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To study the relationship between the dynamics of the COVID-19 pandemic and the state of the economy, the authors developed a stochastic model that covers the epidemic and economic blocks and the system of relationships between them. Based on the three-sector Lorenz model, they have proposed an approach to analyze the interrelationships between economic sectors related to food production, food supply chains, and the healthcare industry. The model combines the above economic sectors in a single structure and considers each sector in terms of productivity, the number of jobs, and the structural disturbance levels. The authors carried out model studies to analyze the dependence of the dynamics of the pandemic on the initial conditions and parameters for managing the epidemic process and the impact of the pandemic on changes in the balance of supply and demand in sectors of the economy related to food production, food transportation, and the healthcare industry. [ABSTRACT FROM AUTHOR]

To study the relationship between the dynamics of the COVID-19 pandemic and the state of the economy, a stochastic model is developed that covers the epidemic and economic blocks and the system of relationships between them. An approach is proposed to analyze the interrelations of economic sectors related to food production, transport chains of their delivery, and the medical sector, using the three-sector Lorenz model. This model combines the described sectors of the economy into a single structure, each of which is considered from the point of view of the level of productivity, the number of jobs, and structural disturbances. Model studies are carried out to analyze the dependence of the dynamics of the pandemic on the initial conditions and parameters for managing the epidemic process and the impact of the pandemic on changes in the balance of supply and demand in sectors of the economy related to food production, transport, and the medical sector. [ABSTRACT FROM AUTHOR]

Current earthquake forecasting approaches are mainly based on probabilistic assumptions, as earthquakes seem to occur randomly. Such apparent randomness can however be caused by deterministic chaos, rendering deterministic short‐term forecasts possible. Due to the short historical and instrumental record of earthquakes, chaos detection has proven challenging, but more frequently occurring slow slip events (SSE) are promising candidates to probe for determinism. Here, we characterize the SSE signatures obtained from GNSS position time series in the Hikurangi Subduction Zone (New Zealand) to investigate whether the seemingly random SSE occurrence is governed by chaotic determinism. We find evidence for deterministic chaos for stations recording shallow SSEs, suggesting that short‐term deterministic forecasting of SSEs, similar to weather forecasts, might indeed be possible over timescales of a few weeks. We anticipate that our findings could open the door for next‐generation SSE forecasting, adding new tools to existing probabilistic approaches. Plain Language Summary: Since earthquakes appear to occur randomly, the currently available probabilistic predictions are based on past earthquake records. These predictions estimate the likelihood of an earthquake of a given magnitude occurring within a defined time period. In contrast to such probabilistic approaches, deterministic systems are fully predictable, albeit often confined to short time scales due to their potential chaotic behavior. Probing for deterministic predictability in the earthquake cycle is intractable due to the limited historical instrumental record. However, frequently occurring slow slip events ‐ captured by transient GNSS displacements that can last several weeks ‐ provide a unique opportunity to explore deterministic predictability in these types of slow earthquakes. By studying GNSS time series from various stations on New Zealand's North Island, we have discovered evidence suggesting that these irregularly occurring slow slip events might be governed by chaotic determinism. This implies the potential to forecast both timing and magnitude of slow slip events a few weeks in advance using deterministic methods, much like we predict weather patterns. Consequently, our theoretical findings could therefore pave the way for innovative approaches to short‐term slow slip forecasting. Key Points: Nonlinear analysis of GNSS displacement time series unveils evidence for deterministic chaos in slow slip events in New ZealandOur theoretical findings imply that irregularly occurring slow slip events could potentially be forecasted a few weeks in advance [ABSTRACT FROM AUTHOR]

Markets, I argue, matter significantly for macroeconomic development. The reason for this strange statement is that policy practice in industrialized market economies is routinely governed by models that ignore the role of markets. Policy makers are therefore at risk of misunderstanding the economic situation and of mismanaging the economy. Some macroeconomic models with sufficient nonlinearities and a production lag, two pioneering articles by Day (1982,1983) demonstrated, exhibit phases of erratic fluctuations (or deterministic chaos). Shortly thereafter Eliasson (1983, 1984) observed the endogenous emergence of similar unpredictable irregularities in simulating a complex, agent based, market self-coordinated economy wide model implemented on Swedish data. The irregularities originated in a failure of markets to coordinate production, employment, and investment decisions as they were pushed excessively by policy to become more efficient. In this agent based model of an Experimentally Organized Economy , called MOSES, progress is governed by a dominant flow of successful business experiments conducted by firms, trailed by a flow of mistaken projects. The coordination of these flows by autonomous market agents was shown to be easily disturbed by external interferences, for instance unwitting policies. Markets, simulation experiments however also showed have great endogenous capacities to restore disturbed order. Large private investment mistakes at the agent level, experiments furthermore indicate, incur only minor social costs if production on the site is promptly terminated, and markets organized to efficiently reallocate resources. Bold private investments should therefore be encouraged, corrections of identified mistakes made without delay, and the efficiency of markets to reallocate resources paid attention to. I conclude that central policy in "Sweden like industrial economies" can only effectively be pursued by taking advantage of the endogenous capacity of functioning markets of effective self-coordination. Coordination failure being created by policy failure masquerading as market failure should be considered before market failure is blamed. [ABSTRACT FROM AUTHOR]

Abstract Current earthquake forecasting approaches are mainly based on probabilistic assumptions, as earthquakes seem to occur randomly. Such apparent randomness can however be caused by deterministic chaos, rendering deterministic short‐term forecasts possible. Due to the short historical and instrumental record of earthquakes, chaos detection has proven challenging, but more frequently occurring slow slip events (SSE) are promising candidates to probe for determinism. Here, we characterize the SSE signatures obtained from GNSS position time series in the Hikurangi Subduction Zone (New Zealand) to investigate whether the seemingly random SSE occurrence is governed by chaotic determinism. We find evidence for deterministic chaos for stations recording shallow SSEs, suggesting that short‐term deterministic forecasting of SSEs, similar to weather forecasts, might indeed be possible over timescales of a few weeks. We anticipate that our findings could open the door for next‐generation SSE forecasting, adding new tools to existing probabilistic approaches.

In this paper, the applied problems of designing a bridge resonant converter with a high-voltage arc ignition system in the channel of an electric arc motor is solved. The development of a bridge resonant power supply is presented, a review and analysis of power sources of arcjet thruster of various capacities is carried out. The theoretical aspects of calculating the main parameters of the source are considered. The transfer function describing the gain of the resonant circuit of the power supply is given. With the help of the transfer function, gain diagrams are constructed for various values of the Q-factor of the contour. A prototype of a power supply, a starting system and an arcjet thruster is presented. A series of experiments aimed at identifying the operability of the source is carried out. According to the experimental results, the maximum output power of the source wiz 182,4 watts, and the maximum efficiency is 87,5 %. Frequency spectra of the output current of the power supply are obtained. The current-voltage characteristics for the Ar and N2 are constructed. In the course of the study, phenomena related to the chaotic dynamics of the arc in the engine channel are discovered. Chaotic fluctuations are observed at the moment of starting the engine and disappeared after steady-state operation.

This paper delves into the intersection between discrete dynamical systems and bio-inspired metaheuristic algorithms. Chaos, an inherent phenomenon manifested by both continuous and discrete dynamic nonlinear systems has become an essential component of engineering design. This study examines the relationship between chaos and bio-inspired metaheuristic algorithms from two distinct angles: the presence of chaos within the realm of algorithms, and the application of bio-inspired algorithms for the identification, control, or synthesis of complex systems. Furthermore, this paper offers an in-depth exploration of the interplay between metaheuristics and complex nonlinear dynamics, including potential avenues for future research. The presented simulation studies, along with the design of objective functions for optimization and the implementation of metaheuristics, serve as a valuable foundation for subsequent experiments involving discrete nonlinear systems and complex systems with time delays. Such systems may benefit from multiparametric or multicriteria optimization approaches, paving the way for novel advancements in the field. [ABSTRACT FROM AUTHOR]

Purpose of the article is to illustrate the genesis, meaning and significance of the functional Schroder equation, introduced in the theory of iterations of rational functions, for the theory of deterministic chaos by analytical calculations of exact trajectory solutions, invariant densities and Lyapunov exponents of one-dimensional chaotic maps. We demonstrate the method for solving the functional Schroder equation for various chaotic maps by passing to a topologically conjugate mappings, for which finding the exact trajectory solution is a simpler mathematical procedure. Results of the analytical solution of the Schroder equation for 12 chaotic mappings of various types and the calculation of the corresponding expressions for exact trajectory solutions, invariant densities and Lyapunov exponents are presented. Conclusion is made about the methodological expediency of formulating and solving the Schroder equations by the study of the dynamics of one-dimensional chaotic mappings.

The authors have developed a mathematical model to determine the conditions whereby the rate of increase of greenhouse gas (GHG) concentrations in the atmosphere is minimized, and the given level of life quality of the population is maintained. The model combines in a holistic structure the economic sectors described in the same way. Each of them is considered in terms of the level of productivity, number of jobs, and the level of structural disturbances (the six-sector Lorenz model with variable coefficients that depend on the GHG level). The model makes it possible to investigate how changes in the GHG level and the relationships between different sectors of the economy affect the risks to sustainable development and the general level of the life quality of the population. The authors have determined conditions for the emergence of turbulent operating modes, which lead to an increase in the total number of structural disturbances and a decrease in the aggregate level of productivity. They also proposed an approach to assessing the level of threats to sustainable development under climate changes. [ABSTRACT FROM AUTHOR]

A mathematical model has been developed to determine the conditions whereby the growth rate of the concentration of greenhouse gases (GHG) in the atmosphere is minimized and the given standard of living of the population is maintained. The model combines in a single structure identically described sectors of the economy, each being considered in terms of productivity levels, number of jobs and structural violations (six-sector Lorenz model with variable coefficients that depend on the GHG level). The model makes it possible to investigate how changes in the level of GHG and the interrelationships between different sectors of the economy affect the risks of sustainable development and the general level of the quality of life of the population. The conditions for the emergence of turbulent modes of operation, which lead to an increase in the total number of structural violations and a decrease in the total level of productivity, have been determined. An approach to assessing the level of threats to sustainable development under climate change is proposed. [ABSTRACT FROM AUTHOR]

The subject of the article is a variant of an efficient algorithm for synthesizing a dual discrete model and controller for tracking a given trajectory of a dynamic nonlinear, nonstationary black box object, using standard procedures for diagonalizing the state matrix, which makes it possible to simplify obtaining control values in numerical form and reduce the number of calculations. The current article presents one the possible solutions to the problem of regulator synthesis to ensure stable development of a given trajectory of motion of a nonlinear, non-stationary object of "black box" type using the concept of dual control. The task was set to simplify the previously proposed synthesis algorithm for the adaptive control of dynamic nonlinear, non-stationary objects using the example of first-order object of the "black box" type, using standard procedures for the diagonalization of the state matrix. An extended state matrix is the basis for obtaining a control model and predicting the behavior of a nonlinear non-stationary object, which in turn makes it possible to effectively use the concept of dual control. Methods used in the work are based on concept of dual control, nonlinear dynamic models, matrix theory, difference equations. Obtained results of this work consist of the development of a version of a dual nonparametric controller of nonstationary nonlinear processes, which has adaptive properties, does not require knowledge of the physics of functioning of the control object, is presented in the form of a simple algebraic formula that does not contain coefficients that require adjustment. Conclusion. Scientific novelty lies in the application of each interval matrix operator control for the diagonalization of the state submatrix. This operator is used for subsequent calculation of the control action. This approach enables the use of a standard diagonalization procedure using mathematical applications. The results are presented in the form of a final formula that does not require use of matrix operations during control, which makes it possible to simplify the synthesis of the controller using standard mathematical procedures.

Modeling of ecological processes is demonstrated using a newly developed nonlinear dynamics model of microbial populations, consisting of a 4-variable system of coupled ordinary differential equations. The system also includes a modified version of the Monod kinetics equation. The model is designed to simulate the temporal behavior of a microbiological system containing a nutrient, two feeding microbes and a microbe predator. Three types of modeling scenarios were numerically simulated to assess the instability caused by (a) variations of the nutrient flux into the system, with fixed initial microbial concentrations and parameters, (b) variations in initial conditions, with fixed other parameters, and (c) variations in selected parameters. A modeling framework, using the high-level statistical computing languages MATLAB and R, was developed to conduct the time series analysis in the time domain and phase space. In the time domain, the Hurst exponent, the information measure–Shannon’s entropy, and the time delay of temporal oscillations of nutrient and microbe concentrations were calculated. In the phase domain, we calculated a set of diagnostic criteria of deterministic chaos: global and local embedding dimensions, correlation dimension, information dimension, and a spectrum of Lyapunov exponents. The time series data are used to plot the phase space attractors to express the dependence between the system’s state parameters, i.e., microbe concentrations, and pseudo-phase space attractors, in which the attractor axes are used to compare the observations from a single time series, which are separated by the time delay. Like classical Lorenz or Rossler systems of equations, which generate a deterministic chaotic behavior for a certain range of input parameters, the developed mathematical model generates a deterministic chaotic behavior for a particular set of input parameters. Even a slight variation of the system’s input data might result in vastly different predictions of the temporal oscillations of the system. As the nutrient influx increases, the system exhibits a sharp transition from a steady state to deterministic chaotic to quasi-periodic and again to steady state behavior. For small changes in initial conditions, resulting attractors are bounded (contrary to that of a random system), i.e., may represent a ‘sustainable state’ (i.e., resilience) of the ecological system.

An approach to the estimation of distribution parameters with a priori specified constraints is developed. The Lorentz six-sector model is generated and used to analyze on the relationships between food, energy, and water resources, supply systems, epidemic and social situation. It is investigated how the successive transition of interconnected sectors into the mode of deterministic chaos affects the mode of system’s operation as a whole. A method of risk assessment for food safety and their management was created. [ABSTRACT FROM AUTHOR]

Simple Summary: Currently, knowledge on chaos has developed rapidly, and the link between cancer and "genomic chaos" seems obvious. Hopes for a deeper understanding of cancer, allowing cancer modeling, therefore relate to the meaning of the term "chaos". It has many meanings, however. Chaos theory and medicine are conceptually quite distant, requiring the comparison and agreement of terms. This article was written by three authors whose fields cover both medical problems and complex dynamic networks suitable for modeling cancer, including chaotic phenomena. The article provides, first of all, a coherent, common interpretative basis linking chaos with modeling tools, which should significantly facilitate teams of specialists from various fields to undertake specific work on simulating cancer-related phenomena. In the search of theoretical models describing cancer, one of promising directions is chaos. It is connected to ideas of "genome chaos" and "life on the edge of chaos", but they profoundly differ in the meaning of the term "chaos". To build any coherent models, notions used by both ideas should be firstly brought closer. The hypothesis "life on the edge of chaos" using deterministic chaos has been radically deepened developed in recent years by the discovery of half-chaos. This new view requires a deeper interpretation within the range of the cell and the organism. It has impacts on understanding "chaos" in the term "genome chaos". This study intends to present such an interpretation on the basis of which such searches will be easier and closer to intuition. We interpret genome chaos as deterministic chaos in a large module of half-chaotic network modeling the cell. We observed such chaotic modules in simulations of evolution controlled by weaker variant of natural selection. We also discuss differences between free and somatic cells in modeling their disturbance using half-chaotic networks. [ABSTRACT FROM AUTHOR]

The approach to estimating distribution parameters with a priori constraints has been developed. The Lorentz six-sector model has been constructed. Based on this model, the authors have studied the relationships between food, energy, water resources, supply systems, epidemic, and social situations. They also have investigated the impact of the successive transitions of interconnected sectors to the deterministic chaos regime on the operation mode of the whole system. The method of risk assessment for food security and risk management has been elaborated. [ABSTRACT FROM AUTHOR]

We establish the mixing property for a family of Lebesgue measure preserving toral maps composed of two piecewise linear shears, the first of which is non-monotonic. The maps serve as a basic model for the ‘stretching and folding’ action in laminar fluid mixing, in particular flows where boundary conditions give rise to non-monotonic flow profiles. The family can be viewed as the parameter space between two well-known systems, Arnold’s Cat Map and a map due to Cerbelli and Giona, both of which possess finite Markov partitions and straightforward to prove mixing properties. However, no such finite Markov partitions appear to exist for the present family, so establishing mixing properties requires a different approach. In particular, we follow a scheme of Katok and Strelcyn, proving strong mixing properties with respect to the Lebesgue measure on two open parameter spaces. Finally, we comment on the challenges in extending these mixing windows and the potential for using the same approach to prove mixing properties in similar systems. [ABSTRACT FROM AUTHOR]

This paper presents the generalized information system theory, which is enlarged into pure quantum systems using wave probability functions. The novelty of this approach is based on analogies with electrical circuits and quantum physics. Information power was chosen as the relevant parameter, which guarantees the balance of both components—information flow and information content. Next, the principles of quantum resonance between individual information components, which can lead to emergent behavior, are analyzed. For such a system, adding more and more probabilistic information elements can lead to better convergence of the whole to the resulting trajectory due to phase parameters. The paper also offers an original interpretation of information "source–recipient" or "resource–demand" models, including not yet implemented "unused resources" and "unmet demands". Finally, possible applications of these principles are shown in several examples from the quantum gyrator to the hypothetical possibility of explaining some properties of the consciousness. [ABSTRACT FROM AUTHOR]

In this paper, we utilize a numerical morphological simulation approach to study braiding channel dynamics under steady flow. We conduct 11 different runs in a laboratory configuration, with each initial condition featuring a unique small‐amplitude perturbation pattern of the bed topography. From these infinitesimal initial differences, braided channels evolve into patterns that are macroscopically different and unique to each run, thus showing sensitive dependence on the initial condition and hence a chaotic behavior. Leveraging the analogy of braiding and fluid turbulence put forward by Paola (1996), by introducing a Reynolds‐type decomposition of bed elevations into a reference and a fluctuating component we characterize braided channel dynamics through statistics of bed fluctuations. We observe that, over the simulation time frame, braiding is not statistically homogeneous, but is stationary in a spatially averaged sense. We prove that braiding is anisotropic and extract two distinct length scales associated with the correlations of bed fluctuations in the streamwise and cross‐channel direction. We observe that, under the time averaging window that yields stationary braiding, braiding shows hints of ergodic behavior, as the time statistics of one run converge to the ensemble statistics computed across all the other runs. Plain Language Summary: We study the dynamics of braided channels using numerical morphological simulations. We conduct 11 different simulation runs, each starting from a slightly different initial bed morphology. These imperceptible differences amplify and, in a rather short time, produce rivers that have visibly different channel patterns. Hence, braiding dynamics are chaotic and channel configuration changes happening over the medium or long term (say, in a few flood events in a real river) are unpredictable. We devise a statistical framework to analyze this chaotic behavior, leveraging the analogy between turbulent flows and braiding put forward by Paola (1996). We decompose the bed elevations of braided channels into a reference component and a fluctuating component, in a similar way as, in the study of turbulence, velocity is decomposed into a mean and a fluctuating component. With a statistical analysis of bed fluctuations, we find that braiding dynamics shows spatial variability (nonhomogeneous character), with bursts of intense evolution and areas of less activity; is unchanging in time (stationary); is direction‐dependent (anisotropic), as it possesses characteristics that depend on the direction of the analysis; shows hints of ergodic behavior, meaning that the statistics of one simulation well approximate the statistics computed across all the other simulations. Key Points: Numerically simulated braided channels show chaotic behavior (sensitive dependence on the initial conditions)We adapt the turbulence theory and utilize statistics of bed elevation fluctuations to characterize river braidingBraiding shows hints of ergodicity as time statistics of one run converge to ensemble statistics over multiple runs [ABSTRACT FROM AUTHOR]

Complex systems such as river flow do not obey the law of linearity but most often than not behave in a nonlinear manner. In this study, the dynamics of the River Niger discharge along two stations located in Nigeria were investigated using the methods of phase space reconstruction, correlation dimension and Lyapunov exponent. The analysis was carried out by using the daily data for three different periods — 1914–1939 (period 1), 1940–1964 (period 2), and 1966–1991 (period 3) spanning before and after the Kainji dam construction. Time delay embedding values at Baro and Lokoja for the three time periods were used to reconstruct the phase space of the river discharge at both locations. The embedding dimension represents the number of variables to completely describe the system. The presence of chaos was confirmed with positive values of Lyapunov exponents in both locations at the three different periods considered. The Lyapunov exponents at Baro were in the range 0.0014–0.0150 while the range at Lokoja was 0.007–0.0145. Significant correlation dimension values obtained at Baro (2.11–2.81) and Lokoja (2.28–4.51) are indicative of low dimensional chaos. A possible explanation for this could be the temporal sequence in the dam installation. The forecast horizon, the inverse of the largest Lyapunov exponents, gives a prediction boundary on a chaotic time series. In this study, the forecast horizon for both locations is expected to be in the range of 40–58 days into the future. [ABSTRACT FROM AUTHOR]

In the article are presented general principles of modeling vibrations in discrete structures formed in the form of special matrix forms of the Latin square (Sudoku type) are presented. The signs of structural and functional self-similarity for the matrix structures of standard Sudoku grids are formulated. It is shown that the structural principle can be interpreted as the implementation of the second iteration in the scale scaling algorithm characteristic of fractal objects. The signs of functional self-similarity of structures include the property of additive conservation of grid shapes to the requirements of Sudoku, which is formulated as a theorem. It is proved that the matrix sums of Sudoku constants and grids of arbitrary sizes, obtained taking into account the introduced cyclic ranking rule, will satisfy the three required Sudoku requirements. It is determined that by performing a given sequence of group shift operators, it is possible to establish a specific scenario for dynamically changing the state of a structure on a discrete time scale. It has been established that the evolution operators of linear-type group translations lead to matrix transformations of Sudoku grids from the set of equivalent ones (concerning the original ones), and the vortex-type group shifts operators to matrix transformations from many essentially different networks. The modes of harmonic, chaotic, and hybrid vibrations for structures of arbitrary size are considered. The requirements for transformations of the operators of the evolution of structures that provide the implementation of the considered modes are formulated. The results of modeling chaotic oscillatory processes by cycles of states of a discrete system that form similarities of attractor paths are analyzed. The principle of synchronization of chaotic states of matrix structures is established. The possibility of simulating the modes of beatings of oscillations in discrete cellular structures organized in the form of two-level matrix forms is substantiated. Specific examples show the results of simulating beatings of oscillations in cycles of changing states of a discrete system for two types of beats: similar to the result of a superposition of harmonic vibrations at multiple frequencies in the theory of radio signals, as well as noise-like beats.

In the search of theoretical models describing cancer, one of promising directions is chaos. It is connected to ideas of “genome chaos” and “life on the edge of chaos”, but they profoundly differ in the meaning of the term “chaos”. To build any coherent models, notions used by both ideas should be firstly brought closer. The hypothesis “life on the edge of chaos” using deterministic chaos has been radically deepened developed in recent years by the discovery of half-chaos. This new view requires a deeper interpretation within the range of the cell and the organism. It has impacts on understanding “chaos” in the term “genome chaos”. This study intends to present such an interpretation on the basis of which such searches will be easier and closer to intuition. We interpret genome chaos as deterministic chaos in a large module of half-chaotic network modeling the cell. We observed such chaotic modules in simulations of evolution controlled by weaker variant of natural selection. We also discuss differences between free and somatic cells in modeling their disturbance using half-chaotic networks.

Nowadays interest of the deterministic differential system of Lorentz equations is still primarily due to the problem of gas and fluid turbulence. Despite numerous existing systems for calculating turbulent flows, new modifications of already known models are constantly being investigated. In this paper we consider the effect of stochastic additive perturbations on the Lorentz convective turbulence model. To implement this and subsequent interpretation of the results obtained, a numerical simulation of the Lorentz system perturbed by adding a stochastic differential to its right side is carried out using the programming capabilities of the MATLAB programming environment.

This paper presents the generalized information system theory, which is enlarged into pure quantum systems using wave probability functions. The novelty of this approach is based on analogies with electrical circuits and quantum physics. Information power was chosen as the relevant parameter, which guarantees the balance of both components—information flow and information content. Next, the principles of quantum resonance between individual information components, which can lead to emergent behavior, are analyzed. For such a system, adding more and more probabilistic information elements can lead to better convergence of the whole to the resulting trajectory due to phase parameters. The paper also offers an original interpretation of information “source–recipient” or “resource–demand” models, including not yet implemented “unused resources” and “unmet demands”. Finally, possible applications of these principles are shown in several examples from the quantum gyrator to the hypothetical possibility of explaining some properties of the consciousness.

Background: In the present study, we analyzed stimulus-induced emotional changes in fetuses and mothers by using the Lyapunov exponent (hereafter λ1) and attractors. Objective: We also investigated how emotions change in fetuses and mothers over the course of pregnancy. Methodology: We measured heart sounds under five conditions: without external stimulation, while the mother's recorded voice was being heard, after the mother's recorded voice had finished being heard,while a stranger's recorded voice was being heard and after the stranger's recorded voice had finished being heard. We used one-way analysis of variance to analyze differences among the λ1 of fetal and maternal heart sounds under these five conditions. To observe the time-series characteristics of emotional responses, we calculated the moving average of the numbers of significantly different condition pairs. Results: Based on moving average results, we confirmed three emotional phases in fetuses and mothers. Fetal phase I was around 20 weeks of pregnancy, phase II was from around 21 weeks of pregnancy to around 26 weeks, and phase III was from around 27 weeks to before birth. Maternal phase I occurred at around 19 weeks of pregnancy, phase II was around 20 weeks to 26 weeks, and phase III was from around 27 weeks to before delivery. Fetal attractors did not change in response to strangers' vocal stimuli but did change dramatically in response to mothers' vocal stimuli from Phase I to Phase III, from nearly structureless attractors to chaotic attractors. Maternal attractors were consistently chaotic attractors from Phase I to Phase III and exhibited little change over the course of pregnancy. Changes in fetal and maternal attractors under different conditions suggested the occurrence of entrainment. Conclusions: The above results conceivably arise from the self-organization ability of fetuses and mothers. [ABSTRACT FROM AUTHOR]

This work is an adaptation of the Katok Closing Lemma for non-singular, endomorphisms on a compact Riemannian manifold. In special, it appears the idea of the proof of the hyperbolicity as well as the admissibility of local stable-unstable manifolds of a proper periodic point in adapted Katok Closing Lemma. [ABSTRACT FROM AUTHOR]