with the mentioned concept of multifractality and entropy, we would like to present irreversibility based on the multifractal approach [ 105 ] and permutation patterns [ 104 ]. The last mentioned approach could be taught within the section of entropy approaches if we were teaching students. However, the calculation of irreversibility based on graph theory is also possible [ 100, 101, 102 ].

Figures 9a to 9c show the mentioned measures of irreversibility for the studied signals.

4. Conclusions

The analysis of the adaptive nature of many complex systems led to the creation of methods and the development of concepts that were successfully applied to describe formally similar phenomena in chemical, biological, social and other systems of agents of non-physical nature. It is sometimes argued that if physics is the science of the four fundamental forces that matter interacts with.

Remembering figures in the field of social sciences, it is still relevant to adapt theoretical material and practical tasks of various fields of physics and higher mathematics to those disciplines that students already study in the framework of social sciences.

In this paper, we have presented some of the most significant approaches on the example of SEI, GW, and Dst, but even more can be shown and much can be taught [ 106, 107 ]. The most important thing is not just to convey information, but to interest the student in the right way. of the Yaroslav the Wise National University 3 (2019) 71–80. URL: http://fil.nlu.edu.ua/ article/view/170335. doi:10.21564/2075-7190.42.170335. [16] V. Andrushchenko, D. Svyrydenko, Akademichna mobilnist v ukrayinskomu prostori vyshchoyi shkoly: realiyi, vyklyky ta perspektyvy rozvytku, Vyshcha osvita Ukrayiny 2 (2016) 5–11. [17] O. Rubanets, Kohnityvnyy aspekt metodolohichnoho onovlennya vyshchoyi shkoly,

Vyshcha osvita Ukrayiny 3 (2016) 24–29. [18] V. H. Kremen, V. V. Illyin, Synerhetyka v osviti: kontekst lyudynotsentryzmu, 2012. [19] N. Kochubei, Neliniine myslennia v osviti, in: I. Predborska (Ed.), Filosofski abrysy suchasnoi osvity, volume 11, Universytetska kniha, 2006, pp. 29–41. [20] J. Davis-Seaver, D. Leflore, T. Smith, Promoting critical thinking at the university level, National forum of teacher educational journal 10E (2000) 1–11. URL: http://www.nationalforum.com/Electronic%20Journal%20Volumes/Davis-Seaver% 20Jane%20Promoting%20Critical%20Thinking%20at%20the%20University%20Level.pdf. [21] V. Kremen, Pedahohichna synerhetyka: poniatiino-katehorialnyi syntez, Teoriia i praktyka upravlinnia sotsialnymy systemamy 3 (2013) 3–19. [22] T. M. Taranenko, Synerhetychnyi pidkhid v orhanizatsii navchalno-vykhovnoho protsessu, Tavriiskyi visnyk osvity 1 (2014) 10–15. [23] M. J. Jacobson, U. Wilensky, Complex systems in education: Scientific and educational importance and implications for the learning sciences, Journal of the Learning Sciences 15 (2006) 11–34. doi:10.1207/s15327809jls1501\_4. [24] U. Wilensky, M. Jacobson, Complex systems and the learning sciences, in: The Cambridge Handbook of the Learning Sciences, Second Edition, Cambridge University Press, 2014, pp. 319–338. doi:10.1017/CBO9781139519526.020. [25] N. R. Council, How People Learn: Brain, Mind, Experience, and School: Expanded Edition, The National Academies Press, Washington, DC, 2000. URL: https://www.nap.edu/catalog/ 9853/how-people-learn-brain-mind-experience-and-school-expanded-edition. doi:10. 17226/9853. [26] National Research Council, Inquiry and the National Science Education Standards: A Guide for Teaching and Learning, The National Academies Press, Washington, DC, 2000. URL: https://www.nap.edu/catalog/9596/ inquiry-and-the-national-science-education-standards-a-guide-for. doi:10.17226/ 9596. [27] E. A. Jackson, The second metamorphosis of science: A second view, SFI Working Papers 96-05-059, Santa Fe Institute, 1995. URL: https://EconPapers.repec.org/RePEc:wop:safiwp: 95-01-001. [28] H. R. Pagels, The Dreams of Reason: The Computer and the Rise of the Sciences of

Complexity, Simon & Schuster, 1988. [29] J. J. West, H. Dowlatabadi, On assessing the economic impacts of sea-level rise on developed coasts, in: T. E. Downing, A. A. Olsthoorn, R. S. J. Tol (Eds.), Climate, Change and Risk, 1 ed., 1998, p. 16. [30] M. Balmer, K. Nagel, B. Raney, Large-scale multi-agent simulations for transportation applications, Journal of Intelligent Transportation Systems 8 (2004) 205–221. doi:10. 1080/15472450490523892. S. J. Waldman, M. Walker, L. Wallace, S. Walsh, G. Wang, H. Wang, M. Wang, X. Wang, Y. Wang, H. Ward, R. L. Ward, J. Warner, M. Was, B. Weaver, L.-W. Wei, M. Weinert, A. J. Weinstein, R. Weiss, T. Welborn, L. Wen, P. Weßels, T. Westphal, K. Wette, J. T. Whelan, S. E. Whitcomb, D. J. White, B. F. Whiting, K. Wiesner, C. Wilkinson, P. A. Willems, L. Williams, R. D. Williams, A. R. Williamson, J. L. Willis, B. Willke, M. H. Wimmer, L. Winkelmann, W. Winkler, C. C. Wipf, A. G. Wiseman, H. Wittel, G. Woan, J. Worden, J. L. Wright, G. Wu, J. Yablon, I. Yakushin, W. Yam, H. Yamamoto, C. C. Yancey, M. J. Yap, H. Yu, M. Yvert, A. Zadrożny, L. Zangrando, M. Zanolin, J.-P. Zendri, M. Zevin, F. Zhang, L. Zhang, M. Zhang, Y. Zhang, C. Zhao, M. Zhou, Z. Zhou, X. J. Zhu, M. E. Zucker, S. E. Zuraw, J. Zweizig (LIGO Scientific Collaboration and Virgo Collaboration), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016) 061102. doi:10.1103/PhysRevLett.116.061102. [50] The disturbance storm time index, 2021. URL: https://www.ngdc.noaa.gov/geomag/ indices/dst.html. [51] V. Soloviev, A. Belinskij, Methods of nonlinear dynamics and the construction of cryptocurrency crisis phenomena precursors, CEUR Workshop Proceedings 2104 (2018) 116–127. [52] G. K. Zipf, Human behavior and the principle of least efort, The Economic Journal 60 (1950) 808–810. URL: http://www.jstor.org/stable/2226729. [53] V. Pareto, Cours d’Économie Politique, volume 1, 1896. [54] F. Auerbach, Das gesetz der bevölkerungskonzentration, petermanns, Geographische

Mitteilungen 59 (1913) 74–76. [55] L. A. Saravia, S. R. Doyle, B. Bond-Lamberty, Power laws and critical fragmentation in global forests, Scientific Reports 8 (2018) 17766. doi: 10.1038/s41598-018-36120-w. [56] M. Brzeziński, Power laws in citation distributions: Evidence from Scopus, Working Papers 2014-05, Faculty of Economic Sciences, University of Warsaw, 2014. URL: https: //ideas.repec.org/p/war/wpaper/2014-05.html. [57] C. Domp, The Critical Point: A Historical Introduction To The Modern Theory Of Critical

Phenomena, 1 ed., CRC Press, 1996. doi:10.1201/9781482295269. [58] D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer Series in Synergetics, 2 ed., Springer-Verlag, 2006. doi:10.1007/3-540-33182-4. [59] P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality: An explanation of the 1/f noise,

Phys. Rev. Lett. 59 (1987) 381–384. doi:10.1103/PhysRevLett.59.381. [60] B. B. Mandelbrot, An informational theory of the statistical structure of languages,

Communication Theory (1953) 486–502. [61] B. Corominas-Murtra, R. Hanel, S. Thurner, Understanding scaling through historydependent processes with collapsing sample space, Proceedings of the National Academy of Sciences 112 (2015) 5348–5353. URL: https://www.pnas.org/content/112/17/5348. doi:10. 1073/pnas.1420946112. [62] A. Bielinskyi, S. Semerikov, V. Solovieva, V. Soloviev, Levy’s stable distribution for stock crash detecting, SHS Web of Conferences 65 (2019) 06006. doi:10.1051/shsconf/ 20196506006. [63] A. Bielinskyi, I. Khvostina, A. Mamanazarov, A. Matviychuk, S. Semerikov, O. Serdyuk, [78] V. N. Soloviev, A. Belinskiy, Complex systems theory and crashes of cryptocurrency market, in: V. Ermolayev, M. C. Suárez-Figueroa, V. Yakovyna, H. C. Mayr, M. Nikitchenko, A. Spivakovsky (Eds.), Information and Communication Technologies in Education, Research, and Industrial Applications, Springer International Publishing, Cham, 2019, pp. 276–297. doi:10.1007/978-3-030-13929-2_14. [79] R. V. Donner, M. Small, J. F. Donges, N. Marwan, Y. Zou, R. Xiang, J. Kurths, Recurrencebased time series analysis by means of complex network methods, International Journal of Bifurcation and Chaos 21 (2011) 1019–1046. doi:10.1142/S0218127411029021. [80] H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta Math 13 (1890) 1–270. [81] J.-P. Eckmann, S. O. Kamphorst, D. Ruelle, Recurrence plots of dynamical systems,

Europhysics Letters (EPL) 4 (1987) 973–977. doi:10.1209/0295-5075/4/9/004. [82] J. P. Zbilut, C. L. Webber, Embeddings and delays as derived from quantification of recurrence plots, Physics Letters A 171 (1992) 199–203. doi:10.1016/0375-9601(92) 90426-M. [83] C. L. Webber, J. P. Zbilut, Dynamical assessment of physiological systems and states using recurrence plot strategies, Journal of Applied Physiology 76 (1994) 965–973. doi:10. 1152/jappl.1994.76.2.965. [84] N. Marwan, J. Kurths, Nonlinear analysis of bivariate data with cross recurrence plots,

Physics Letters A 302 (2002) 299–307. doi:10.1016/s0375-9601(02)01170-2. [85] V. Derbentsev, S. Semerikov, O. Serdyuk, V. Solovieva, V. Soloviev, Recurrence based entropies for sustainability indices, E3S Web of Conferences 166 (2020) 13031. doi:10. 1051/e3sconf/202016613031. [86] Z. Fan, Q. Chen, G. Sun, N. Mastorakis, X. Zhuang, Nonlinear analysis of gravitational wave signals based on recurrence quantification analysis, MATEC Web of Conferences 210 (2018) 05011. doi:10.1051/matecconf/201821005011. [87] M. Lin, G. Zhao, G. Wang, Recurrence quantification analysis for detecting dynamical changes in earthquake magnitude time series, International Journal of Modern Physics C 26 (2015) 1550077. doi:10.1142/S0129183115500771. [88] A. Banerjee, B. Goswami, Y. Hirata, D. Eroglu, B. Merz, J. Kurths, N. Marwan, Recurrence analysis of extreme event-like data, Nonlinear Processes in Geophysics 28 (2021) 213–229. URL: https://npg.copernicus.org/articles/28/213/2021/. doi:10.5194/ npg-28-213-2021. [89] R. Clausius, XVI. On a mechanical theorem applicable to heat, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 40 (1870) 122–127. doi:10. 1080/14786447008640370. [90] L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen, volume 67, Vieweg+Teubner Verlag, 1970, pp. 115–225. doi:10.1007/ 978-3-322-84986-1_3. [91] C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal 27 (1948) 379–423. doi:10.1002/j.1538-7305.1948.tb01338.x. [92] L. P. Karakatsanis, G. P. Pavlos, M. N. Xenakis, Tsallis non-extensive statistics, intermittent turbulence, SOC and chaos in the solar plasma. Part two: Solar flares dynamics, Physica A: Statistical Mechanics and its Applications 392 (2013) 3920–3944. doi:10.1016/j.physa.

References

[1] Nobel Foundation , Press release: The nobel prize in physics 2021 , 2021 . URL: https: //www.nobelprize.org/prizes/physics/2021/press-release/.

[2] N. V. Karlov , Preobrazovaniye obrazovaniya, Voprosy filosofii 11 ( 1998 ) 3 - 20 .

[3] E. N. Knyazeva , S. P. Kurdyumov , Zakony evolyutsii i samoorganizatsii slozhnykh system , Nauka , Moscow, 1994 .

[4] S. Thurner (Ed.), 43 Visions for Complexity, volume 3 of Exploring Complexity, World Scientific, Singapore, 2017 . URL: http://pure.iiasa.ac.at/id/eprint/14000/.

[5] H. Haken , Synergetics: Introduction and Advanced Topics , 1 ed., Springer, Berlin, Heidelberg, 2004 . doi: 10 .1007/978-3- 662 -10184-1.

[6] H. Haken , Synergetics, Physics Bulletin 28 ( 1977 ) 412 .

[7] H. Haken , Synergetics: Formation of ordered structures out of chaos , Leonardo 15 ( 1982 ) 66 - 67 . URL: http://www.jstor.org/stable/1574350.

[8] H. Haken , Can synergetics be of use to management theory?, in: Self-organization and management of social systems , Springer, 1984 , pp. 33 - 41 .

[9] H. Haken , G. Schiepek, Synergetik in der Psychologie: Selbstorganisation verstehen und gestalten , Hogrefe Göttingen, 2006 .

[10] G. Nicolis , I. Prigogine , Exploring complexity: an introduction , W.H. Freeman New York, 1989 .

[11] I. Prigogine , I. Stengers , Order Out of Chaos Man's New Dialogue with Nature , Bantam Books , 1984 .

[12] I. Prigogine , I. Stengers , The End of Certainty, Free Press, 1997 .

[13] I. Prigogine , The philosophy of instability, Futures 21 ( 1989 ) 396 - 400 . doi: 10 .1016/ S0016- 3287 ( 89 ) 80009 - 6 .

[14] I. Prigogine , From Being to Becoming Time and Complexity in the Physical Sciences , W.H. Freeman , 1980 .

[15] O. Y. Panfilov , I. V. Romanova , Synerhetychnyy pidkhid v osmyslenni osvity , Bulletin

[31] D. Helbing , K. Nagel , The physics of trafic and regional development , Contemporary Physics 45 ( 2004 ) 405 - 426 . doi: 10 .1080/00107510410001715944.

[32] M. Noth , A. Borning , P. Waddell , An extensible, modular architecture for simulating urban development, transportation, and environmental impacts , Computers, Environment and Urban Systems 27 ( 2003 ) 181 - 203 . doi: 10 .1016/S0198- 9715 ( 01 ) 00030 - 8 .

[33] P. W. Anderson , K. J. Arrow , D. Pines (Eds.), The Economy As An Evolving Complex System , 1 ed ., Addison-Wesley , 1988 .

[34] W. B. Arthur , S. N. Durlauf , D. A. Lane (Eds.), The Economy As An Evolving Complex System II , 1 ed., Addison-Wesley , 1997 .

[35] R. Axelrod , The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration , Princeton Studies in Complexity, Princeton University Press, 1997 .

[36] J. M. Epstein , R. Axtell , Growing Artificial Societies: Social Science from the Bottom Up , volume 1 , 1 ed., The MIT Press, 1996 . URL: https://EconPapers.repec.org/RePEc:mtp:titles: 0262550253 .

[37] C. Langton (Ed.), Artificial life, Addison-Wesley , 1989 .

[38] C. Langton (Ed.), Artificial life: Overview , MIT, 1995 .

[39] J. M. Epstein , R. Axtell , Growing artificial societies: Social science from the bottom up , MIT, 1996 .

[40] V. M. Vandyshev (Ed.), Problemy koevolyutsiyi, Philosophical Sciences, Sums'kyy Derzhavnyy Pedahohichnyy Universytet Imeni A. S. Makarenka , 2000 .

[41] D. Abrahamson , U. Wilensky, Collaboration and equity in classroom activities using Statistics As Multi-Participant Learning-Environment Resource (S.A . M.P.L.E.R .), in: W. Stroup, U. Wilensky , C. D. Lee (Eds.), Patterns in group learning with next-generation network technology , 2005 .

[42] D. Abrahamson , U. Wilensky, The stratified learning zone: Examining collaborativelearning design in demographically diverse mathematics classrooms , in: D. Y. White , E. H. Gutstein (Eds.), Equity and diversity studies in mathematics learning and instruction . Conference: annual meeting of the American Educational Research Associationannual meeting of the American Educational Research Association , 2005 .

[43] S. Jackson , J. Krajcik , E. Soloway, Model-it: A design retrospective , in: M. J. Jacobson , R. B. Kozma (Eds.), Innovations in science and mathematics education: Advanced designs for technologies of learning , Lawrence Erlbaum Associates, Inc, 2000 , pp. 77 - 115 .

[44] M. J. Wheatley , Leadership and the New Science: Discovering order in a chaotic world, 3 ed ., Berrett-Koehler Publishers , 2006 .

[45] R. H. Shumway , D. S. Stofer , Time Series Analysis and Its Applications. With R Examples , Springer Texts in Statistics, 4 ed., Springer, 2016 .

[46] B. D. Fulcher , M. A. Little , N. S. Jones , Highly comparative time-series analysis: the empirical structure of time series and their methods , Journal of The Royal Society Interface 10 ( 2013 ) 20130048 . doi: 10 .1098/rsif. 2013 . 0048 .

[47] B. Davis , D. Sumara , Complexity and education: Inquiries into learning, teaching, and research , Journal of Contemporary Issues in Education 1 ( 2006 ) 54 - 55 .

[48] R. R. Bladford , Discrimination of Earthquakes and Explosions, Report AFTAC -TR- 93 -044 HQ , Air Force Technical Applications Center, Patrick Air Force Base FL 32925-6001 , 1993 . URL: https://apps.dtic.mil/sti/pdfs/ADA267638.pdf. V. Solovieva , V. N. Soloviev , Predictors of oil shocks. econophysical approach in environmental science , IOP Conference Series: Earth and Environmental Science 628 ( 2021 ) 012019 . doi: 10 .1088/ 1755 -1315/628/1/012019.

[64] A. Bielinskyi , O. Serdyuk , S. Semerikov , V. Soloviev , Econophysics of cryptocurrency crashes: an overview , SHS Web of Conferences 107 ( 2021 ) 03001 . doi: 10 .1051/shsconf/ 202110703001.

[65] H. E. Stanley , P. Meakin , Multifractal phenomena in physics and chemistry , Nature 335 ( 1988 ) 405 - 409 . doi: 10 .1038/335405a0.

[66] K. R. Sreenivasan , C. Meneveau , The fractal facets of turbulence , Journal of Fluid Mechanics 173 ( 1986 ) 357 - 386 . doi: 10 .1017/S0022112086001209.

[67] J. W. Kantelhardt , S. A. Zschiegner , E. Koscielny-Bunde , S. Havlin , A. Bunde , H. Stanley , Multifractal detrended fluctuation analysis of nonstationary time series , Physica A: Statistical Mechanics and its Applications 316 ( 2002 ) 87 - 114 . doi: 10 .1016/S0378- 4371 ( 02 ) 01383 - 3 .

[68] D. B. de Freitas , M. M. F. Nepomuceno , J. R. D. Medeiros , Multifractal signatures of gravitational waves detected by ligo , 2019 . arXiv: 1912 .12967.

[69] I. Eghdami , H. Panahi , S. M. S. Movahed , Multifractal analysis of pulsar timing residuals: Assessment of gravitational wave detection , The Astrophysical Journal 864 ( 2018 ) 162 . doi: 10 .3847/ 1538 -4357/aad7b9.

[70] Y. Ashkenazy , D. R. Baker , H. Gildor , S. Havlin , Nonlinearity and multifractality of climate change in the past 420,000 years , Geophysical Research Letters 30 ( 2003 ) 4 . doi: 10 .1029/2003GL018099.

[71] M. E. J. Newman , The structure and function of complex networks , SIAM Review 45 ( 2003 ) 167 - 256 . doi: 10 .1137/s003614450342480.

[72] S. Boccaletti , G. Bianconi, R. Criado , C. I. del Genio , J. Gómez-Gardeñes , M. Romance , I. Sendiña-Nadal , Z. Wang , M. Zanin , The structure and dynamics of multilayer networks , Physics Reports 544 ( 2014 ) 1 - 122 . doi: 10 .1016/j.physrep. 2014 . 07 .001, the structure and dynamics of multilayer networks .

[73] S. Boccaletti , V. Latora , Y. Moreno , M. Chavez , D.-U. Hwang, Complex networks: Structure and dynamics , Physics Reports 424 ( 2006 ) 175 - 308 . doi: 10 .1016/j.physrep. 2005 . 10 .009.

[74] M. Baiesi , M. Paczuski , Scale-free networks of earthquakes and aftershocks , Physical Review E 69 ( 2004 ). doi: 10 .1103/physreve.69.066106.

[75] B. A.-S. Juan , L. Guzmán-Vargas , Earthquake magnitude time series: scaling behavior of visibility networks , The European Physical Journal B: Condensed Matter and Complex Systems 86 ( 2013 ) 1 - 10 . URL: https://ideas.repec.org/a/spr/eurphb/v86y2013i11p1- 1010 . 1140 - epjb- e2013- 40762 -2.html. doi: 10 .1140/epjb/e2013-40762-2.

[76] S. Kundu , A. Opris , Y. Yukutake , T. Hatano, Extracting correlations in earthquake time series using visibility graph analysis , Frontiers in Physics 9 ( 2021 ) 179 . doi: 10 .3389/ fphy. 2021 . 656310 .

[77] A. Bielinskyi , S. Hushko , A. Matviychuk , O. Serdyuk , S. Semerikov , V. Soloviev , The lack of reversibility during financial crisis and its identification , SHS Web of Conferences 107 ( 2021 ) 03002 . URL: https://doi.org/10.1051/shsconf/202110703002. doi: 10 .1051/ shsconf/202110703002. 2013 . 05 .010.

[93] M. Javaherian , S. Mollaei , Multiscale Entropy Analysis of Gravitational Waves, Adv. High Energy Phys . 2021 ( 2021 ) 6643546 . doi: 10 .1155/ 2021 /6643546.

[94] Y. E. Litvinenko , A maximum entropy argument for the slopes of power-law particle spectra in solar flares , The Astrophysical Journal 880 ( 2019 ) 20 . doi: 10 .3847/ 1538 -4357/ ab2760.

[95] A. Posadas , J. Morales , A. Posadas-Garzon , Earthquakes and entropy: Characterization of occurrence of earthquakes in Southern Spain and Alboran Sea , Chaos: An Interdisciplinary Journal of Nonlinear Science 31 ( 2021 ) 043124 . doi: 10 .1063/5.0031844.

[96] A. J. Lawrance , Directionality and reversibility in time series, International Statistical Review / Revue Internationale de Statistique 59 ( 1991 ) 67 - 79 . URL: http://www.jstor.org/ stable/1403575.

[97] L. Stone , G. Landan, R. M. May , Detecting TimeArrow: a method for identifying nonlinearity and deterministic chaos in time-series data , Proceedings of the Royal Society of London. Series B: Biological Sciences 263 ( 1996 ) 1509 - 1513 . doi: 10 .1098/rspb. 1996 . 0220 .

[98] C. S. Daw , C. E. A. Finney , M. B. Kennel , Symbolic approach for measuring temporal “irreversibility” , Phys. Rev. E 62 ( 2000 ) 1912 - 1921 . doi: 10 .1103/PhysRevE.62. 1912 .

[99] M. B. Kennel , Testing time symmetry in time series using data compression dictionaries , Phys. Rev. E 69 ( 2004 ) 056208 . doi: 10 .1103/PhysRevE.69.056208.

[100] L. Lacasa , A. Nuñez , E. Roldán , J. M. R. Parrondo , B. Luque , Time series irreversibility: a visibility graph approach , Eur. Phys. J. B 85 ( 2012 ) 217 . doi: 10 .1140/epjb/ e2012-20809-8.

[101] J. F. Donges , R. V. Donner , J. Kurths , Testing time series irreversibility using complex network methods , EPL (Europhysics Letters) 102 ( 2013 ) 10004 . doi: 10 .1209/ 0295 -5075/ 102/10004.

[102] R. Flanagan , L. Lacasa, Irreversibility of financial time series: A graph-theoretical approach , Physics Letters A 380 ( 2016 ) 1689 - 1697 . doi: 10 .1016/j.physleta. 2016 . 03 . 011.

[103] M. Costa , A. L. Goldberger , C. - K. Peng, Broken asymmetry of the human heartbeat: Loss of time irreversibility in aging and disease , Phys. Rev. Lett . 95 ( 2005 ) 198102 . doi: 10 .1103/PhysRevLett.95.198102.

[104] M. Zanin , A. Rodríguez-González , E. Menasalvas Ruiz , D. Papo , Assessing time series reversibility through permutation patterns , Entropy 20 ( 2018 ). URL: https://www.mdpi. com/1099-4300/20/9/665. doi: 10 .3390/e20090665.

[105] C. Jiang , P. Shang , W. Shi , Multiscale multifractal time irreversibility analysis of stock markets , Physica A: Statistical Mechanics and its Applications 462 ( 2016 ) 492 - 507 . doi: 10 .1016/j.physa. 2016 . 06 .092.

[106] Y. Chen , W. B. Manchester , A. O. Hero , G. Toth , B. DuFumier, T. Zhou , X. Wang , H. Zhu , Z. Sun , T. I. Gombosi , Identifying solar flare precursors using time series of SDO/HMI images and SHARP parameters , Space Weather 17 ( 2019 ) 1404 - 1426 . doi: 10 .1029/ 2019sw002214.

[107] H. Liu , C. Liu, J. T. L. Wang , H. Wang , Predicting solar flares using a long short-term memory network , The Astrophysical Journal 877 ( 2019 ) 121 . doi: 10 .3847/ 1538 -4357/ ab1b3c.