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In this paper, an output dynamic game model of intertwined supply chains operating in two different countries is established. The Nash equilibrium point of the model and its stable region are obtained using nonlinear dynamic principles. The complex properties of the system, such as stability, period-doubling bifurcations, and chaos, are investigated using numerical simulations. Our results suggest that the level of output and the system's profits undergo bifurcation and chaos with an increase in the output adjustment speed. An interesting phenomenon occurs in that higher tariffs lead to the expansion of the stable range of the supply chain in the product-exporting country. The chaotic behavior of the system is sensitive to the value of the initial level of output. In supply chain competition, each supply chain firm should make suitable adjustments to the speed of output. To maintain the stability of domestic markets, excessive tariffs should be avoided. It is essential that each supply chain firm evaluates the potential impacts of different initial output values when making initial decisions. Using the method of delayed feedback control, the chaotic behavior of the system can effectively be controlled. These findings offer valuable and novel insight into inter-chain competition in supply chain networks. [ABSTRACT FROM AUTHOR]

Deep reinforcement learning (DRL) algorithms are suitable for modeling and controlling complex systems. Methods for controlling chaos, a difficult task, require improvement. In this article, we present a DRL‐based control method that can control a nonlinear chaotic system without any prior knowledge of the system's equations. We use proximal policy optimization (PPO) to train an agent. The environment is a Lorenz chaotic system, and our goal is to stabilize this chaotic system as quickly as possible and minimize the error by adding extra control terms to the chaotic system. Therefore, the reward function accounts for the total triaxial error. The experimental results demonstrated that the trained agent can rapidly suppress chaos in the system, regardless of the system's random initial conditions. A comprehensive comparison of different DRL algorithms indicated that PPO is the most efficient and effective algorithm for controlling the chaotic system. Moreover, different maximum control forces were applied to determine the relationship between the control forces and controller performance. To verify the robustness of the controller, random disturbances were introduced during training and testing, and the empirical results indicated that the agent trained with random noise performed better. The chaotic system has highly nonlinear characteristics and is extremely sensitive to initial conditions, and DRL is suitable for modeling such systems. [ABSTRACT FROM AUTHOR]

The reservoir computing (RC) is increasingly used to learn the synchronization behavior of chaotic systems as well as the dynamical behavior of complex systems, but it is scarcely applied in studying synchronization of non-smooth chaotic systems likely due to its complexity leading to the unimpressive effect. Here proposes a simulated annealing-based differential evolution (SADE) algorithm for the optimal parameter selection in the reservoir, and constructs an improved RC model for synchronization, which can work well not only for non-smooth chaotic systems but for smooth ones. Extensive simulations show that the trained RC model with optimal parameters has far longer prediction time than those with empirical and random parameters. More importantly, the well-trained RC system can be well synchronized to its original chaotic system as well as its replicate RC system via one shared signal, whereas the traditional RC system with empirical or random parameters fails for some chaotic systems, particularly for some non-smooth chaotic systems. [ABSTRACT FROM AUTHOR]

Tinnitus is a heterogeneous condition of hearing a rattling sound when there is no auditory stimulus. This rattling sound is associated with abnormal synchronous oscillations in auditory and non-auditory cortical areas. Since tinnitus is a highly heterogeneous condition with no objective detection criteria, it is necessary to search for indicators that can be compared between and within participants for diagnostic purposes. This study introduces heterogeneous though comparable indicators of tinnitus through investigation of spontaneous fluctuations in resting-state brain dynamics. The proposed approach uses nonlinear measures of chaos theory, to detect tinnitus and cross correlation patterns to reflect many of the previously reported neural correlates of tinnitus. These indicators may serve as effective measures of tinnitus risk even at early ages before any symptom is reported. The approach quantifies differences in oscillatory brain dynamics of tinnitus and normal subjects. It demonstrates that the left temporal areas of subjects with tinnitus exhibit larger lyapunov exponent indicating irregularity of brain dynamics in these regions. More complex dynamics is further recognized in tinnitus cases through entropy. We use this evidence to distinguish tinnitus patients from normal participants. Besides, we illustrate that certain anticorrelation patterns appear in these nonlinear measures across temporal and frontal areas in the brain perhaps corresponding to increased/decreased connectivity in certain brain networks and a shift in the balance of excitation and inhibition in tinnitus. Additionally, the main correlations are lost in tinnitus participants compared to control group suggesting involvement of distinct neural mechanisms in generation and persistence of tinnitus. [ABSTRACT FROM AUTHOR]

This paper introduces a new 3D chaotic attractor in a gene regulatory network. The proposed model has eighteen parameters. Formulas for characteristic numbers of critical points for three-dimensional systems were considered. We show that the three equilibrium points of the new chaotic 3D system are unstable and deduce that the three-dimensional system exhibits chaotic behavior. The possible outcomes of this 3D model were compared with the results of the Chua circuit. The bifurcation structures of the proposed 3D system are investigated numerically, showing periodic solutions and chaotic solutions. Lyapunov exponents and Kaplan-Yorke dimension are calculated. For calculations, the Wolfram Mathematica is used. [ABSTRACT FROM AUTHOR]

Starting from the contribution of such thinkers as the famous Giordano Bruno (1583) and the great mathematician and physicist Henri Poincaré (1889) and the surprising discovery of the meteorologist Edward Lorenz (1963), we consider the expansion of the mathematics of chaos in this article, paying attention to topology, qualitative geometry, and Catastrophe Theory, on the one hand, and addressing the possibilities derived from the new Computer Science as Quantum Algorithms and the advances in Artificial Intelligence, on the other. We especially highlight the section on computing chaos, which we consider to be new calculation and analysis instruments, such as machine learning and its algorithm called reservoir computing, through which we can know the dynamics of a chaotic system. With past data, with equations like Karamoto–Sivashinsky, one can improve predictions of the system eight times further ahead than in previous methods. Integrating the machine learning approach and traditional model-based prediction, one could obtain accurate predictions twelve Lyapunov times. As we know, in the framework of chaos theory, it is habitually accepted that the idea of long-term prediction seems impossible because we live under a veil of uncertainty. But with technological advances, the landscape begins to change, both in chaos theory and in its applications, especially in the field of economics, to which we devote particular attention, carrying out as an example the analysis of the evolution of the Madrid Stock Exchange in the 2006–2013 crisis. Above all this, a reflection of a general nature is necessary to enlighten us on the possibility of opening a new horizon. [ABSTRACT FROM AUTHOR]

Why do fractals appear in so many domains of science? What is the physical principle that generates them? While it is true that fractals naturally appear in many physical systems, it has so far been impossible to derive them from first physical principles. However, a proposed interpretation could shed light on the inherent principle behind the creation of fractals. This is the multiscale thermodynamic perspective, which states that an increase in external energy could initiate energy transport mechanisms that facilitate the dissipation or release of excess energy at different scales. Within this framework, it is revealed that power law patterns, and to a lesser extent, fractals, can emerge as a geometric manifestation to dissipate energy in response to external forces. In this context, the exponent of these power law patterns (thermodynamic fractal dimension D ) serves as an indicator of the balance between entropy production at small and large scales. Thus, when a system is more efficient at releasing excess energy at the microscopic (macroscopic) level, D tends to increase (decrease). While this principle, known as Principium luxuriæ, may sound promising for describing both multiscale and complex systems, there is still uncertainty about its true applicability. Thus, this work explores different physical, astrophysical, sociological, and biological systems to attempt to describe and interpret them through the lens of the Principium luxuriæ. The analyzed physical systems correspond to emergent behaviors, chaos theory, and turbulence. To a lesser extent, the cosmic evolution of the universe and geomorphology are examined. Biological systems such as the geometry of human organs, aging, human brain development and cognition, moral evolution, Natural Selection, and biological death are also analyzed. It is found that these systems can be reinterpreted and described through the thermodynamic fractal dimension. Therefore, it is proposed that the physical principle that could be behind the creation of fractals is the Principium luxuriæ, which can be defined as "Systems that interact with each other can trigger responses at multiple scales as a manner to dissipate the excess energy that comes from this interaction". That is why this framework has the potential to uncover new discoveries in various fields. For example, it is suggested that the reduction in D in the universe could generate emergent behavior and the proliferation of complexity in numerous fields or the reinterpretation of Natural Selection. [ABSTRACT FROM AUTHOR]

In text-based authentication, the passwords along with user names are maintained in the Authentication Data Table (ADT). It is necessary to preserve the privacy of passwords in ADT to avoid offline attacks like brute force attacks, lookup table attacks, etc. In this paper, three password protection schemes, namely Encrypted Image Password (EIP), Dynamic Authentication Data Table (D-ADT), and Extended Encrypted Image Password (EEIP) are proposed for secure authentication. In EIP, the input passwords are first converted to hashed passwords and then transformed into images. Next, these image passwords are encrypted using a novel image password encryption system using chaos functions and confusion-diffusion mechanisms. In D-ADT, the hashed passwords are encrypted using a random key. The major highlight of this scheme is that during every log, the hashed password is encrypted with a new random key while keeping the plain password same as it is. So, during each login of the user, the old encrypted password is replaced with a new encrypted password in the authentication data table. The EEIP scheme combines both approaches. Passwords are converted to images and image passwords are encrypted with the new random key at every login. Performance and security analysis are carried out for the proposed algorithm concerning correlation analysis, differential analysis, entropy analysis, computation time, keyspace, and offline attack analysis. [ABSTRACT FROM AUTHOR]

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of standard results in matrices. A number of suggestions for classroom use are given. [ABSTRACT FROM AUTHOR]

Discrete time models, one linear and one non-linear, are investigated, both with a herbivore species that consumes a basal food source species. Results are presented for coexistence of the species and to illustrate chaotic behaviour as parameters are varied in the non-linear model. The results indicate the benefit of fertilization in terms of the region of parameter space for which coexistence occurs. Possible extensions from these models for independent investigations are provided alongside classroom exercises. [ABSTRACT FROM AUTHOR]

Chaotic systems have proven highly beneficial in engineering applications. Pseudo-random numbers produced by chaotic systems have been used for secure communication, notably image encryption. Specific characteristics can increase the chaotic behavior of the system by adding complexity and nonlinearity. The three most well-known characteristics are memristive properties, multistability (coexisting attractors), and hidden attractors. These characteristics strengthen the produced time series' unpredictability and randomness, strengthening an encryption algorithm's resistance to many attacks. This study introduces a unique four-dimensional chaotic system with extreme multistability with respect to three initial conditions (including the memristor initial condition) and all previously known properties. It is rare to find an extreme multistable system like this. This system is coupled with a quadratic flux-controlled memristor based on the well-known Sprott J system. This system has a line of unstable equilibrium points with hidden attractors. The memristor displays the characteristic pinched hysteresis loops, where the area inside a loop and the voltage frequency are inversely related. A comprehensive dynamical analysis thoroughly examines all system characteristics and initial conditions. The numerical findings are carefully verified, and an analog circuit is successfully built and simulated. The chaotic sequences generated by this system are combined with deoxyribonucleic acid (DNA) operations and the global bit scrambling (GBS) technique to create an image encryption algorithm that has strong resistance to a variety of potential attacks, including noise, statistical, exhaustive, differential, and cropping attacks. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive predictive control for controlling and stabilizing fractional order chaotic systems around the equilibrium point is provided. The stability of fractional order chaotic systems around equilibrium points in the presence of parameter uncertainty has been demonstrated using Lyapunov's stability theorem. In addition, the uncertain parameters of fractional order chaotic systems are calculated using appropriate adaptive methods based on the proposed predictive controller structure. Rossler and Chen systems were considered to numerical simulations. The results demonstrated the adaptive predictive control method's usefulness and performance. [ABSTRACT FROM AUTHOR]

Chaos theory studies the irregular, the discontinuous, the non-linear. Shifting away from determinism and scientific explanations rooted in reductionistic models, it brings a revolution by revealing that the universe has an inherent contradictory nature. Chaos raises philosophical questions and confronts us with paradox. How can we live in a universe ruled by physical laws and yet is unpredictable? How can certainty and uncertainty, stability and turbulence, simplicity and complexity co-exist? The insights of chaos are inherent to existentialism. Two radical theories – one scientific, one philosophical – reach similar conclusions by questioning our current understanding and our connection to the familiar. I would like to highlight how they can be an inspiration and a paradigm as models that challenge assumptions and the dominant scientific traditions using creative doubt and humble curiosity, and immerse into their mutual offerings and perceptions. [ABSTRACT FROM AUTHOR]

This article is aimed at studying the cosmological problem, the concepts of "nothingness" and "chaos" as interrelated elements; solving this issue in an ontological system. The leading methods of studying this problem are methods of comparative analysis, descriptive method, basic principles of cognitive methodology, principles of dialectical logic, synergetic analysis, induction, deduction. The article identifies aspects of the development of cosmogenesis as a philosophical problem of modernity; analyses the specific features of nothingness and chaos and their harmony. The materials of the article are of practical and theoretical value for philosophers, theologians, historians, specialists in the field of ontological anthropology. [ABSTRACT FROM AUTHOR]

Purpose: Online question-and-answer (Q&A) communities serve as important channels for knowledge diffusion. The purpose of this study is to investigate the dynamic development process of online knowledge systems and explore the final or progressive state of system development. By measuring the nonlinear characteristics of knowledge systems from the perspective of complexity science, the authors aim to enrich the perspective and method of the research on the dynamics of knowledge systems, and to deeply understand the behavior rules of knowledge systems. Design/methodology/approach: The authors collected data from the programming-related Q&A site Stack Overflow for a ten-year period (2008–2017) and included 48,373 tags in the analyses. The number of tags is taken as the time series, the correlation dimension and the maximum Lyapunov index are used to examine the chaos of the system and the Volterra series multistep forecast method is used to predict the system state. Findings: There are strange attractors in the system, the whole system is complex but bounded and its evolution is bound to approach a relatively stable range. Empirical analyses indicate that chaos exists in the process of knowledge sharing in this social labeling system, and the period of change over time is about one week. Originality/value: This study contributes to revealing the evolutionary cycle of knowledge stock in online knowledge systems and further indicates how this dynamic evolution can help in the setting of platform mechanics and resource inputs. [ABSTRACT FROM AUTHOR]

Today, creating a smart grid that is resistant to cyberattacks is a subject of utmost significance. One of the components of the smart grid that is most susceptible to a cyber-attack is the phasor measuring unit (PMU). The reason is that PMU employs IEEE C37.118.2 communication standards, which specify the structure and sequencing of data packets but offer no security measures. Users must implement the security techniques to ensure the protection of PMU data. Additionally, PMU communicates via a public wide-area network, raising the risk to security. In addition, PMU is a crucial component of the smart grid, enabling different crucial choices for the reliable functioning of the smart grid to be made using its data. This research suggests a chaos-based data encryption solution to close the knowledge gap and reduce the confidentiality assault on PMU data. For this, a brand-new boost converter logarithmic map (BCLM), a one-dimensional (1D) chaotic map, has been presented. The research demonstrates how the suggested chaotic map has better chaotic qualities than conventional chaotic maps. The pseudorandom generator is the chaotic BCLM system. The PMU data are encrypted using the random sequence produced by the BCLM chaotic system. The suggested chaotic map is not computationally demanding, making it simple to implement in a PMU device with limited resources. [ABSTRACT FROM AUTHOR]

The objective of this study was to determine whether multi-microRNA analysis using a combination of four microRNA biomarkers (miR-1246, 202, 21, and 219B) could improve the diagnostic performance of mammography in determining breast cancer risk by age group (under 50 vs. over 50) and distinguish breast cancer from benign breast diseases and other cancers (thyroid, colon, stomach, lung, liver, and cervix cancers). To verify breast cancer classification performance of the four miRNA biomarkers and whether the model providing breast cancer risk score could distinguish between benign breast disease and other cancers, the model was verified using nonlinear support vector machine (SVM) and generalized linear model (GLM) and age and four miRNA qRT-PCR analysis values (dCt) were input to these models. Breast cancer risk scores for each Breast Imaging-Reporting and Data System (BI-RADS) category in multi-microRNA analysis were analyzed to examine the correlation between breast cancer risk scores and mammography categories. We generated two models using two classification algorithms, SVM and GLM, with a combination of four miRNA biomarkers showing high performance and sensitivities of 84.5% and 82.1%, a specificity of 85%, and areas under the curve (AUCs) of 0.967 and 0.965, respectively, which showed consistent performance across all stages of breast cancer and patient ages. The results of this study showed that this multi-microRNA analysis using the four miRNA biomarkers was effective in classifying breast cancer in patients under the age of 50, which is challenging to accurately diagnose. In addition, breast cancer and benign breast diseases can be classified, showing the possibility of helping with diagnosis by mammography. Verification of the performance of the four miRNA biomarkers confirmed that multi-microRNA analysis could be used as a new breast cancer screening aid to improve the accuracy of mammography. However, many factors must be considered for clinical use. Further validation with an appropriate screening population in large clinical trials is required. This trial is registered with (KNUCH 2022-04-036). [ABSTRACT FROM AUTHOR]

We present a new methodology demonstrating that specific small‐magnitude regional perturbations can cause large systematic responses in subseasonal predictions. We show this with ensemble forecasts of the January 2013 Sudden Stratospheric Warming (SSW) from an operational long‐range global prediction system. In forecast members which predict the SSW, tropospheric ridging over the North Atlantic is strengthened 10 days prior to the event. This subsequently enhances planetary wave forcing and weakens the stratospheric polar vortex. Transplanting tropospheric conditions in this region from "correct" to "incorrect" forecasts (and vice versa) significantly alters the SSW forecast probability. The occurrence of this SSW is therefore strongly controlled by the troposphere several days prior. Tipping more members into a sudden warming also significantly affects surface predictions the following month. Despite chaotic behavior of the climate system, small‐magnitude synoptic‐scale perturbations can drive different dynamical states and systematically impact medium and long‐range predictions. Plain Language Summary: Subseasonal forecast ensembles exhibit a large spread in conditions after several days. Some ensemble members go on to produce "correct" forecasts whilst others do not: is this unorganized chaotic noise, or can specific perturbations drive systematic responses? We investigate this with forecasts of the January 2013 Sudden Stratospheric Warming (SSW). Comparing "correct" and "incorrect" SSW forecasts we identify key differences developing over the North Atlantic shortly after initialization. To demonstrate causality, we take atmospheric conditions for this region from a "correct" forecast member and insert them into all other forecast members. This change significantly increases the number of "correct" forecasts of the SSW 10 days later. The reverse experiment, taking conditions from an "incorrect" member, significantly decreases the SSW forecast probability. These results demonstrate how specific atmospheric features can systematically influence predictions at medium and long‐range time scales, potentially providing additional skill at times of uncertainty. Key Points: We identify key atmospheric features over the North Atlantic in "correct" forecasts of the January 2013 Sudden Stratospheric Warming (SSW)Transplanting these conditions between ensemble members can systematically increase or decrease the SSW forecast probabilityThe proposed methodology can demonstrate the causality of specific atmospheric features and their role in medium and long‐range forecasts [ABSTRACT FROM AUTHOR]

This paper proposes a Boolean network model with high dimensional chaos driving and investigates the synchronization of the chaos‐driven Boolean network with a semi‐tensor product. In order to protect the privacy and ensure the security of image transmission, the synchronization results are utilized in the image cryptosystem to achieve compression and encryption. First, the driving chaos system is coupled with multiple local systems and synchronized with the transmitted encrypted signals. Second, the Boolean network is driven and synchronized with derived chaos signals. Finally, images are encrypted and compressed with chaos‐driven Boolean network signals in the transmitter, and then decrypted and recovered with synchronized chaos and Boolean network signals in the recipient. Because of the complexities of high dimensional chaos and Boolean network, the proposed cryptosystem has good security in the secure communication and image process. [ABSTRACT FROM AUTHOR]

Structural change tests aim to identify evidence of a structural break or change in the underlying generating process of a time series. The BDS test has its origins in chaos theory and seeks to test, using the correlation integral, the hypothesis that a time series is generated by an identically and independently distributed (IID) stochastic process over time. The BDS test is already widely used as a powerful tool for testing the hypothesis of white noise in the residuals of time series models. In this paper, we illustrate how the BDS test can be implemented also in a recursive manner to evaluate the hypothesis of structural change in a time series, taking advantage of its ability to test the IID hypothesis. We apply the BDS test repeatedly, starting with a sub-sample of the original time series and incrementally increasing the number of observations until it is applied to the full sample time series. A structural change in the unknown underlying generator model is detected when a change in the trend shown by this recursively computed BDS statistic is detected. The strength of this recursive BDS test lies in the fact that it does not require making any assumptions about the underlying time series generator model. We ilustrate the power and potential of this recursive BDS test through an application to real economic data. In this sense, we apply the test to assess the structural changes caused by the COVID-19 pandemic in international financial markets. Using daily data from the world's top stock indices, we have detected strong and statistically significant evidence of two major structural changes during the period from June 2018 to June 2022. The first occurred in March 2020, coinciding with the onset of economic restrictions in the main Western countries as a result of the pandemic. The second occurred towards the end of August 2020, with the end of the main economic restrictions and the beginning of a new post-pandemic economic scenario. This methodology to test for structural changes in a time series is easy to implement and can detect changes in any system or process behind the time series even when this generating system is not known, and without the need to specify or estimate any a priori generating model. In this sense, the recursive BDS test could be incorporated as an initial preliminary step to any exercise of time series modeling. If a structural change is detected in a time series, rather than estimating a single predictive model for the full-sample time series, efforts should be made to estimate different predictive models, one for the time before and one for the time after the detected structural change. [ABSTRACT FROM AUTHOR]

The new way of thinking science from Newtonian determinism to nonlinear unpredictability and the dawn of advanced computer science and technology can be summarized in the words of the theoretical physicist Michel Baranger, who, in 2000, said in a conference: "Twenty-first-century theoretical physics is coming out of the chaos revolution; it will be about complexity and its principal tool will be the computer.". This can be extended to natural sciences in general. Modelling and predicting ionosphere variables have been considered since many decades as a paramount objective of research by scientists and engineers. The new approach to natural sciences influenced also ionosphere research. Ionosphere as a part of the solar–terrestrial environment is recognized to be a complex chaotic system, and its study under this new way of thinking should become an important area of ionospheric research. After discussing the new context, this paper will try to review recent advances in the exploration of ionosphere parameter time series in terms of chaos theory and the use of machine-learning algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a study of a discrete prey–predator model with additive Allee effect. The model is discretized using the forward Euler method, and the system is analyzed using bifurcation theory and chaos control. The equilibrium point of the discrete system is obtained through equilibrium analysis, and the stability of the equilibrium point is determined by analyzing the parameter conditions. The study also establishes the existence and direction of Neimark–Sacker bifurcation at the positive equilibrium point. The paper proposes two control strategies to manage chaotic behavior and Neimark–Sacker bifurcation: exponential control and hybrid feedback control. These control methods are demonstrated to be effective in controlling the chaotic behavior and bifurcation of the system through numerical simulation. Overall, the results of the study provide important insights into the dynamics of the discrete prey–predator model with additive Allee effect, as well as effective methods for controlling chaos and bifurcation in the system. [ABSTRACT FROM AUTHOR]

This article explores the gendered afterlife of slave rebellions, building on historical and sociological scholarship that takes death and haunting as central analytics. It focuses on a moment of state repression in Cuba in 1844, in the aftermath of two substantive slave insurgencies, and the movement of enslaved and free people of color known as La Escalera. While a voluminous colonial record has documented the spectacle of public punishment in 1844, I trace an invisible, largely unarchived form of punishment that haunted Black people who were forced to witness acts of torture and execution, stalking their psychic worlds. Alongside the masculinization of this public punishment, this article explores a phenomenon that I call encumberment, which encompasses the absorption of state terror on the most intimate of levels; the act of contending and coping with torture's impact on wider communities; and the labor of rebuilding and caring for traumatized loved ones -- all of which were deeply gendered processes. Yet if haunting was a central feature of moments like 1844, these moments were also characterized by a counter-haunting, illustrating how the aftermath of slave insurgencies also stalked and plagued the colonial state, and disrupted the fantasy of the rebellion's undoing. Encompassing the memory of collective Black rage, the colonial pursuit of public tranquility, the care for traumatized Black bodies and the attention to the dead, counter-haunting became central to the afterlife of 1844. The deep entanglement of haunting and counterlife was a critical legacy of this and other slave rebellions. [ABSTRACT FROM AUTHOR]

Chaos is an important dynamic feature, which generally occurs in deterministic and stochastic nonlinear systems and is an inherent characteristic that is ubiquitous. Many difficulties have been solved and new research perspectives have been provided in many fields. The control of chaos is another problem that has been studied. In recent years, a recurrent neural network has emerged, which is widely used to solve many problems in nonlinear dynamics and has fast and accurate computational speed. In this paper, we employ reservoir computing to control chaos in dynamic systems. The results show that the reservoir calculation algorithm with a control term can control the chaotic phenomenon in a dynamic system. Meanwhile, the method is applicable to dynamic systems with random noise. In addition, we investigate the problem of different values for neurons and leakage rates in the algorithm. The findings indicate that the performance of machine learning techniques can be improved by appropriately constructing neural networks. [ABSTRACT FROM AUTHOR]

In this study, collaboration has been revisited between an American faculty member and an international graduate teaching assistant during a unique faculty‐student mentoring model within principles of chaos theory. The role of culture, beliefs about language learning and teaching had been studied closely to improve faculty/international students' cooperation in academic settings. The research focused on the nature of the collaborative experience of a faculty member and a graduate teaching assistant in designing and implementing an authentic language learning experience into a graduate level linguistic course and the role of dynamism and chaos theory evident in this mentoring model. Participants engaged in collaborative planning of a language learning experience embedded into a graduate level linguistic course. Reflective journals of participants and assignments were analyzed to better understand the experiences of the collaborators and the ways their culture, beliefs, and mentoring model impacted andragogical choices for instruction and learning of participants and students. Through self‐study, the researchers explored their professional identity and their academic flexibility within the organizational boundaries. The results of the study unveiled the complexity and chaos of each phase of cooperation and how the participants applied flexibility to achieve a more linear teaching/learning process. The collaboration between American faculty and international GTA under the shadow of chaos twisted by culture, beliefs, and hierarchy could still generate novice andragogical practices in higher education to contribute implications for collaborative work across cultures and hierarchies. [ABSTRACT FROM AUTHOR]

This article aims to describe the qualitative behavior of a discrete‐time prey–predator model including prey refuge. It is assumed that the fraction of prey in refuge is a monotonically increasing but self‐limiting function of prey population. The fixed points of the system and their local stability are investigated. The criteria for Neimark–Sacker bifurcation and period‐doubling bifurcation are presented. The sufficient conditions for global asymptotic stability of the interior fixed point are derived. The chaotic behavior of the system is stabilized by applying a hybrid control strategy. We conclude with a numerical simulation that strengthens our theoretical discussion and provides a way to observe the complex behavior of the system. [ABSTRACT FROM AUTHOR]

The dynamics of food consumption by higher trophic level is complicated as its availability, choice, and consequently their development. The top predator interference has a strong influence on the dynamic structure of the tritrophic food chain system. The proposed article deals with the characteristic features of a three‐tire modified Leslie–Gower model with a one‐prey two‐predator system with the assumption of multiple gestation delays. The intermediate predator of the food chain is taken as a specialist type, whereas the growth of top predator is assumed by sexual reproduction. Two different response functions named Monod–Haldane (M‐H) and Beddington–DeAngelis (B‐D) are considered for the modeling of system dynamics. The population densities of intermediate and top predators are assumed to be affected by two different gestation delays. The boundedness and positive equilibria and their stability conditions are examined theoretically for the delay‐free system. The local and global stability conditions of the delayed system are also determined. With the help of normal form theory and central manifold arguments, the properties of bifurcating periodic solutions are carried out. The numerical computation is performed to validate our analytical findings that show the different dynamical outcomes such as periodic and chaotic dynamics. The Hopf‐bifurcation scenario for different parameters of the model system is well studied. Further, the stability behavior of multiple delays for the different cases is investigated. The system shows chaotic behavior when the gestation delay of intermediate predators is large enough. It is also observed that the top predator density is highly fluctuating through the creation of chaos by the inclusion of time delay. Finally, the time delay destabilizing effect is noticed for three‐way interactions among prey, intermediate predator, and top predator, highlighting its significance in chaotic dynamics. [ABSTRACT FROM AUTHOR]

From Cournot, Walras, and Pareto's research to what followed in the form of marginalist economics, chaos theory, agent-based modeling, game theory, and econophysics, the interpretation and analysis of economic systems have been carried out using a broad range of higher mathematics methods. The evolution of mathematical economics is associated with the most productive and influential authors, sources, and countries, as well as the identification of interactions between the authors and research topics. Bibliometric analysis provides journal-, author-, document-, and country-level metrics. In the present study, a bibliometric overview of mathematical economics came from a screening performed in September 2023, covering the timespan 1898–2023. About 6477 documents on mathematical economics were retrieved and extracted from the Scopus academic database for analysis. The Bibliometrix package in the statistical programming language R was employed to perform a bibliometric analysis of scientific literature and citation data indexed in the Scopus database. VOSviewer (version 1.6.19) was used for the visualization of similarities using several bibliometric techniques, including bibliographic coupling, co-citation, and co-occurrence of keywords. The analysis traced the most influential papers, keywords, countries, and journals among high-quality studies in mathematical economics. [ABSTRACT FROM AUTHOR]

Service systems for children and families have been shaped by standard approaches to knowledge‐building, which reflect a reductionist approach and assume linearity and/or that individuals and experiences are normally distributed. Yet, these approaches may be inadequate for clients most at‐risk, especially those who would be analytic 'outliers'. A complexity lens focuses on the whole system and seeks to identify patterns, including the dynamic interactions between components of the system. Social work scholars have begun to apply complexity theory to social work research efforts, demonstrating the conceptual potential of incorporating this theoretical approach into social work theories and models such as the person‐in‐environment framework and the ecosystems perspective. Yet, frameworks informed by complexity theory may require ontological and epistemological shifts in thinking and new methodological approaches in order to fully embody a complexity approach. Complexity theory offers the opportunity to consider social work clients who are most at‐risk, as it is better suited for power law distributions. We can, therefore, reconceptualize the most 'at‐risk' clients as being in a state of transition, which is also the space of most creativity and possibility. [ABSTRACT FROM AUTHOR]

Purpose: This study aims to develop the first Theory of Technological Response and Progress in Chaos (TRPC) and examine the case of technological development during the COVID-19 pandemic. The research objectives of this study were to: identify the key technologies that act as a response mechanism during the chaos event, specifically in the case of COVID-19; examine how technologies evolve, develop and diffuse in an immediate crisis and a chaotic environment; theorise various types and periods of technological response and progress during the emergence of chaos and the stages that unfold; and develop policy-oriented recommendations and establish technological foundations to address subsequent chaos events. Design/methodology/approach: This study used the grounded theory as a methodology with a mixed-method approach that included quantitative and qualitative methods. The authors used the quantitative method to assist with the qualitative step to build the TRPC theory. Accordingly, this study integrated machine learning and text mining approaches to the qualitative data analysis following the steps of the grounded theory approach. Findings: As a result of the TRPC theory development process, the authors identified three types of technologies (survival, essential and enhancement technologies) and five types of periods (stable, initial, survival-dominant, essential-dominant and enhancement-dominant periods) that are specific to chaos-technology interactions. The policy implications of this study demonstrate that a required technological base and know-how must be established before a chaotic event emerges. Research limitations/implications: Concerning the limitations of this study, social media data has advantages over other data sources, such as the examination of dynamic areas and analyses of immediate responses to chaos. However, other researchers can examine publications and patent sources to augment the findings concerning scientific approaches and new inventions in relation to COVID-19 and other chaos-specific developments. The authors developed the TRPC theory by studying the COVID-19 pandemic, however, other researchers can utilise it to study other chaos-related conditions, such as chaotic events that are caused by natural disasters. Other scholars can investigate the technological response and progress pattern in other rapidly emerging chaotic events of an uncertain and complex nature to augment these findings. Practical implications: Following the indications of the OECD (2021a) and considering the study conducted by the European Parliamentary Research Service (Kritikos, 2020), the authors identified the key technologies that are significant for chaos and COVID-19 response using machine learning and text intelligence approach. Accordingly, the authors mapped all technological developments using clustering approaches, and examined the technological progress within the immediate chaos period using social media data. Social implications: The key policy implication of this study concerns the need for policymakers to develop policies that will help to establish the required technological base and know-how before chaos emerges. As a result, a rapid response can be implemented to mitigate the chaos and transform it into a competitive advantage. The authors also revealed that this recommendation overlaps with the model of dynamic capabilities in the literature (Teece and Pisano, 2003). Furthermore, this study recommends that nations and organisations establish a technological base that specifically includes technologies that bear 3A characteristics. These are the most crucial technologies for the survival- and essential-dominant stages. Moreover, the results of this study demonstrate that chaos accelerates technological progress through the rapid adoption and diffusion of technologies into different fields. Hence, nations and organisations should regard this rapid progress as an opportunity and establish the prior knowledge base and technologies before chaos emerges. Originality/value: The authors have contributed to the chaos studies and the relationship between chaos and technological development by establishing the first theoretical foundation using the grounded theory approach, hereafter referred to as the TRPC theory. As part of the TRPC theory, the authors present three periods of technological response in the following sequence: survival technology, essential technology and enhancement technology. Moreover, this study illustrates the evolving technological importance and priorities as the periods of technological progress proceed under rapidly developing chaos. [ABSTRACT FROM AUTHOR]

Principal Component Analysis (PCA) is a widely used technique for fault detection and diagnosis. PCA works well when the data set has linear characteristics. However, most industrial processes have nonlinear characteristics in their data. Kernel PCA (KPCA) is an alternative solution for such types of data sets. This solution doesn't come without a cost since one of KPCA's disadvantages is a large number of observations which results in more occupied storage space and more execution time than the PCA technique. Furthermore, if the data is too large it may minimize the monitoring performance of the KPCA model. Reduced KPCA (RKPCA) is a solution for the conventional KPCA limitations. Firstly, RKPCA can deal with nonlinear characteristics without crucial problems because it is based on the KPCA algorithm with a data reduction part where it keeps most of the data's information. Thus, by reducing the number of observations RKPCA reduces the occupied storage space and execution time while preserving tolerable monitoring performance. The proposed RKPCA algorithm consists of two parts. First, the large-sized training data set is reduced using the fractal dimension technique (correlation dimension). Afterward, the KPCA model is developed through the obtained reduced training data set. The proposed scheme is applied to the Tennessee Eastman Process and the Cement Plant Rotary Kiln data sets to evaluate its performance in comparison with other algorithms. [ABSTRACT FROM AUTHOR]

The complexity of hydrological data requires an understanding of its spatiotemporal evolution for effective modeling and prediction. In this study, the underlying spatiotemporal, chaotic, and nonlinear dynamics of rainfall and streamflow data at different timescales in Ceará were assessed by applying chaos theory concepts. This assessment included phase space reconstruction (PSR), correlation dimension, the largest Lyapunov exponent (LLE), nonlinear methods such as recurrence plots, and recurrence quantification analysis techniques. The results indicated that as the timescale increased, the required dimension for PSR decreased for most stations, indicating a shift in the dynamics of the variables. The presence of chaos was confirmed in 78% of the rainfall stations at a monthly timescale by the correlation dimension and positive values of LLE. For streamflow, 73% of monthly data showed indications of chaos using both methods. Values of LLE ranged from 0.02 to 0.24 and − 0.32 to 3.4 for monthly data of rainfall and streamflow, respectively. Most methods showed that the northwestern area of Ceará exhibited higher complexity. Furthermore, the extent of data complexity varied temporally, with the monthly timescale being more complex than the annual timescale. The study revealed that long-term predictions for streamflow may be ineffective for water resources in the region. The results suggested a potential for rainfall predictions up to six years in advance. These findings have implications for developing an integrated water management plan in the region and provide insight into how hydrological variables evolve over time and space. [ABSTRACT FROM AUTHOR]

The influence of orientation on the control of chaotic behaviors of the system in a porous medium is studied in this article based on the theory of dynamical systems. A three-dimensional system model is obtained using the spectral method. The results showed that chaos can be suppressed in the case of a low or moderate Prandtl number by choosing appropriate tilt angles. [ABSTRACT FROM AUTHOR]

This paper discusses the complex dynamic behavior of a novel chaotic system, which was firstly established by introducing a memristor into a similar Chen's system. Then by choosing a as the key parameter, we analyze the stability of memristor system based on eigenvalue theory. It is also found that when a cross some critical values, the system can exhibit Neimark–Sacker bifurcation and chaos behaviors. Some numerical simulations including phase diagrams and maximum Lyapunov exponent graph of the memristor-based systems are presented to verify the existence of chaos attractors. Finally, to make the results of this paper useful in the actual situation, such as the design of chaos security algorithm, analog electronic circuit of memristor chaotic system is designed. [ABSTRACT FROM AUTHOR]

This paper presents the temporal and spatiotemporal dynamics of a delayed prey–predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter β , which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter β and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species. [ABSTRACT FROM AUTHOR]

It has been reported that the quick progression and abnormal division of cells give rise to tumors. New cells are produced to fulfill new duties and replace older cells in the body of the host. It is obvious that tumors are deadly and affect practically every aspect of human life. Businesses, families, patients, and society as a whole lose financial resources and opportunities as a result of cancer and its treatment. Therefore, it is crucial to look at how tumors behave dynamically as a result of therapy and to highlight for policymakers and health officials the key elements of the system. In this research work, we formulate the transmission phenomena of tumor growth with the effect of virotherapy treatment in the fractional framework. Besides that, we focussed in our work on qualitative analysis and dynamic behavior of tumor dynamics. Using Banach's and Schaefer's theorems and the fixed-point theorem, the uniqueness and existence of the solution to the suggested model of tumor are examined. The necessary circumstances are established for our tumor system's Ulam-Hyers stability. Laplace Adomian decomposition approach is used to examine the solution pathways and show the contribution of input factors on the dynamics of tumor growth. More specifically, we focused on the time series and chaotic behavior of the suggested system with change in input factors. The system's most crucial components are recommended for regulating the dynamics of the tumor system. [ABSTRACT FROM AUTHOR]

Entropy serves as a measure of chaos in systems by representing the average rate of information loss about a phase point's position on the attractor. When dealing with a multifractal system, a single exponent cannot fully describe its dynamics, necessitating a continuous spectrum of exponents, known as the singularity spectrum. From an investor's point of view, a rise in entropy is a signal of abnormal and possibly negative returns. This means he has to expect the unexpected and prepare for it. To explore this, we analyse the New York Stock Exchange (NYSE) U.S. Index as well as its constituents. Through this examination, we assess their multifractal characteristics and identify market conditions (bearish/bullish markets) using entropy, an effective method for recognizing fluctuating fractal markets. Our findings challenge conventional beliefs by demonstrating that price declines lead to increased entropy, contrary to some studies in the literature that suggest that reduced entropy in market crises implies more determinism. Instead, we propose that bear markets are likely to exhibit higher entropy, indicating a greater chance of unexpected extreme events. Moreover, our study reveals a power-law behaviour and indicates the absence of variance. [ABSTRACT FROM AUTHOR]

Protecting digital data, especially digital images, from unauthorized access and malicious activities is crucial in today's digital era. This paper introduces a novel approach to enhance image encryption by combining the strengths of the RSA algorithm, homomorphic encryption, and chaotic maps, specifically the sine and logistic map, alongside the self-similar properties of the fractal Sierpinski triangle. The proposed fractal-based hybrid cryptosystem leverages Paillier encryption for maintaining security and privacy, while the chaotic maps introduce randomness, periodicity, and robustness. Simultaneously, the fractal Sierpinski triangle generates intricate shapes at different scales, resulting in a substantially expanded key space and heightened sensitivity through randomly selected initial points. The secret keys derived from the chaotic maps and Sierpinski triangle are employed for image encryption. The proposed scheme offers simplicity, efficiency, and robust security, effectively safeguarding against statistical, differential, and brute-force attacks. Through comprehensive experimental evaluations, we demonstrate the superior performance of the proposed scheme compared to existing methods in terms of both security and efficiency. This paper makes a significant contribution to the field of digital image encryption, paving the way for further exploration and optimization in the future. [ABSTRACT FROM AUTHOR]

Meta-heuristics have been successfully applied to many complex optimization problems. One of the main reasons for its success is its ability to handle non-convex, nonlinear, multimodal, multi-variable, and multi-objective problems with easy implementation. However, the quality of the response of these algorithms to an optimization problem is highly susceptible to the control parameters, and few works aim to tune them or find tools that can improve the algorithms. The literature is rich in proposals for new algorithms, but not for improving existing ones. This paper presents different strategies for tuning and accelerating meta-heuristics using the first hybrid algorithm in the literature. The Lichtenberg algorithm is inspired by lightning and Lichtenberg figures and has been increasingly successfully applied to various optimization problems. However, a study of its best parameters has never been presented until now. After a discussion of the best tuning tools, its tuning parameters are performed using response surface methodology. Then, 14 versions are studied through 10 test functions using chaos theory and Lévy flights scenarios. After 13,500 simulations, the chaotic Lichtenberg algorithm equipped with the piecewise function and tuned parameters proved the best version with only 16% similarity to the original algorithm. Then, it was compared to the genetic algorithm, particle swarm optimization, gray wolf optimizer, salp swarm optimization, whale optimization algorithm, and dragonfly algorithm. The proposed algorithm had both the best average accuracy, lower computational cost, and the smallest standard deviation. [ABSTRACT FROM AUTHOR]

The dynamics of a discrete Holling–Tanner model with Beddington–DeAngelis functional response is studied. The permanence and local stability of fixed points for the model are derived. The center manifold theorem and bifurcation theory are used to show that the model can undergo flip and Hopf bifurcations. Codimension-two bifurcation associated with 1:2 resonance is analyzed by applying the bifurcation theory. Numerical simulations are performed not only to verify the correctness of theoretical analysis but to explore complex dynamical behaviors such as period-6, 7, 10, 12 orbits, a cascade of period-doubling, quasi-periodic orbits, and the chaotic sets. The maximum Lyapunov exponents validate the chaotic dynamical behaviors of the system. The feedback control method is considered to stabilize the chaotic orbits. These complex dynamical behaviors imply that the coexistence of predator and prey may produce very complex patterns. [ABSTRACT FROM AUTHOR]

Dynamical chaos is a fundamental manifestation of gravity in astrophysical, many-body systems. The spectrum of Lyapunov exponents quantifies the associated exponential response to small perturbations. Analytical derivations of these exponents are critical for understanding the stability and predictability of observed systems. This paper presents a new model for chaos in systems with eccentric and crossing orbits. Here, exponential divergence is not a continuous process but rather the cumulative effect of an ever-increasing linear response driven by discrete events at regular intervals, i.e. punctuated chaos. We show that long-lived systems with punctuated chaos can magnify Planck length perturbations to astronomical scales within their lifetime, rendering them fundamentally indeterministic. [ABSTRACT FROM AUTHOR]

This article introduces a novel model of asymmetric neural networks combined with fractional difference memristors, which has both theoretical and practical implications in the rapidly evolving field of computational intelligence. The proposed model includes two types of fractional difference memristor elements: one with hyperbolic tangent memductance and the other with periodic memductance and memristor state described by sine functions. The authenticity of the constructed memristor is confirmed through fingerprint verification. The research extensively investigates the dynamics of a coupled neural network model, analyzing its stability at equilibrium states, studying bifurcation diagrams, and calculating the largest Lyapunov exponents. The results suggest that when incorporating sine memristors, the model demonstrates coexisting state variables depending on the initial conditions, revealing the emergence of multi-layer attractors. The article further demonstrates how the memristor state shifts through numerical simulations with varying memductance values. Notably, the study emphasizes the crucial role of memductance (synaptic weight) in determining the complex dynamical characteristics of neural network systems. To support the analytical results and demonstrate the chaotic response of state variables, the article includes appropriate numerical simulations. These simulations effectively validate the presented findings and provide concrete evidence of the system's chaotic behavior. [ABSTRACT FROM AUTHOR]

With the development of sharing economy, a joint distribution mode has been increasingly adopted as the preferred cooperation mode of third-party logistics enterprises to achieve the efficient, resource-saving, and profit-optimal business goals of enterprises. In the joint distribution mode, the distribution price is one of key factors that influences the operation of the joint distribution. Thus, to acquire the optimal pricing for the logistics enterprises, we establish a pricing model based on dynamic game theory for a joint distribution system including one joint distribution company and two express enterprises. In the proposed model, two dimensions of games exist simultaneously, including the game between express competitors and the game between express and distribution enterprises. The multidimensional game leads to more complex system characteristics. Through the stability analysis, we find the Nash equilibrium point and its stability conditions. Numerical simulations are conducted to investigate the complex dynamical behaviors of the game model, such as the system stability region, the bifurcation diagram, the largest Lyapunov exponent, strange attractors, etc. The simulation results indicate that different price adjustment speeds and ranges have a significant impact on the system stability and the profits of all participants in the game. The parameter adjustment control can well dominate the chaotic behaviors of the system. Enterprises should make pricing decisions based on their market positions to promote the continuous and stable development of the operation mode of the multi-agent joint sharing distribution center. [ABSTRACT FROM AUTHOR]

Adaptive network models arise when describing processes in a wide range of fields and are characterized by some specific effects. One of them is mixed dynamics, which is the third type of chaos in addition to the conservative and dissipative types. In this work, we consider a more complex type of connections between network elements—simplex, or higher-order adaptive interactions. Using numerical simulation methods, we analyze various characteristics of mixed dynamics and compare them with the case of pairwise couplings. We found that mixed dynamics in the case of simplex interactions is characterized by a very high similarity of a chaotic attractor to a chaotic repeller, as well as a stronger closeness of the sum of the Lyapunov exponents of the attractor and repeller to zero. This means that in the case of three elements, the conservative properties of the system are more pronounced than in the case of two. [ABSTRACT FROM AUTHOR]

The generalized complex Ginzburg–Landau equation is considered. An analytical condition for the existence of horseshoe chaos is obtained for the traveling wave reduction of the investigated equation by using the Melnikov method. A way to control chaos in the dynamical system is proposed. An analytical prediction is tested numerically by plotting attraction basins of attraction of the Poincare section of the studied system. [ABSTRACT FROM AUTHOR]

This study focuses on analyzing a coupled space-time fractional nonlinear Schrodinger equation, which has applications in describing non-relativistic quantum mechanical behavior. The investigation covers various aspects, including the examination of dynamical behaviors and the exploration of optical soliton solutions. The modified F-expansion method is employed to derive these soliton solutions. To visualize and interpret the physical characteristics of the solutions, they are plotted in 2D, 3D, and density plots with appropriate parameter settings. The dynamical behaviors of the equation are discussed by investigating bifurcations at equilibrium points, and the chaotic behavior of the perturbed dynamical system is demonstrated using chaos theory. Phase portraits illustrating bifurcation and chaotic patterns are generated using the RK4 algorithm in Matlab. These findings provide a dynamic and powerful mathematical tool to address a range of nonlinear wave phenomena. Key discoveries include the identification of new solitary wave forms, as well as bifurcation and chaotic solutions. These unique and intriguing solutions have theoretical significance in understanding energy transfer and diffusion processes in mathematical models. [ABSTRACT FROM AUTHOR]

The fact that very small changes cause unpredictable big changes in nature has led to the chaos theory. Many researchers focusing on chaos have improved this theory in several directions. One of the prominent areas in chaos theory is controlled chaos. Due to the usefulness of controlled chaos, it has been used in many working areas as the iteration methods of fixed point theory. In this study, we have developed a chaos controlling mechanism using the Ishikawa iteration method. Firstly, Ishikawa fixed point type controlling mechanism is defined and the theorems determining the control parameters are proved. Afterward, the results of Ishikawa fixed point type controlling mechanism were obtained by taking the logistic map as an example. In addition, by using the MATLAB software program, the convergence, periodical behavior and chaotic regions occurring outside the range of the parameters forming the controlling mechanism have been investigated and numerical simulations studied. Subsequently, the effectiveness of the stability intervals obtained for the fixed point and periodic fixed points of the Ishikawa fixed point controlling mechanism by applying the Lyapunov exponent is demonstrated. Finally, we have given an application as control chaos of Ishikawa traffic model. [ABSTRACT FROM AUTHOR]

The emergent global economy depends on financial growth. Social and economic development is achieved by a stable, growing, and secure financial chain system. Financial instability poses huge financial inflation and economic decline. The financial crisis and the turbulence of the finance market also result in deterministic instability. This fluctuation during the financial operation may also critically affect the development of the economic system and other sociological and financial stabilities. For the chaotic behavior of the finance system, the mathematical formulation is developed along with controlling terms in this paper. Fitting the controlling parameters, the financial model can be made secure and safe from periodic behavior and will run in chaotic conditions. At the first attempt, the dynamical system and the controlling parameters are adjusted through assigned values and their ranges. Second, the impact of these parameters is studied along with the feasible techniques by graphical representations. The said financial dynamics are investigated in the sense of fractal–fractional derivatives. The concept of Ulam–Hyers stability is also developed for the considered model. [ABSTRACT FROM AUTHOR]

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents' properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors' capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems' exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains. [ABSTRACT FROM AUTHOR]

In this paper, spikes adding to infinity on bifurcation trees of period-1 orbits to chaos in the Rössler system is studied. The spikes adding on the periodic orbits is completed through a saddle-node bifurcation. With onset of a period-1 orbit, there is 1-spike on such a period-1 orbit, followed by the development from 1-spike to 2 l -spikes and the period-1 to period- 2 l orbits have 1-spike to 2 l -spikes. For a spike bifurcation of a period-1 orbit with m -spikes (m = 1 , 2 , ... , ∞), a new spike is added on such a period-1 orbit. Thus, the period-1 orbit has (m + 1) -spikes. Such a period-1 to period- 2 l orbits (l = 1 , 2 , ... , ∞) have (m + 1) -spikes to 2 l (m + 1) -spikes. The three bifurcation trees of period-1 orbits with m -spikes (m = 1 , 2 , 3) to period-4 orbits with 4 m -spikes are presented numerically. The phase trajectories and responses of z -component for period-1 to period-4 orbits with different spikes are given for illustrations of spikes adding on periodic orbits. The spikes adding generating the complexity of period-1 orbits to chaos can be developed. [ABSTRACT FROM AUTHOR]