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This literature review explores the role of attractor neural networks (ANNs) in modeling psychological processes in artificial and biological systems. By synthesizing research from dynamical systems theory, psychology, and computational neuroscience, the review provides an overview of the current understanding of ANN function in memory formation, memory reinforcement, retrieval, and forgetting. Key mathematical foundations of ANNs, including dynamical systems theory and energy functions, are discussed to explain the behavior and stability of these networks. The review also examines empirical applications of ANNs in cognitive processes such as semantic memory and episodic recall, as well as highlighting the hippocampus' role in pattern separation and completion. The review addresses challenges like catastrophic forgetting and noise effects on memory retrieval. By identifying gaps between theoretical models and empirical findings, it highlights the interdisciplinary nature of ANN research and suggests future areas for exploration.

Keywords: Attractors; homogenization; reaction-diffusion system; nonlinear equations; weak convergence; randomly perforated domains; random porous medium; topological space; compactness.GraphBy Kuanysh A. Bekmaganbetov; Gregory A. Chechkin and Vladimir V. ChepyzhovReported by Author; Author; Author [Extracted from the article]

In this paper, I review our current understanding of the applicability of hydrodynamics to modeling the quark–gluon plasma (QGP), focusing on the question of hydrodynamization/thermalization of the QGP and the anisotropic hydrodynamics (aHydro) far-from-equilibrium hydrodynamic framework. I discuss the existence of far-from-equilibrium hydrodynamic attractors and methods for determining attractors within different hydrodynamical frameworks. I also discuss the determination of attractors from exact solutions to the Boltzmann equation in relaxation time approximation and effective kinetic field theory applied to quantum chromodynamics. I then present comparisons of the kinetic attractors with the attractors obtained in standard second-viscous hydrodynamics frameworks and aHydro. I demonstrate that, due to the resummation of terms to all orders in the inverse Reynolds number, the aHydro framework can describe both the weak- and strong-interaction limits. I then review the phenomenological application of aHydro to relativistic heavy-ion collisions using both quasiparticle aHydro and second-order viscous aHydro. The phenomenological results indicate that aHydro provides a controlled extension of dissipative relativistic hydrodynamics to the early-time far-from-equilibrium stage of heavy-ion collisions. This allows one to better describe the data and to extract the temperature dependence of transport coefficients at much higher temperatures than linearized second-order viscous hydrodynamics. [ABSTRACT FROM AUTHOR]

In this article we study the long-time dynamics of the dynamical system generated by the Ericksen–Leslie model. More precisely, we discretize the Ericksen–Leslie equations in time using the implicit Euler scheme, and with the aid of the discrete Gronwall lemmas we prove that the scheme is uniformly bounded. Moreover, using the theory of the multi-valued attractors we prove in a particular case the convergence of the global attractors generated by the numerical scheme to the global attractor of the continuous system as the time-step approaches zero. [ABSTRACT FROM AUTHOR]

Making accurate and reliable predictions for monthly runoff in data scarce basins is still a major challenge. In this study, a new model, the CL-NDM, is developed by combining Convolutional Neural Network-Long Short-term Memory (CNN-LSTM) and a nonlinear dynamic model. The CL-NDM can overcome the deficiency of observed data by fusing spatial and temporal dependencies in runoff sequences at different stations. First, phase space reconstruction is used to enlarge the dimensions of the runoff sequences and reconstruct the attractors of the runoff sequences. Then, the CNN-LSTM is employed to construct the mapping between non-delay and delay attractors. Finally, the prediction set of the target variable is obtained by embedding multiple times. The CL-NDM is performed for monthly runoff prediction at eleven hydrological stations in the Weihe River, China. Compared with the CNN, LSTM and CNN-LSTM models, which require a large amount of training samples, the CL-NDM behaves much better, especially in situations with small training sample sizes. The maximum increase in R is 74%, and the maximum NSE is as large as 0.8. The maximum improvement in RMSE and MAPE is 53% and 88%, respectively. The CL-NDM has stronger ability to capture peak value while LSTM, CNN-LSTM and CNN models show obvious time lag in the prediction of peak point. The improved nonlinear dynamical model may provide a valuable method for runoff prediction in data-scarce regions. [ABSTRACT FROM AUTHOR]

We present some results on the absence of a wide class of invariant measures for dynamical systems possessing attractors. We then consider a generalization of the classical nonholonomic Suslov problem which shows how previous investigations of existence of invariant measures for nonholonomic systems should necessarily be extended beyond the class of measures with strictly positive densities if one wishes to determine dynamical obstructions to the presence of attractors. [ABSTRACT FROM AUTHOR]

The relevance of the topic presented in the article is determined primarily by the current foreign policy situation, namely the hybrid war unleashed by the collective West against the Russian Federation, a significant component of which are the so-called nonkinetic wars: informational, organizational, mental, cognitive, etc. In the article, the author focuses specifically on cognitive wars, which are the least studied in the social sciences. Hence, the main purpose of the article is to consider possible methodological foundations for the analysis of cognitive wars in the context of the high turbulence of the world order and the growing dynamics of digital transformations in the economy, politics, and social spheres. As the core of such a methodology, the author suggests considering social synergetics, the key postulates of which are openness, non-linearity of self-development, and the nonequilibrium nature of systems of various natures, which are most consistent with the current context of public relations. Special attention is paid to the role of the system of spiritual and moral values, or the social value basis, in ensuring the stability of the political system and effective counteraction to cognitive aggression. The author revealed its main functions, such as ensuring communicative effectiveness, overcoming the social atomization of modern society, and ensuring the cognitive stability of society by increasing resistance to cognitive and mental aggression.

Gain a comprehensive understanding of the environmental and tourism aspects of the formation of the territorial structure of the region. The authors advocate for an ecological framework approach to pinpoint key natural areas crucial for establishing a tourist framework. In the southern low-mountain region of Altai Krai, mountain ranges are integral to the ecological framework, shaping its diverse landscape. This territory harbors specially protected natural areas (SPNA) that comprise natural heritage sites and tourist attractions. Different elements of the ecological framework, including nature reserves and protected zones within natural parks, require varying levels of protection. The study emphasizes that the main river valleys serve as focal points of the tourist framework. Through the development of schematic maps, the research identifies nine distinct tourist destinations in the foothill-low-mountain zone of Altai Krai, each characterized by unique ecological and tourist attributes.

We study the stability properties and the long-term dynamics of chimeras in systems of globally coupled maps. In particular, we establish a formula for the transverse Lyapunov exponent of the states of the system containing synchronized units. We use this formula to present numerical evidence of attracting chimeras having chaotic dynamics as well as periodic behaviors. We also show that, at least for polynomial local maps, attracting periodic cycles tend to belong to cluster spaces, and, more generally, limit sets of chimera orbits have zero Lebesgue measure for strong coupling regimes. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic dynamics for a class of semilinear parabolic problems with Dirichlet boundary conditions in domains with a small hole of size proportional to a positive parameter ε. In other words, we prove that the family of attractors behaves continuously as ε → 0. We will also provide the convergence rates in terms of the parameter. [ABSTRACT FROM AUTHOR]

Sea surface temperature (SST), with its complex and dynamic behavior, is a major driver of ocean–atmosphere interactions. The purpose of this study is to investigate the behavior of SST and its prediction using a chaotic approach. Average mutual information (AMI) and Cao methods were used to reconstruct the phase space. The Lyapunov exponent and correlation dimension were used to investigate chaos. The Lyapunov exponent index was used to predict SST with a 5-year average prediction horizon using the local prediction method between 2023 and 2027. The results showed a 3-month delay time for the Pacific and Antarctic Oceans, and a 2-month delay time for the Atlantic, Indian, and Arctic Oceans. The optimal embedding dimension for all oceans is between 6 and 7. Our analysis reveals that the dynamics of SST in all oceans exhibit varying degrees of chaos, as indicated by the correlation dimension. The local prediction method achieves relatively accurate short-term SST predictions due to the clustering of SST points around specific attractors in the phase space. However, in the long term, the accuracy of this method decreases as the points in the phase space of SST can spread randomly. The model performance ranking with a Percent Mean Relative Absolute Error shows that the Indian Ocean has the best performance compared to other oceans, while the Atlantic, Pacific, and Antarctic and Arctic Oceans are in the next ranks. This study contributes to understanding the dynamics of SST and has practical value for use in the development of climate models. [ABSTRACT FROM AUTHOR]

By employing mathematical procedures specific to non-linear dynamics, we devised a method for analyzing all the stages of an epileptic crisis. It can be seen that a specific attractor (reconstructed in the states space by time delay method) can be associated with each stage of the brain dynamics: normal functioning and epileptic crisis (pre-crisis, epileptic seizure, post-crisis). Moreover, since for each attractor a Hurst coefficient can be determined, we can state that each stage of an epileptic seizure can be characterized either by calculating the specific Hurst coefficient, or by calculating its fractal dimension. [ABSTRACT FROM AUTHOR]

To curb the spread of fake news, I propose an alternative to the current trend of implementing coercive measures. This approach would preserve freedom of speech while neutralizing the social impact of fake news. The proposal relies on creating an environment to naturally sequestrate fake news within quite small networks of people. I illustrate the process using a stylized model of opinion dynamics. In particular, I explore the effect of a simultaneous activation of prejudice tie breaking and contrarian behavior, on the spread of fake news. The results show that indeed most pieces of fake news do not propagate beyond quite small groups of people and thus pose no global threat. However, some peculiar sets of parameters are found to boost fake news so that it "naturally" invades an entire community with no resistance, even if initially shared by only a handful of agents. These findings identify the modifications of the parameters required to reverse the boosting effect into a sequestration effect by an appropriate reshaping of the social geometry of the opinion dynamics landscape. Then, all fake news items become "naturally" trapped inside limited networks of people. No prohibition is required. The next significant challenge is implementing this groundbreaking scheme within social media. [ABSTRACT FROM AUTHOR]

In this article, we study a class of differential quasi-variational–hemivariational inequalities involving time delays. We establish new systems and prove the solvability and the existence of decay solutions. Moreover, we are concerned with long-time behavior of solutions by showing the existence of a compact global attractor to m-semiflow associated with delay differential quasi-variational–hemivariational inequalities. An application to the contact problems driven by dynamic systems is discussed to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

A new method for obtaining lower bounds for the dimension of attractors for the Navier–Stokes equations is presented, which does not use Kolmogorov flows. By applying this method, exact estimates of the dimension are obtained for the case of equations on a plane with Ekman damping. Similar estimates were previously known only for the case of periodic boundary conditions. In addition, similar lower bounds are obtained for the classical Navier–Stokes system in a two-dimensional bounded domain with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

Early shifts in network activity constitute an important first step in the transition from a healthy towards a disease state. These network states are often characterized by neuronal hyperactivity – putatively the start of activity-induced neurodegeneration – and are associated with subtle, yet discernable, behavioral dysregulations. Local network changes can also occur in regions and brain circuits that are not associated with the clinical symptoms of later-stage disease. It can be argued that from the viewpoint of the neural network, early changes in network activity are not necessarily governed by the aim of preserving long-term functionality, but rather by the attempt to achieve a state that is temporarily stable (a notion we refer to as the selfish network). Early neural network state transitions follow attractor-like dynamics and may represent a distinct entity that could be leveraged for new therapies and for fostering resilience. Neuronal networks possess the ability to regulate their activity states in response to disruptions. How and when neuronal networks turn from physiological into pathological states, leading to the manifestation of neuropsychiatric disorders, remains largely unknown. Here, we propose that neuronal networks intrinsically maintain network stability even at the cost of neuronal loss. Despite the new stable state being potentially maladaptive, neural networks may not reverse back to states associated with better long-term outcomes. These maladaptive states are often associated with hyperactive neurons, marking the starting point for activity-dependent neurodegeneration. Transitions between network states may occur rapidly, and in discrete steps rather than continuously, particularly in neurodegenerative disorders. The self-stabilizing, metastable, and noncontinuous characteristics of these network states can be mathematically described as attractors. Maladaptive attractors may represent a distinct pathophysiological entity that could serve as a target for new therapies and for fostering resilience. [ABSTRACT FROM AUTHOR]

Based on the existence of pullback attractors for the non-autonomous stochastic plate equations with multiplicative noise and nonlinear damping defined in the entire space Rn by Xiaobin Yao in [15], in the paper, we further investigate the upper semicontinuity of pullback attractors for the problem. [ABSTRACT FROM AUTHOR]

Gain a comprehensive understanding of the environmental and tourism aspects of the formation of the territorial structure of the region. The authors advocate for an ecological framework approach to pinpoint key natural areas crucial for establishing a tourist framework. In the southern low-mountain region of Altai Krai, mountain ranges are integral to the ecological framework, shaping its diverse landscape. This territory harbors specially protected natural areas (SPNA) that comprise natural heritage sites and tourist attractions. Different elements of the ecological framework, including nature reserves and protected zones within natural parks, require varying levels of protection. The study emphasizes that the main river valleys serve as focal points of the tourist framework. Through the development of schematic maps, the research identifies nine distinct tourist destinations in the foothill-low-mountain zone of Altai Krai, each characterized by unique ecological and tourist attributes. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained. [ABSTRACT FROM AUTHOR]

We present a dynamical approach to the study of unordered, attracting manifolds of retrotone maps commonly known as carrying simplices. Our approach is novel in that it uses the radial representation of unordered manifolds over the probability simplex coupled with distances between these manifolds measured by way of the Harnack and Hausdorff metrics. We establish Kuratowski convergence of radial representations of unordered manifolds to a unique function which then provides the locally Lipschitz radial representation of the carrying simplex. [ABSTRACT FROM AUTHOR]

Our aim in this article is to study the well-posedness of the generalized logarithmic nonlinear Cahn-Hilliard equation with regularization and proliferation terms. We are interested in studying the asymptotic behavior, in terms of finite-dimensional attractors, of the dynamical system associated with the problem and majorate the rate of convergence between the solutions of the Cahn-Hilliard equation and the regularized one. Additionally, we present some further regularity results and subsequently prove a strict separation property of the solution. Finally, we provide some numerical simulations to compare the solution with and without the regularization term, and more. [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]

The Ginzburg-Landau equation with rapidly oscillating terms in the equation and boundary conditions in a perforated domain was considered. Proof was given that the trajectory attractors of this equation converge weakly to the trajectory attractors of the homogenized Ginzburg-Landau equation. To do this, we use the approach from the articles and monographs of V.V. Chepyzhov and M.I. Vishik about trajectory attractors of evolutionary equations, and we also use homogenization methods that appeared at the end of the 20th century. First, we use asymptotic methods to construct asymptotics formally, and then we justify the form of the main terms of the asymptotic series using functional analysis and integral estimates. By defining the corresponding auxiliary function spaces with weak topology, we derive a limit (homogenized) equation and prove the existence of a trajectory attractor for this equation. Then, we formulate the main theorems and prove them by using auxiliary lemmas. We prove that the trajectory attractors of this equation tend in a weak sense to the trajectory attractors of the homogenized Ginzburg-Landau equation in the subcritical case, and they disappear in the supercritical case.

This paper contains new results about the well-posedness and the asymptotic dynamics of solutions for a general abstract coupled system that arises in connection with thermoelastic Balakrishnan–Taylor beam models with fractional operators. [ABSTRACT FROM AUTHOR]

Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining the 'global stability' of a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline 'resilience'. Unfortunately, there are many different variants and explanations of resilience, and often, the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalised to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain nonautonomous perturbations to demonstrate how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-dimensional system of coupled nonlinear wave equations, one of which is subject to fractional Laplacian damping, the linear part of the system is based on Alabau-Boussouira et al. (J Evol Equ 2(2):127–150, 2002. https://doi.org/10.1007/s00028-002-8083-0). In addition, we consider the thermal effect according to Fourier's Law acting on the system. We prove that the (two-parameters) dynamical system associates the solution of the system is quasi-stable and gradient, which implies the existence of a (two-parameters) family of compact global attractors. A result of regularity is also proven for the attractors, as well as showing that the family of attractors is upper semicontinuous, on the set of parameters. [ABSTRACT FROM AUTHOR]

We consider the Ginzburg–Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg–Landau equation with a 'strange term' (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor. [ABSTRACT FROM AUTHOR]

In the theory of gene networks, the mathematical apparatus that uses dynamical systems is fruitfully used. The same is true for the theory of neural networks. In both cases, the purpose of the simulation is to study the properties of phase space, as well as the types and the properties of attractors. The paper compares both models, notes their similarities and considers a number of illustrative examples. A local analysis is carried out in the vicinity of critical points and the necessary formulas are derived. [ABSTRACT FROM AUTHOR]

This case study identified the subjective change in a patient diagnosed with Borderline Personality Disorder (BPD) by analyzing the dynamic patterns that emerged during discursive interaction with her therapist during a successful long-term psychotherapy. A qualitative analysis was conducted by applying the Model of Analysis of Discursive Positioning in Psychotherapy (MAPP), tracking voices and the personal positions of the patient and therapist in all sessions. Subsequently, dynamic patterns were identified and the hypothetical attractors were defined (i.e., the most stable patterns in the interaction of voices) using the Space State Grid (SSG) technique. Five sessions (representing the initial, middle and final stages) were selected to describe the trajectory of the patient's subjective change. These sessions highlight the presence of different attractors and depict the intersubjective interaction that developed during the process. The results suggest a process of subjective transformation from a monological to a dialogical dimension, a change supported by therapeutic interaction based on propositional and reflective discursive aspects; a transition from a state of dissociation of the patient to a reorganization of her subjectivity. The most relevant characteristics of this process as an emergent quality of psychotherapy and its relation to the patient's positive outcomes are discussed. [ABSTRACT FROM AUTHOR]

В статье рассматриваются языковые средства проявления авторской лингвокреативности в названиях немецких фудтраков. Устанавливается, что в целях оригинальности, креативности и нестандартности номинаторы используют целенаправленное нарушение орфографических, грамматических, стилистических норм немецкого и английского языков. Авторы названий фудтраков используют языковую игру, продуктами которой в некоторых случаях являются названия с затемненной и/или пейоративной семантикой, названия, демонстрирующие «семантическое противоречие», названия с отрицательной коннотацией и лингвотоксины. Исследование может быть полезно специалистам, занимающимся теорией и практикой немецкого языка, а также всем интересующимся.

We describe a strategy for the development of a rational approach of neoplastic disease therapy based on the demonstration that scale-free networks are susceptible to specific attacks directed against its connective hubs. This strategy involves the (i) selection of up-regulated hubs of connectivity in the tumors interactome, (ii) drug repurposing of these hubs, (iii) RNA silencing of non-druggable hubs, (iv) in vitro hub validation, (v) tumor-on-a-chip, (vi) in vivo validation, and (vii) clinical trial. Hubs are protein targets that are assessed as targets for rational therapy of cancer in the context of personalized oncology. We confirmed the existence of a negative correlation between malignant cell aggressivity and the target number needed for specific drugs or RNA interference (RNAi) to maximize the benefit to the patient's overall survival. Interestingly, we found that some additional proteins not generally targeted by drug treatments might justify the addition of inhibitors designed against them in order to improve therapeutic outcomes. However, many proteins are not druggable, or the available pharmacopeia for these targets is limited, which justifies a therapy based on encapsulated RNAi. [ABSTRACT FROM AUTHOR]

This paper considers a periodic boundary value problem for a nonlinear partial differential equation with a deviating spatial variable. It is called the nonlocal erosion equation and was proposed as a model for the formation of dynamic patterns on the semiconductor surface. As is demonstrated below, the formation of a spatially inhomogeneous relief is a self-organization process. An inhomogeneous relief appears due to local bifurcations in the neighborhood of homogeneous equilibria when they change their stability. The analysis of this problem is based on modern methods of the theory of infinite-dimensional dynamic systems, including such branches as the theory of invariant manifolds, the apparatus of normal forms, and asymptotic methods for studying dynamic systems. [ABSTRACT FROM AUTHOR]

Summary: Gene regulatory networks (GRN) are dynamic models in time and space. These models are used to predict diseases and in drugs research. GRN models are discrete, and Boolean graphs can efficiently represent them. However, GRN algorithms explore a large solution space with high computational complexity. This work proposes efficient FPGA‐based accelerators to implement two GRN algorithms: attractor computation and Derrida plot. Nevertheless, FPGA accelerator design and deployment are still a challenge. This work presents an accelerator design framework for AWS Amazon FPGA cloud. The framework simplifies the software (SW) and hardware (HW) generation for GRN accelerators. The user provides a high‐level model for the Boolean GRN, and our tool automatically creates AWS‐ready‐to‐deploy software and hardware components. For the attractor and the Derrida plot computation, the proposed FPGA accelerators are on average 44×$$ 44\times $$ and 18×$$ 18\times $$ faster than a V100 GPU. [ABSTRACT FROM AUTHOR]

In the paper we consider a problem for complex Ginzburg–Landau equations in a medium with locally periodic small obstacles. It is assumed that the obstacle surface can have different conductivity coefficients. We prove that the trajectory attractors of this system converge in a certain weak topology to the trajectory attractors of the homogenized Ginzburg–Landau equations with an additional potential (in the critical case), without an additional potential (in the subcritical case) in the medium without obstacles, or disappear (in the supercritical case). [ABSTRACT FROM AUTHOR]

We study the three‐dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. First, we study the existence, uniqueness, and regularity of global weak solutions. Then, we investigate the asymptotic behavior of weak solutions via attractors. Since the system might not always have regular solutions, we use a new framework developed by Cheskidov and Lu which is called evolutionary system to obtain various attractors and their properties. [ABSTRACT FROM AUTHOR]

We study the classical Hénon family f a , b : (x , y) ↦ (1 - a x 2 + y , b x) , 0 < a < 2 , 0 < b < 1 , and prove that given an integer k ≥ 1 , there is a set of parameters E k of positive two-dimensional Lebesgue measure so that f a , b , for (a , b) ∈ E k , has at least k attractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73–169, 1991). A corresponding statement also holds for the Hénon-like families of Mora and Viana (Acta Math 171:1–71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original Hénon maps. The final main result of the paper is the existence, within the classical Hénon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors. [ABSTRACT FROM AUTHOR]

Boolean networks are popular tools for the exploration of qualitative dynamical properties of biological systems. Several dynamical interpretations have been proposed based on the same logical structure that captures the interactions between Boolean components. They reproduce, in different degrees, the behaviours emerging in more quantitative models. In particular, regulatory conflicts can prevent the standard asynchronous dynamics from reproducing some trajectories that might be expected upon inspection of more detailed models. We introduce and study the class of networks with linear cuts, where linear components—intermediates with a single regulator and a single target—eliminate the aforementioned regulatory conflicts. The interaction graph of a Boolean network admits a linear cut when a linear component occurs in each cycle and in each path from components with multiple targets to components with multiple regulators. Under this structural condition the attractors are in one-to-one correspondence with the minimal trap spaces, and the reachability of attractors can also be easily characterized. Linear cuts provide the base for a new interpretation of the Boolean semantics that captures all behaviours of multi-valued refinements with regulatory thresholds that are uniquely defined for each interaction, and contribute a new approach for the investigation of behaviour of logical models. [ABSTRACT FROM AUTHOR]

Extensive research has previously been carried out on the existence of strange attractors in 2D piecewise linear maps, including the renowned Lozi map. However, the rigorous analysis of strange attractors in 2D nonlinear maps remains an underdeveloped area of study. In this paper, we introduce a 2D map with a single piecewise-smooth nonlinear function that is suitable for analytic studies of its strange attractor. This 2D piecewise-smooth nonlinear map represents a broad range of chaotic maps, including a hybrid Lozi-Hénon map and the Belykh map. To prove the existence of a strange attractor in the 2D map, we (i) construct its trapping region that contains all limit sets and (ii) demonstrate that the invariant set's trajectories have negative and positive Lyapunov exponents. We develop an invariant cone approach to establish the latter property, which involves constructing expanding and contracting cones as bounds for the eigenvectors of the variational equations along the chaotic trajectories. We apply our approach to analyze the chaotic hybrid Lozi-Hénon map, the original Lozi map, and the Belykh map. [ABSTRACT FROM AUTHOR]

Change in music therapy is often modeled linearly. In linear analysis, change is represented as the difference between the scores recorded before and after treatment, where changes in the input are proportional to the output. However, changes in complex systems are often not linear and depend on time. We propose Dynamic Systems Theory (DST) as a means to overcome the shortcomings of linear analysis and enrich the study of change in music therapy. This article aims to introduce and critically discuss the applications of DST in music therapy, focusing on its theoretical and methodological aspects. DST offers a meta-framework to model nonlinear change in music therapy, considering time as continuous. The application of DST can further enhance the understanding of how music therapy works, the shape of the change, and how the relevant therapeutic processes within music therapy support therapeutic change. An introduction to DST theory is provided along with its history, implications, assessment methods, statistical analyses, mathematical modeling, and implementation examples in music therapy research. [ABSTRACT FROM AUTHOR]

The research paradigm of 21st century linguistics links the study of discourse with the use of language as a public practice. The subject of the study is linguistic features of different levels, characterizing the phenomenon of cancel culture. The new discourse practice or speech reality functions thanks to the media and the Internet in various areas of modern speech — in political and business communication, in professional and everyday life. The purpose of the study is to consider cancel culture as a linguistic object that reflects the speech life of modern society. The relevance of the study is determined by a modern request for regulation and control of speech behavior in the public sphere within the framework of constructive criticism. The article implements the following research tasks: to describe the linguistic essence of the cancel culture and characterize the latter as a phenomenon characteristic of the communicative behavior of representatives of the linguocultural community. The linguistic method of discourse analysis is used as the main tool for analyzing texts with functioning examples of lexical units representing the cancellation action, which makes it possible to identify features of the communication and evaluation activities of speakers. As a result, new language facts in the system of linguistic coordinates are localized and a possible vector for the study of aggressive communication in the language-environment category is determined. It allows for an expansion of the field of scientific understanding of the modern speech environment. The author concludes that the cancel culture as a new discourse practice has a thematic and semiotic dominant in the form of destructive criticism. Research of new discourse practice on the identified attractors of criticism and aggressive communication is set as a research perspective.

In this paper the Ginzburg-Landau equation is considered in locally periodic porous medium, with rapidly oscillating terms in the equation and boundary conditions. It is proved that the trajectory attractors of this equation converge in a weak sense to the trajectory attractors of the limit Ginzburg-Landau equation with an additional potential term. For this aim we use an approach from the papers and monographs of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of evolution equations. Also we apply homogenization methods appeared at the end of the XX-th century. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) equation and prove the existence of trajectory attractors for this equation. Then we formulate the main theorem and prove it with the help of axillary lemmas.