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The stochastic dynamical ρ4 equation is utilized as a robust framework for modeling the behavior of complex systems characterized by randomness and nonlinearity, with applications spanning various scientific fields. The aim of this paper is to employ an analytical method to identify stochastic traveling wave solutions of the dynamical ρ4 equation. Novel hyperbolic and rational functions are investigated through this method. A Galilean transformation is applied to reformulate the model into a planar dynamical system, which enables a comprehensive qualitative analysis. Additionally, the emergence of chaotic and quasi-periodic patterns following the introduction of a perturbation term is addressed. Simulation results indicate that significant changes in the systems’ dynamic behavior are caused by adjusting the amplitude and frequency parameters. Our findings indicate the impact of the method on system dynamics and its efficacy in analyzing solitons and phase behavior in nonlinear models. These discoveries provide fresh perspectives on how the suggested method can lead to notable shifts in the systems’ dynamic behavior. The effectiveness and practicality of the proposed methodology in scrutinizing soliton solutions and phase visualizations across diverse nonlinear models are underscored by these revelations. [ABSTRACT FROM AUTHOR]

Discretization schemes such as Euler method and Runge–Kutta techniques are extensively used to find approximate solutions of Continuous-Time (CT) dynamical system. While the approximation is good for small discretization step sizes, as pointed out by Lorenz, when the step size increases, computational chaos and computational instability are frequently observed, the former phenomenon being a precursor to the latter. By computational instability, it is meant that there is a blow up of trajectories for the Discrete-Time (DT) system. Computational chaos instead means that for certain step sizes, the DT system displays chaos while the CT counterpart is not chaotic. This paper studies the dynamics of a class of second-order maps obtained via the discretization of a Memristor Murali–Lakshmanan–Chua Circuit (MMLCC). The discretization, which is based on the DT Flux–Charge Analysis Method (FCAM), guarantees that the first integrals of a CT-MMLCC are preserved exactly for the DT system. Hence the dynamics of DT-MMLCC evolves on invariant manifolds and it is characterized by the coexistence of infinitely many different attractors (extreme multistability). The paper uses analytic techniques introduced by Marotto, based on the concept of snapback repellers and transverse homoclinic orbits, to study the chaotic behaviors of the maps. Regions of the parameter space are singled out where there exist snapback repellers for DT-MMLCC, thus implying that the maps display chaos in the Marotto and in the Li–Yorke sense. Since the corresponding CT-MMLCC does not display chaos, the observed chaos of DT-MMLCC is genuinely a consequence of the discretization scheme used in the paper, i.e. it can be actually considered as computational chaos. It is also verified that computational chaos is a precursor to computational instability of the DT-MMLCC maps. Finally, the paper analyzes the effect on chaos obtained by changing the invariant manifold where the dynamics evolves. [ABSTRACT FROM AUTHOR]

In this paper, we present a new class of extended oscillators in light of chaos theory. It is based on dynamical complex systems built on the concept of self-describing with a stopping criterion process. We offer an effective studying approach with a specific focus on learning, provoking students' thinking through the triad of enigmatics–creativity–acmeology. Dynamic processes are the basis of mathematical modeling; thus, we can reach the goal of the above-mentioned triad by the proposed differential systems. The results we derive strongly confirm the presence of symmetry in the outcomes of the proposed models. We suggest a stochastic approach to structuring the proposed dynamical systems by modeling the coefficients that drive them by some discrete probability distribution that exhibits symmetry or asymmetry. We propose specific tools for researching the behavior of these systems. [ABSTRACT FROM AUTHOR]

This paper is dedicated to exterminate the ( 2 + 1 )-dimensional integro-differential Jaulent–Miodek equation, a prominent model linked to energy-dependent Schrödinger potential. This equation is employed in a wide array of disciplines, including fluid dynamics, condensed matter physics, optics, and various engineering systems. First, we are given to derive exact wave solutions for the ( 2 + 1 )-dimensional integro-differential Jaulent–Miodek equation using an innovative approach known as the new modified unified auxiliary equation method. We offer a comprehensive visual representation and some exact solutions propagate of these solutions in 2D and 3D plots, using various parametric values for a comprehensive analysis. In addition, we employed the planar dynamical system method to study phase portraits and chaotic behavior of the governing equation. Through the analysis of phase portraits, the sensitivity of dynamic system is examined. The chaotic behavior of dynamic system is examined by time series, Poincaré section, and 2D, 3D phase portraits. [ABSTRACT FROM AUTHOR]

Dynamic systems theory and complexity theory (DST/CT) is a framework explaining how complex systems change and adapt over time. In psychotherapy, DST/CT can be used to understand how a person's mental and emotional state changes during therapy incorporating higher levels of complexity. This study aimed to systematically review the variability of DST/CT methods applied in psychotherapy research. A primary studies search was conducted in the EBSCO and Web of Knowledge databases, extracting information about the analyzed DST/CT phenomena, employed mathematical methods to investigate these phenomena, descriptions of specified dynamic models, psychotherapy phenomena, and other information regarding studies with empirical data (e.g., measurement granularity). After screening 38,216 abstracts and 4,194 full texts, N = 41 studies published from 1990 to 2021 were identified. The employed methods typically included measures of dynamic complexity or chaoticity. Computational and simulation studies most often employed first-order ordinary differential equations and typically focused on describing the time evolution of client-therapist dyadic influences. Eligible studies with empirical data were usually based on case studies and focused on data with high time intensity of within-session dynamics. This review provides a descriptive synthesis of the current state of the proliferation of DST/CT methods in the psychotherapy research field. [ABSTRACT FROM AUTHOR]

The dynamical behavior of the double-chain deoxyribonucleic acid (DNA) system holds significant implications for advancing the understanding of DNA transmission laws in the realms of biology and medicine. This study delves into the investigation of chaos patterns and solitary wave solutions for the (2+1) Beta-fractional double-chain DNA system, employing the theory of planar dynamical systems and the method of complete discrimination system for polynomials (CDSP). The results demonstrate a diverse spectrum of solitary wave solutions, sensitivity to perturbations, and manifestations of chaotic behavior within the system. Through the utilization of the complete discrimination system for polynomials, a multitude of novel solitary wave solutions, encompassing periodic, solitary wave, and Jacobian elliptic function solutions, were systematically constructed. The influence of Beta derivatives on the solutions was elucidated through parameter comparison analysis, emphasizing the innovative nature of this study. These findings underscore the potential of this system in unraveling various biologically significant DNA transmission mechanisms. [ABSTRACT FROM AUTHOR]

Many stochastic continuous-state dynamical systems can be modeled as probabilistic programs with nonlinear non-polynomial updates in non-nested loops. We present two methods, one approximate and one exact, to automatically compute, without sampling, moment-based invariants for such probabilistic programs as closed-form solutions parameterized by the loop iteration. The exact method applies to probabilistic programs with trigonometric and exponential updates and is embedded in the Polar tool. The approximate method for moment computation applies to any nonlinear random function as it exploits the theory of polynomial chaos expansion to approximate non-polynomial updates as the sum of orthogonal polynomials. This translates the dynamical system to a non-nested loop with polynomial updates, and thus renders it conformable with the Polar tool that computes the moments of any order of the state variables. We evaluate our methods on an extensive number of examples ranging from modeling monetary policy to several physical motion systems in uncertain environments. The experimental results demonstrate the advantages of our approach with respect to the current state-of-the-art. [ABSTRACT FROM AUTHOR]

Purpose. To provide a procedure for identifying chaotic processes in a dynamic system and to examine time series, describing the chemical composition of cast iron at the blast furnace output with the purpose of identifying the nonlinearity of the investigated system and detecting the presence of chaotic processes in it. Methodology. The determination of the unique characteristics of the attractor of a dynamic chaotic system based on the time series of cast iron’s chemical composition values was carried out using methods of nonlinear dynamics and dynamic chaos theory, such as the autocorrelation function method, correlation and fractal dimensions. Findings. The methods of nonlinear dynamics and dynamic chaos theory were used to study the behavior of time series data on the chemical composition of cast iron at the blast furnace output. The presence was identified of chaotic processes with a fractal structure in the studied dynamic system, leading to the inefficiency of traditional analysis methods based on the Gaussian properties of stochastic processes. Originality. For the first time, the possibility and feasibility of applying chaos theory methods for the analysis and prediction of time series data on the chemical composition of cast iron at the blast furnace output were substantiated. For the first time, the nonlinearity of the studied dynamic system was identified, and chaotic processes were discovered within it by determining the unique characteristics of the strange attractor of the system using the analyzed time series, such as embedding dimension, time delay, and the largest Lyapunov exponent. Practical value. The obtained results open up the possibility for more effective and qualitative analysis of the behavior of the studied dynamic system by developing new tools for assessment and prediction that are adequate to the nature of the ongoing processes. [ABSTRACT FROM AUTHOR]

This paper proposed two new generalizations of the logistic function, each drawing on non-extensive thermodynamics, the q-logistic Equation and the logistic Equation of arbitrary order, respectively. It demonstrated the impact of chaos theory by integrating it with logistics Equations and revealed how minor parameter variations will change system behavior from deterministic to non-deterministic behavior. Moreover, this work presented BifDraw – a Python program for drawing bifurcation diagrams using classical logistic function and its generalizations illustrating the diversity of the system’s response to the changes in the conditions. The research gave a pivotal role to the place of the logistic Equation in chaos theory by looking at its complicated dynamics and offering new generalizations that may be new in terms of thermodynamic basic states and entropy. Also, the paper investigated dynamics nature of the Equations and bifurcation diagrams in it which present complexity and the surprising dynamic systems features. The development of the BifDraw tool exemplifies the practical application of theoretical concepts, facilitating further exploration and understanding of logistic Equations within chaos theory. This study not only deepens the comprehension of logistic Equations and chaos theory, but also introduces practical tools for visualizing and analyzing their behaviors. [ABSTRACT FROM AUTHOR]

This research explores the incorporation of variable order (VO) fractional calculus into the hyperchaotic Lorenz system and studies various chaotic features and attractors. Initially, we propose a variable fractional order hyperchaotic Lorenz system and numerically solve it. The solutions are obtained for multiple choices of control parameters, and these results serve as reference solutions for exploring chaos with the artificial intelligence tool radial basis function network (RBFN). We rebuild phase spaces and trajectories of system states to exhibit chaotic behavior at various levels. To further assess the sensitivity of chaotic attractors, Lyapunov exponents are calculated. The efficacy of the designed computational RBFN is validated through the RMSE and extensive error analysis. The proposed research on AI capabilities aims to introduce an innovative methodology for modeling and analyzing hyperchaotic dynamical systems with variable orders. [ABSTRACT FROM AUTHOR]

This study delves into the exploration of the dynamics of (3 + 1) -dimensional Painlevé integrable generalized model from different perspectives, which delineates the evolution of nonlinear phenomena in three spatial dimensions and one temporal dimension, displaying the remarkable Painlevé integrability property. The ϕ 6 -expansion technique is applied to extract traveling wave solutions in the form of Jacobi elliptic functions. To give physical insights of obtained solutions, we present these solutions through graphs such as 2D, 3D and contour plots. Further, to understand the planar dynamical system, we employ the concepts of bifurcation, chaos theory and sensitivity analysis. Bifurcation analysis reveals the dependence on the solution of a planar dynamical system at critical points. Additionally, the detection of chaotic movements in the perturbed dynamical system is achieved by detecting tools. Also the sensitivity analysis of the model is investigate by three distinct initial conditions. The findings are innovative, valuable and captivating for the readers in exploring this model. [ABSTRACT FROM AUTHOR]

As a nonlinear dynamic system, the dynamic equation of triboelectric nanogenerator system is not only difficult to accurately establish, but also difficult to solve. The output signals of triboelectric nanogenerator system depends on the initial conditions of the system, which reflects the dynamic behavior of the system. Currently, there is little research on using the output signals to study the dynamic behavior of triboelectric nanogenerator system. Therefore, a dynamic simulation model of a freestanding triboelectric nanogenerator (F-TENG) system was established using COMSOL software. The mathematical model of the F-TENG system was fitted. Using chaos theory and its phase-space reconstruction technology, the phase-space trajectories of the F-TENG system were reconstructed through voltage signals under external factors, and the dynamic behavior of the F-TENG system was discussed and studied. Finally, validation experiments were carried out. The research shows that with the variation of internal factors and operating frequency, the mathematical model of the F-TENG system remains basically unchanged, and the variation law of output characteristics also remains basically unchanged. The F-TENG system is in periodic motion; with the variation of external factors such as resistance and friction layer spacing, the mathematical model of the F-TENG system will evolve in a complex direction, and the output characteristics will gradually distort. The F-TENG system will evolve from periodic motion to chaotic motion. The research results not only reveal the dynamic behavior and enrich the nonlinear dynamic theory of triboelectric nanogenerator system, but also provide theoretical guidance for increasing the working bandwidth and stability of triboelectric nanogenerator system. [ABSTRACT FROM AUTHOR]

One technique, especially in chaos-based cryptographic applications, is to include the message in the evolution of the dynamical system. This paper aims to find out if and to what extent the statistical behavior of the chaotic system is affected by the message inclusion in its dynamic evolution. The study is illustrated by the dynamical system described by the logistic map in cryptographic applications based on images. The evaluation of the statistical behavior was performed on an original scheme proposed. The Monte Carlo analysis of the applied Kolmogorov–Smirnov statistical test revealed that the dynamical system in the processing scheme with message inclusion does not modify its proper statistical behavior (revealed by definition relation). This was possible due to the proposed scheme designed. Namely, this scheme contains a decision switch which, supported by an appropriate choice of the magnitude of the scaling factor, ensures that the values of the dynamical system are maintained in the definition domain. The proposed framework for analyzing the statistical properties and for preserving the dynamical system behavior is one main contribution of this research. The message inclusion scheme also provides an enhancement with cryptographic mixing functions applied internally; the statistical behavior of the dynamical system is also analyzed in this case. Thus, the paper contributes to the theoretical complex characterization of the dynamical system considering also the message inclusion case. [ABSTRACT FROM AUTHOR]

This narrative review, the second of 2 parts about a new approach to the diagnosis and treatment of psychiatric disorders that is based on dynamical systems theory, describes the implications of evidence for the practical applicability of this theory and its quantitative tools. Importance: Dynamical systems theory is widely used to explain tipping points, cycles, and chaos in complex systems ranging from the climate to ecosystems. It has been suggested that the same theory may be used to explain the nature and dynamics of psychiatric disorders, which may come and go with symptoms changing over a lifetime. Here we review evidence for the practical applicability of this theory and its quantitative tools in psychiatry. Observations: Emerging results suggest that time series of mood and behavior may be used to monitor the resilience of patients using the same generic dynamical indicators that are now employed globally to monitor the risks of collapse of complex systems, such as tropical rainforest and tipping elements of the climate system. Other dynamical systems tools used in ecology and climate science open ways to infer personalized webs of causality for patients that may be used to identify targets for intervention. Meanwhile, experiences in ecological restoration help make sense of the occasional long-term success of short interventions. Conclusions and Relevance: Those observations, while promising, evoke follow-up questions on how best to collect dynamic data, infer informative timescales, construct mechanistic models, and measure the effect of interventions on resilience. Done well, monitoring resilience to inform well-timed interventions may be integrated into approaches that give patients an active role in the lifelong challenge of managing their resilience and knowing when to seek professional help. [ABSTRACT FROM AUTHOR]

This narrative review describes a new approach to the diagnosis and treatment of psychiatric disorders that is based on dynamical systems theory, which addresses the concepts of tipping points, cycles, and chaos in complex systems. Importance: Psychiatric disorders may come and go with symptoms changing over a lifetime. This suggests the need for a paradigm shift in diagnosis and treatment. Here we present a fresh look inspired by dynamical systems theory. This theory is used widely to explain tipping points, cycles, and chaos in complex systems ranging from the climate to ecosystems. Observations: In the dynamical systems view, we propose the healthy state has a basin of attraction representing its resilience, while disorders are alternative attractors in which the system can become trapped. Rather than an immutable trait, resilience in this approach is a dynamical property. Recent work has demonstrated the universality of generic dynamical indicators of resilience that are now employed globally to monitor the risks of collapse of complex systems, such as tropical rainforests and tipping elements of the climate system. Other dynamical systems tools are used in ecology and climate science to infer causality from time series. Moreover, experiences in ecological restoration confirm the theoretical prediction that under some conditions, short interventions may invoke long-term success when they flip the system into an alternative basin of attraction. All this implies practical applications for psychiatry, as are discussed in part 2 of this article. Conclusions and Relevance: Work in the field of dynamical systems points to novel ways of inferring causality and quantifying resilience from time series. Those approaches have now been tried and tested in a range of complex systems. The same tools may help monitoring and managing resilience of the healthy state as well as psychiatric disorders. [ABSTRACT FROM AUTHOR]

Nonlinear dynamical problems, characterized by unpredictable and chaotic changes among variables over time, pose unique challenges in understanding. This paper explores the coupled nonlinear (2+1)-dimensional complex modified Korteweg-de-Vries (cmKdV) equation-a fundamental equation in applied magnetism and nanophysics. The study focuses on dynamic behaviors, specifically examining bifurcations and equilibrium points leading to chaotic phenomena by introducing an external term to the system. Employing chaos theory, we showcase the chaotic tendencies of the perturbed dynamical system. Additionally, a sensitivity analysis using the Runge-Kutta method reveals the solution's stability under slight variations in initial conditions. Innovatively, the paper utilizes the planar dynamical system technique to construct various solitons within the governing model. This research provides novel insights into the behavior of the (2+1)-dimensional cmKdV equation and its applications in applied magnetism and nanophysics. [ABSTRACT FROM AUTHOR]

In the work, we prove the presence of chaotic dynamics, for suitable values of the model parameters, for the discrete-time system, composed of two coupled logistic maps, as formulated in Yousefi et al. [Discrete Dyn. Nat. Soc. 5, 161–177 (2000)], which describes two interdependent economies, characterized by two competitive markets each, with a demand link between them. In particular, we rely on the SAP (Stretching Along the Paths) method, based on a stretching relation for maps defined on sets homeomorphic to the unit square and expanding the paths along one direction. Such technique is topological in nature and allows to establish the existence of a semiconjugacy between the considered dynamical system and the Bernoulli shift, so that the main features related to the chaos of the latter (e.g., the positivity of the topological entropy) are transmitted to the former. In more detail, we apply the SAP method both to the first and to the second iterate of the map associated with our system, and we show how dealing with the second iterate, although being more demanding in terms of computations, allows for a larger freedom in the sign and in the variation range of the linking parameters for which chaos emerges. Moreover, the latter choice guarantees a good agreement with the numerical simulations, which highlight the presence of a chaotic attractor under the conditions derived for the applicability of the SAP technique to the second iterate of the map. [ABSTRACT FROM AUTHOR]

A three-dimensional multiparametric system of ordinary differential equations, arising in the theory of genetic networks, is considered. Examples of chaotic behavior are constructed using the methodology by Shilnikov. This methodology requires the existence of saddle-focus points satisfying some additional conditions. As the result, rich dynamical behavior of solutions can be observed, including chaotic behavior of solutions. [ABSTRACT FROM AUTHOR]

Soil moisture‐Precipitation (SM‐P) coupling, especially the causality from SM to P (SM → P), is an important and highly debated topic. The causal inference from observational data provides a statistical approach for this issue, where the experimental research is infeasible. Various causal inference methods exist, each assuming distinct underlying systems for the targeted variables: pure stochastic or deterministic dynamic system (DDS), which means that these methods detect different types of causality: separable or non‐separable. Acknowledging the inherent deterministic dynamic nature of SM‐P coupling, this study employs a DDS‐based method: Convergent Cross‐mapping (CCM), to detect their non‐separable causality. Centering around SM‐P coupling, we also detect the causality between SMs in shallow and deeper layers, and the causality between evapotranspiration (ET) and SMs in all layers. Notably, before applying CCM, a preconditional procedure is required: verifying the DDS nature of targeted variables P, SM, and ET. Key findings of this research include: (a) In SM‐P coupling, only SMs in shallow layers but not deeper layers could render causalities toward P, while P renders causalities toward SMs in all layers. (b) The time‐delay of causality from SM1 to P in spring/summer is around 4–6 days, and that from P to SM1 is around 2–4 days. (c) Inside SMs, shallow SMs have obvious causalities downwards to deeper SMs, but deeper SMs seem harder to render causalities upwards to shallow SMs, explaining why they hardly render causalities to P. In summary, this study furnishes an indispensable DDS‐based complement to stochastic‐based methodologies for SM‐P coupling. Plain Language Summary: This study aims to investigate the causal interaction between soil moisture (SM) and precipitation (P), especially the impact from SM toward P, based on their observational data (i.e., time series), as experimental approaches are impractical given their large‐scale complex physical mechanisms. Many methods exist in the realm of causal detection, either assuming a stochastic underlying system or a deterministic dynamic system (DDS) for the targeted variables. In stochastic systems, causal interactions among variables are separable, whereas they are non‐separable in DDSs. Given that SM‐P coupling seems aligning with the description of a DDS, we employ a DDS‐based method: Convergent Cross‐mapping (CCM), to detect the non‐separable causality between SM and P. Specifically, we explore whether causality exists both from SM to P (SM → P, SM in different layers) and from P to SM (P → SM), and if so, what is the causal response time and how long could the causality last. Meanwhile, the causalities among SMs in different layers, and those between SMs and evapotranspiration (ET), are also detected. Through these inquiries, our findings highlight the DDS nature of SM‐P coupling and their non‐separable causality, underscoring the indispensable role of CCM as a complement to previous stochastic‐based studies for SM‐P coupling. Key Points: Soil moisture‐Precipitation (SM‐P) coupling is detected by Convergent Cross‐mapping (CCM) to investigate their non‐separable causalityThe deterministic dynamic system nature of SM‐P coupling is verified, as a precondition for applying CCM to detect their causalityWhile P could causally impact SMs in all layers; in turn, however, only shallow SMs but not deeper SMs could causally impact P [ABSTRACT FROM AUTHOR]

In nonlinear dynamics, there is a continuous exploration of introducing systems with evidence of chaotic behavior. The presence of nonlinearity within system equations is crucial, as it allows for the emergence of chaotic dynamics. Given that quadratic terms represent the simplest form of nonlinearity, our study focuses on introducing a novel chaotic system characterized by only quadratic nonlinearities. We conducted an extensive analysis of this system's dynamical properties, encompassing the examination of equilibrium stability, bifurcation phenomena, Lyapunov analysis, and the system's basin of attraction. Our investigations revealed the presence of eight unstable equilibria, the coexistence of symmetrical strange repeller(s), and the potential for multistability in the system. [ABSTRACT FROM AUTHOR]

In a vast area of probabilistic limit theorems for dynamical systems with chaotic behaviors always only functional form (exponential, power, etc.) of the asymptotic laws and of convergence rates were studied. However, for basically all applications, e.g., for computer simulations, development of algorithms to study chaotic dynamical systems numerically, as well as for design and analysis of real (e.g., in physics) experiments, the exact values (or at least estimates) of constants (parameters) of the functions, which appear in the asymptotic laws and rates of convergence, are of primary interest. In this paper, we provide such estimates of constants (parameters) in the central limit theorem, large deviations principle, law of large numbers and the rate of correlations decay for strongly chaotic dynamical systems. [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]

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]

This paper deals with the problem of process faults in the Lorenz system that can affect any of the system parameters and cause the system to exhibit various behaviors. In this paper, the homoclinic orbits in the Lorenz system are described and then the occurrence of process faults in the system is investigated that can cause a homoclinic explosion, bifurcation, change of fixed point, or even instability in the system. In such systems, where a small change in one of the parameters causes large changes in the behavior of the system, to prevent disaster in industrial systems and also to stop the propagation of faults in the system, the faults must be identified as soon as possible. In this paper, the states in the system are estimated by using a reduced-order observer, and the faults are detected. The purpose of this article is to recognize the change in behavior of this system in the face of this type of fault and to express the importance of timely detection and identification of faults in the system so as not to lead to failure and disaster in industrial systems. Finally, the effect of process fault, disturbance, and sensor fault are investigated simultaneously and the states and faults in the system are estimated. [ABSTRACT FROM AUTHOR]

Effectively identifying and characterizing the various dynamics present in complex and chaotic systems is fundamental for chaos control, chaos classification, and behavior-transition forecasting, among others. It is a complicated task that becomes increasingly difficult as systems involve more dimensions and parameters. Here, we extend methods inspired in ordinal patterns to analyze 2D iterative maps to unveil underlying approximate symmetries of their dynamics. We distinguish different families of chaos within the systems, find similarities among chaotic maps, identify approximate temporal and dynamical symmetries, and anticipate sharp transitions in dynamics. We show how this methodology displays the evolution of the spatial correlations in a dynamical system as the control parameter varies. We prove the power of these techniques, which involve simple quantifiers as well as combinations of them, in extracting relevant information from the complex dynamics of 2D systems, where other techniques are less informative or more computationally demanding. [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]

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]

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]

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]

Dynamic Mode Decomposition with Control is a powerful technique for analyzing and modeling complex dynamical systems under the influence of external control inputs. In this paper, we propose a novel approach to implement this technique that offers computational advantages over the existing method. The proposed scheme uses singular value decomposition of a lower order matrix and requires fewer matrix multiplications when determining corresponding approximation matrices. Moreover, the matrix of dynamic modes also has a simpler structure than the corresponding matrix in the standard approach. To demonstrate the efficacy of the proposed implementation, we applied it to a diverse set of numerical examples. The algorithm's flexibility is demonstrated in tests: accurate modeling of ecological systems like Lotka-Volterra, successful control of chaotic behavior in the Lorenz system and efficient handling of large-scale stable linear systems. This showcased its versatility and efficacy across different dynamical systems. [ABSTRACT FROM AUTHOR]

Pseudo‐random number generators (PRNGs) are the cornerstone of various fields including computer science, cryptography, and scientific simulation. To generate high‐quality and unpredictable pseudo‐random sequences, many works draw attention to the use of chaos theory and dynamical systems. However, most of them focus on continuous or low‐dimensional chaotic systems with strong temporal correlations or insufficient nonlinear dynamics. Thus, in this study, a new PRNG based on a four‐dimensional discrete system is proposed to solve the aforementioned flaws. The PRNG has been implemented on Xilinx Artix‐7 xc7a100tfgg484‐2, which can generate pseudo‐random sequences at a high speed of 12811.53 Mb/s without additional post‐processing. All these sequences have successfully passed the NIST SP800.22 standard test. Both throughput and randomness quality of the proposed PRNG outperform the state‐of‐the‐art. [ABSTRACT FROM AUTHOR]

Further exploration into the influence of a memristor on the behavior of chaotic systems deserves attention. When constructing memristor chaotic systems, it is commonly believed that increasing the number of memristors will lead to better system performance. This paper proposes a class of chaotic maps with different discrete memristors, achieved through internal perturbation based on the Sine map. The I-V curve of the discrete memristor has a symmetrical structure. The dynamic characteristics of the designed system are analyzed using the chaotic attractor phase diagram, Lyapunov exponent (LE) spectrum, and bifurcation diagram. Numerical simulations demonstrate that internal perturbations of discrete memristors enhance the Sine map's chaotic characteristics, expand the chaos range, and improve the ergodicity and LE value. Moreover, the type of discrete memristors has a significant impact on the dynamic characteristics of the system, while the number of discrete memristors has little influence. Therefore, in this paper, a direction for the design of a discrete memristor chaotic system is provided. Finally, a discrete memristor chaotic map with a simple structure and better performance is selected. Based on this, a pseudo-random sequence generator is designed, and the generated sequence passes the National Institute of Standards and Technology (NIST) test. [ABSTRACT FROM AUTHOR]

Theoretically, an eight-term chaos system is presented. The effect of changing the initial conditions values on behavior Chen system was studied. The basic dynamical properties of system are analyzed like time series, attractor, FFT spectrum, and bifurcation. Where the system appears steady state behavior at initial condition xi, yi, zi equal (0, 0, 0) respectively and it convert to quasi-chaotic at xi,yi,zi equal (-0.1, 0.5,-0.6). Finally, the system become hyper chaotic at xi,yi, zi equal(- 0.5, 0.5,-0.6 ) that can used it in many applications like secure communication. [ABSTRACT FROM AUTHOR]

Recent work has highlighted the vast array of dynamics possible within both neuronal networks and individual neural models. In this work, we demonstrate the capability of interacting chaotic Hindmarsh–Rose neurons to communicate and transition into periodic dynamics through specific interactions which we call mutual stabilization, despite individual units existing in chaotic parameter regimes. Mutual stabilization has been seen before in other chaotic systems but has yet to be reported in interacting neural models. The process of chaotic stabilization is similar to related previous work, where a control scheme which provides small perturbations on carefully chosen Poincaré surfaces that act as control planes stabilized a chaotic trajectory onto a cupolet. For mutual stabilization to occur, the symbolic dynamics of a cupolet are passed through an interaction function such that the output acts as a control on a second chaotic system. If chosen correctly, the second system stabilizes onto another cupolet. This process can send feedback to the first system, replacing the original control, so that in some cases the two systems are locked into persistent periodic behavior as long as the interaction continues. Here, we demonstrate how this process works in a two-cell network and then extend the results to four cells with potential generalizations to larger networks. We conclude that stabilization of different states may be linked to a type of information storage or memory. [ABSTRACT FROM AUTHOR]

Studies on fractional-order chaotic systems have increased significantly in the last decade. This paper presents Rucklidge chaotic system's dynamical analyses and its fractional-order circuit implementations. Component values required for realizing the circuit of the fractional-order system are calculated for different fractional-orders. The feasibility of the attractor is examined by implementing its electronic circuit with a fractional-order module. The module is constructed based on the Diyi-Chen model since it is easier to implement and cost-effective. In electronic circuit implementations, it is observed that the system's chaotic state disappears as the fractional degree decreases. Numerical and circuit simulation results are consistent well with the hardware experimental results. [ABSTRACT FROM AUTHOR]

Considering that chaotic systems are immersed in multiple areas of science and nature and that their dynamics are governed by a great sensitivity to the initial conditions and variations in their parameters, it is of great interest for the scientific community to have tools to characterize and reproduce these trajectories. Two dynamic chaotic systems whose equations are based on the jerky system are used as benchmarks, i.e., the Memristive Shaking Chaotic System (MSCS) and the Unstable Dissipative System of type I (UDSI). One characteristic common to them is their simple mathematical structure and the complexity of their solutions. Therefore, this paper presents a strategy for identifying chaotic trajectories using a recurrent wavelet first-order neural network (RWFONN) that is trained online with an error filtering algorithm and considering the Morlet-wavelet as an activation function. The parameters of the network are adjusted considering the Euclidean distance between the solutions. Finally, the results depict proper identification of the chaotic systems studied through analysis and numerical simulation to validate the behavior and functionality of the proposed network. [ABSTRACT FROM AUTHOR]

The coexistence of different attractors with fixed parameters affords versatility in the performance of the dynamical system. As a result, recent research has focused on defining nonlinear oscillators with infinite coexisting attractors, the vast majority of which are non-autonomous systems. In this study, we present an infinite number of equilibriums achieved with the simplest autonomous megastable oscillator. To begin, we explore the symmetry property of distinct coexisting attractors, such as periodic, quasi-periodic, and chaotic attractors. The dynamical properties of the proposed system are further investigated using phase portraits, Lyapunov exponents spectrum, and bifurcation diagram using the local maxima of the state variables. The detailed investigation reveals that the proposed system has a wide range of dynamical properties ranging from periodic oscillations to hyperchaos. Linear quadratic regulator (LQR) control-based controllers are designed to control the chaos in the proposed system and also achieve synchronization between proposed systems that have different initial conditions. The effectiveness of the controller designed based on LQR is presented by simulation results. It is shown that the proposed system converges asymptotically to the desired equilibrium point, and the synchronization between the slave and the master systems is achieved by converging the error to zero after controllers are activated. [ABSTRACT FROM AUTHOR]

A hydraulic generator regulating system with electrical, mechanical, and hydraulic constitution is a complex nonlinear system, which is analyzed in this research. In the present study, the dynamical behavior of this system is investigated. Afterward, the input/output feedback linearization theory is exerted to derive the controllable model of the system. Then, the chaotic behavior of the system is controlled using a robust controller that uses a Chebyshev neural network as a disturbance observer in combination with a non-singular robust terminal sliding mode control method. Moreover, the convergence of the system response to the desired output in the presence of uncertainty and unexpected disturbances is demonstrated through the Lyapunov stability theorem. Finally, the effectiveness and appropriate performance of the proposed control scheme in terms of robustness against uncertainty and unexpected disturbances are demonstrated through numerical simulations. [ABSTRACT FROM AUTHOR]

In this research, the ship power system is studied with a fractional-order approach. A 2-D model of a two-generator parallel-connected is considered. A chaotic attractor is observed for particular parameter values. The fractional-order form is calculated with the Adam–Bashforth–Moulton method. The chaotic response is identified even for the order 0.99. Phase portrait is generated using the Caputo derivative approach. Wolf's algorithm is used to calculate Lyapunov exponents. For the considered values of parameters, one positive Lyapunov exponent confirms the existence of chaos. Bifurcation diagrams are presented to analyze the various dynamical behaviors and bifurcation points. Interestingly, the considered system is multistable. Also, antimonotonicity, period-doubling, and period halving are observed in the bifurcation diagram. As the last step, a fractional-order controller is designed to remove chaotic dynamics. Time plots are simulated to show the effectiveness of the controller. [ABSTRACT FROM AUTHOR]

This study explores the intrinsic characteristics of the (2+1)-dimensional Schwarz-Korteweg-de Vries equation used to describe shallow water waves. The multiple solitons are successfully constructed using a technique involving multiple exponential functions. Graphical representations of the results are provided in 3D, 2D, and density plots to assess the compatibility of the solutions. Also, the dynamic nature of studied equation is examined based on bifurcation and chaos theory for nonlinear systems. Bifurcation signifies how our dynamical system is affected by physical parameters in planar dynamical system. After that, we apply the external force on planar dynamical system to show the chaotic like behavior of the studied model. Such behavior is confirmed by utilizing different chaos detecting tools. The obtained results serve to explain the effectiveness and applicability of the utilized methodologies in comprehend the exact solution and qualitative behavior of nonlinear physical models. [ABSTRACT FROM AUTHOR]

Background: Conservative chaos is one of the most interesting objects in modern nonlinear dynamics and chaos theory. Purpose: In this paper, we are focused on a simple symmetric 2–4–6 polynomial oscillatory potential under external periodic excitation. Methods: The Poincaré sections, bifurcation diagrams, and the Lyapunov characteristic exponents for such a dynamical system are obtained and analyzed together with the fixed points. Results: Our numerical results show that the motion of a particle in this potential has rich nonlinear dynamics. Because of two local instabilities, such a system exhibits conservative chaos and can be considered as a simple and efficient source of the chaotic signal. Conclusion: The obtained results show that this system is not only of fundamental interest for the theory of nonlinear oscillations but also can be considered as a promising source of chaos in the field of secure communications and encryption. [ABSTRACT FROM AUTHOR]

The chaotic behavior of the Solar System prevents the prediction of the precise evolution of the planetary secular frequencies. Here we use four cyclostratigraphic sequences, ∼259 Ma Wujiaping Formation, ∼455 Ma Pingliang Formation, ∼655 Ma Datangpo Formation and ∼2,465 Ma Dales Gorge Member, and Bayesian inversion to provide powerful geological constraints to this problem. The inversion provides constraining estimates for the Solar System fundamental secular frequencies g1, g2, g3, g4, and g5, and the Earth's precession rate k from 105.26 ± 2.68 (2σ) arcseconds/year at ∼2,465 Ma to 55.86 ± 2.60 (2σ) arcseconds/year at ∼259 Ma. Stochastic monotone interpolation enables the reconstruction of Earth‐Moon distance from 2.5 billion years to the present. Calculation of tidal dissipation supports the ocean model before 1.4 Ga, but is outside the error range after ∼1.8 Ga, suggesting lower tidal dissipation in the early Proterozoic Eon, and a possible high value in the early Paleozoic. Plain Language Summary: The chaotic history of the Solar System imposes unknown uncertainties on the fundamental secular frequencies of the planets over most of geologic time. This is complicated by a lack of knowledge about the evolution of Earth‐Moon distance, Earth's length of day and tidal dissipation. Geological evidence provides powerful constraints on all of these parameters. To shed light on the parameters, here we carry out Bayesian inversion of fundamental secular frequencies and Earth precession rate on four cyclostratigraphic sequences, and stochastic monotone interpolation to reconstruct Earth‐Moon distance from 2.5 billion years ago to present. From these results we estimate the Earth's tidal dissipation variation, which we compare with theoretical models. Our results provide critical geological constraints for paleotidal simulation, and shine a light on the unique potential of cyclostratigraphy to constrain Earth‐Moon system history across geologic time. Key Points: We obtain new estimates of Earth‐Moon and Solar System dynamical parameters from cyclostratigraphy at four ages from ∼2.5 Ga to ∼259 MaWe interpolate Earth‐Moon distances inferred from the cyclostratigraphy to reconstruct a continuous history from ∼2.5 Ga to the presentWe estimate Earth's tidal dissipation variations from ∼2.5 Ga to the present [ABSTRACT FROM AUTHOR]

This work identifies the influence of chaos theory on fractional calculus by providing a theorem for the existence and stability of solution in fractional-order gyrostat model with the help of a fixed-point theorem. We modified an integer order gyrostat model consisting of three rotors into fractional order by attaching rotatory fuel-filled tank and provided an iterative scheme for our proposed model as a working rule of obtained analytical results. Moreover, this iterative scheme is injected into algorithms for a system of integer order dynamical systems to observe Lyapunov exponents and a bifurcation diagram for our proposed fractional-order dynamical model. Furthermore, we obtained five equilibrium points, including four unstable spirals and one saddle node, using local dynamical analysis which acted as self-exciting attractors and a separatrix in a global domain. [ABSTRACT FROM AUTHOR]

Image encryption based on a chaos system can effectively protect the privacy of digital images. It is said that a 3D chaotic system has a larger parameter range, better unpredictability and more complex behavior compared to low-dimension chaotic systems. Motivated by this fact, we propose a new image cryptosystem that makes use of a 3D chaotic system. There are three main steps in our scheme. In the first step, the chaotic system uses the hash value of the plaintext image to generate three sequences. In step two, one of the sequences is used to dynamically select confusion and diffusion methods, where confusion and diffusion have three algorithms, respectively, and will produce 32n (n > 100) combinations for encryption. In step three, the image is divided into hundreds of overlapping subblocks, along with the other two sequences, and each block is encrypted in the confusion and diffusion process. Information entropy, NPCR, UACI results and various security analysis results show that the algorithm has a better security performance than existing, similar algorithms, and can better resist clipping, noise, statistical analysis and other attacks. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of the study is to obtain explicit formulas to determine the stability of periodic solutions to the new system and study the extent of the stability of those periodic solutions and the direction of bifurcated periodic solutions. More than that, the authors did a numerical simulation to confirm the results that the authors obtained and presented through numerical analysis are the periodic and stable solutions and when the system returns again to the state of out of control. Design/methodology/approach: The authors studied local bifurcation and verified its occurrence after choosing the delay as a parameter of control in Zhou 2019's dynamical system with delayed feedback control. The authors investigated the normal form theory and the center manifold theorem. Findings: The occurrence of local Hopf bifurcations at the Zhou's system is verified. By using the normal form theory and the center manifold theorem, the authors obtain the explicit formulas for determining the stability and direction of bifurcated periodic solutions. The theoretical results obtained and the corresponding numerical simulations showed that the chaos phenomenon in the Zhou's system can be controlled using a method of time-delay auto-synchronization. Originality/value: As the delay increases further, the numerical simulations show that the periodic solution disappears, and the chaos attractor appears again. The obtained results can also be applied to the control and anti-control of chaos phenomena of system (1). There are still abundant and complex dynamical behaviors, and the topological structure of the new system should be completely and thoroughly investigated and exploited. [ABSTRACT FROM AUTHOR]

Bifurcation analysis and dynamical system studies are carried out to find the stability regime and chaotic scenario in electron acoustic waves in relativistic degenerate plasma. We have obtained the quantum hydrodynamic model and obtain the Korteweg–de Vries equation describing the nature and characteristics of solitary structures. The amplitude modulated envelop soliton formation due to external perturbations has been studied by analyzing the nonlinear Schrodinger equation. Further to study the stability factors and the parametric range for such stability, the dynamical system is studied and bifurcation analysis has been carried out. The chaotic behavior of the system is studied through largest Lyapunov exponent (LLE). This work will find application in theoretically predicting the stable modes in many solar plasma and stellar plasma applications and in laser plasma in future. [ABSTRACT FROM AUTHOR]

The Lyapunov exponent is the most-well-known measure for quantifying chaos in a dynamical system. However, its computation for any time series without information regarding a dynamical system is challenging because the Jacobian matrix of the map generating the dynamical system is required. The entropic chaos degree measures the chaos of a dynamical system as an information quantity in the framework of Information Dynamics and can be directly computed for any time series even if the dynamical system is unknown. A recent study introduced the extended entropic chaos degree, which attained the same value as the total sum of the Lyapunov exponents under typical chaotic conditions. Moreover, an improved calculation formula for the extended entropic chaos degree was recently proposed to obtain appropriate numerical computation results for multidimensional chaotic maps. This study shows that all Lyapunov exponents of a chaotic map can be estimated to calculate the extended entropic chaos degree and proposes a computational algorithm for the extended entropic chaos degree; furthermore, this computational algorithm was applied to one and two-dimensional chaotic maps. The results indicate that the extended entropic chaos degree may be a viable alternative to the Lyapunov exponent for both one and two-dimensional chaotic dynamics. [ABSTRACT FROM AUTHOR]