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This paper is motivated by study of long timescale variability of the climate system. We focus on a model of nonlinear behaviour that is used in climate modelling. This is the forced van der Pol oscillator, motivated by examination of the Pleistocene ice age oscillations forced by astronomical orbital variations. We discuss a forced van der Pol oscillator, following the analysis of Guckenheimer et al. for periodically cases. We use a geometric singular perturbation theory (GSPT) approach of Guckenheimer et al. to reduce to the dynamics of the return map and extend to their work to construct return maps for quasiperiodically forced cases. We note this return map can be noninvertible in various values to the parameters. [ABSTRACT FROM AUTHOR]

Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity conditions, also obey the Kirchhoff conditions, expressing the balance of tensions. In this paper, we propose a new look at quantum graphs as temporal networks, which means that the variable parametrizing the edges of a graph is interpreted as time, while each internal vertex is a branching point giving several different scenarios for the further trajectory of a process. Then Kirchhoff‐type conditions may also arise. Namely, they will be satisfied by such a trajectory of the process that is optimal with account of all the scenarios simultaneously. By employing the recent concept of global delay, we extend the problem of damping a first‐order control system with aftereffect, considered earlier only on an interval, to an arbitrary tree graph. The first means that the delay, imposed starting from the initial moment of time, associated with the root of the tree, propagates through all internal vertices. Bringing the system into the equilibrium and minimizing the energy functional with account of the anticipated probability of each scenario, we come to a variational problem. Then, we establish its equivalence to a self‐adjoint boundary value problem on the tree for some second‐order equations involving both the global delay and the global advance. The unique solvability of both problems is proved. We also illustrate that the interval case when the coefficients of the equation are discrete stochastic processes in discrete time can be viewed as the extension to a tree. [ABSTRACT FROM AUTHOR]

Action potential generation underlies some of the most consequential dynamical systems on Earth, from brains to hearts. It is therefore interesting to develop synthetic cell-free systems, based on the same molecular mechanisms, which may allow for the exploration of parameter regions and phenomena not attainable, or not apparent, in the live cell. We previously constructed such a synthetic system, based on biological components, which fires action potentials. We call it "Artificial Axon". The system is minimal in that it relies on a single ion channel species for its dynamics. Here we characterize the Artificial Axon as a dynamical system in time, using a simplified Hodgkin-Huxley model adapted to our experimental context. We construct a phase diagram in parameter space identifying regions corresponding to different temporal behavior, such as Action Potential (AP) trains, single shot APs, or damped oscillations. The main new result is the finding that our system with a single ion channel species, with inactivation, is dynamically equivalent to the system of two channel species without inactivation (the Morris-Lecar system), which exists in nature. We discuss the transitions and bifurcations occurring crossing phase boundaries in the phase diagram, and obtain criteria for the channels' properties necessary to obtain the desired dynamical behavior. In the second part of the paper we present new experimental results obtained with a system of two AAs connected by excitatory and/or inhibitory electronic "synapses". We discuss the feasibility of constructing an autonomous oscillator with this system. [ABSTRACT FROM AUTHOR]

This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring's elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system's efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions. [ABSTRACT FROM AUTHOR]

A physics-informed convolutional neural network (PICNN) is proposed to simulate two-phase flow in porous media with time-varying well controls. While most PICNNs in the existing literature worked on parameter-to-state mapping, our proposed network parameterizes the solutions with time-varying controls to establish a control-to-state regression. Firstly, a finite volume scheme is adopted to discretize flow equations and formulate a loss function that respects mass conservation laws. Neumann boundary conditions are seamlessly incorporated into the semi-discretized equations so no additional loss term is needed. The network architecture comprises two parallel U-Net structures, with network inputs being well controls and outputs being the system states (e.g., oil pressure and water saturation). To capture the time-dependent relationship between inputs and outputs, the network is well designed to mimic discretized state-space equations. We train the network progressively for every time step, enabling it to simultaneously predict oil pressure and water saturation at each timestep. After training the network for one timestep, we leverage transfer learning techniques to expedite the training process for a subsequent time step. The proposed model is used to simulate oil–water porous flow scenarios with varying reservoir model dimensionality, and aspects including computation efficiency and accuracy are compared against corresponding numerical approaches. The comparison with numerical methods demonstrates that a PICNN is highly efficient yet preserves decent accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we study the shrinking target problem regarding Q-Cantor series expansions of the formal Laurent series field. We provide the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the unit disk I. [ABSTRACT FROM AUTHOR]

This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions. [ABSTRACT FROM AUTHOR]

Let S be a discrete semigroup and let S S denote the collection of all functions f : S → S . If (P , ∘) is a subsemigroup of S S by composition operation, then P induces a natural topological dynamical system. In fact, (β S , { T f } f ∈ P) is a topological dynamical system, where β S is the Stone–Čech compactification of S, x ↦ T f (x) = f β (x) : β S → β S and f β is a unique continuous22 extension of f. In this paper, we concentrate on the dynamical system (β S , { T f } f ∈ P) , when S is an arbitrary discrete semigroup and P is a subsemigroup of S S and obtain some relations between subsets of S and subsystems of β S with respect to P. As a consequence, we prove that if (S , +) is an infinite commutative discrete semigroup and C is a finite partition of S, then for every finite number of arbitrary homomorphisms g 1 , ⋯ , g l : N → S , there exist an infinite subset B of the natural numbers and C ∈ C such that for every finite summations n 1 , ⋯ , n k of B there exists s ∈ S such that { s + g i (n 1) , s + g i (n 2) , ⋯ , s + g i (n k) } ⊆ C , ∀ i ∈ { 1 , ⋯ , l }. [ABSTRACT FROM AUTHOR]

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]

The Zika virus has been shown to infect glioblastoma stem cells via the membrane receptor α v β 5 , which is activated by the stem-specific transcription factor SOX2. Since the expression level of SOX2 is an important predictive marker for successful virotherapy, it is important to understand the fundamental mechanisms of the role of SOX2 in the dynamics of cancer stem cells and Zika viruses. In this paper, we develop a mathematical ODE model to investigate the effects of SOX2 expression levels on Zika virotherapy against glioblastoma stem cells. Our study aimed to identify the conditions under which SOX2 expression level, viral infection, and replication can reduce or eradicate the glioblastoma stem cells. Analytic work on the existence and stability conditions of equilibrium points with respect to the basic reproduction number are provided. Numerical results were in good agreement with analytic solutions. Our results show that critical threshold levels of both SOX2 and viral replication, which change the stability of equilibrium points through population dynamics such as transcritical and Hopf bifurcations, were observed. These critical thresholds provide the optimal conditions for SOX2 expression levels and viral bursting sizes to enhance therapeutic efficacy of Zika virotherapy against glioblastoma stem cells. This study provides critical insights into optimizing Zika virus-based treatment for glioblastoma by highlighting the essential role of SOX2 in viral infection and replication. [ABSTRACT FROM AUTHOR]

The paper aims to propose a two-step iterated method, which is derived from a regularized dynamical system of extragradient-type in terms of time discretizing, for solving an equilibrium problem. We prove that the iterative sequence generated by the method converges strongly to a solution of the equilibrium problem. Some numerical experiments are given to illustrate and compare the behavior of the new method with several other methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a forward-backward splitting dynamical system designed to address the inclusion problem of the form 0 ∈ G (x) + F(x), where G is a multi-valued operator and F is a single-valued operator in Hilbert spaces. The involved operators are required to satisfy a generalized monotonicity condition, which is less restrictive than standard monotone assumptions. Also, the maximality property does not impose on our involved operators. With mild conditions on parameters, we demonstrate the finite-time stability of the proposed dynamical system. We also present some applications to other optimization problems, such as Constrained Optimization Problems (COPs), Mixed Variational Inequalities (MVIs), and Variational Inequalities (VIs). [ABSTRACT FROM AUTHOR]

Mathematical models and numerical simulations are necessary to understand the dynamical behaviors of complex systems. The aim of this work is to investigate closed-form solutions for the ball–plate problem considering a system derived from an optimal control problem for ball–plate dynamics. The nonlinear properties of ball and plate control system are presented in this work. To semi-analytically solve this system, we explored a second-order nonlinear differential equation. Consequently, we obtained the approximate closed-form solutions by the Optimal Parametric Iteration Method (OPIM) using only one iteration. A comparison between the analytical and corresponding numerical procedures reflects the advantages of the first one. The accordance between the obtained results and the numerical ones highlights that the procedure used is accurate, effective, and good to implement in applications such as sliding mode control to the ball-and-plate problem. [ABSTRACT FROM AUTHOR]

In this paper, by using the bifurcation theory for dynamical system, we construct traveling wave solutions of a high-order nonlinear Schrödinger equation with a quintic nonlinearity. Firstly, based on wave variables, the equation is transformed into an ordinary differential equation. Then, under the parameter conditions, we obtain the Hamiltonian system and phase portraits. Finally, traveling wave solutions which contains solitary, periodic and kink wave solutions are constructed by integrating along the homoclinic or heteroclinic orbits. In addition, by choosing appropriate values to parameters, different types of structures of solutions can be displayed graphically. Moreover, the computational work and it's figures show that this technique is influential and efficient. [ABSTRACT FROM AUTHOR]

Abstract This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring’s elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system’s efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions.

Abstract This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions.

In this paper we introduce some weak dynamical properties by using subbases for the phase space. Among them, the notion of light chaos is the most significant. Several examples, which clarify the relationships between this kind of chaos and some classical notions, are given. Particular attention is also devoted to the connections between the dynamical properties of a system and the dynamical properties of the associated functional envelope. We show, among other things, that a continuous map f : X → X , where X is a metric space, is chaotic (in the sense of Devaney) if and only if the associated functional dynamical system is lightly chaotic.

We have examined collisional degenerate plasma composed of charged state of heavy positive ion and light positive as well as negative ion. Employing the reductive perturbation method, we derived the damped Korteweg-de Vries-Burgers (dKdV-B) equation and by using its standard solution we analyze the characteristics of the solitary-shock profile under varying parameters. Furthermore, with the application of planar dynamical systems bifurcation theory, the phase portraits have been analyzed. This dynamical system analysis allowed us to extract important information on the stability of these structures as represented by the dKdV-B equation.

The Riemann wave equation presents appealing nonlinear equations applicable in sea-water and tsunami wave propagation, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, and homogeneous stationary media. This study focuses on deriving soliton solutions in optics and exploring their physical properties. A wave transformation is used to convert a partial differential equation into an ordinary differential equation, from which soliton solutions are obtained using the generalized Riccati equation mapping approach. The solutions encompass various types of solitons, including bright, dark, periodic, and kink solitons. A comparison of solutions from this analytical method enhances the understanding of the nonlinear model’s behavior, with implications in plasma physics, fluid dynamics, optics, and communication technology. Additionally, 2D and 3D graphs illustrate the physical phenomena of the solutions using appropriate constant parameters. The qualitative analysis of the undisturbed planar system involves examining phase portraits in bifurcation theory, followed by introducing an outward force to induce disruption and reveal chaotic phenomena. Chaotic trajectories in the perturbed system are detected through various plots, including 3D, 2D, power spectrum, and chaotic attractor, alongside Lyapunov exponents. Stability analysis under different initial conditions is conducted, and sensitivity assessments are performed using the Runge–Kutta method. The findings are innovative and have not been previously explored for this system, highlighting the reliability, simplicity, and effectiveness of these techniques in analyzing nonlinear models in mathematical physics and engineering.

AbstractThe HIV/AIDS model was dynamically analyzed in this study. The model has seven compartments: the uneducated, the educated, the HIV-positive who take antiretroviral therapy (ART), the HIV-positive who do not take ART, people receiving ART treatment, people with AIDS who do not receive any treatment (full-blown AIDS), and the recovered. This model takes into account the analysis of the multiple interactions between the uneducated and the educated subpopulations, the HIV-positive who take and who do not take ART. The free-disease and endemic equilibrium points, as well as the basic reproduction number [Formula: see text] as a limit condition for infection-free and endemic occurrence, were produced by a mathematical analysis. The center-manifold hypothesis was used to prove that a backward bifurcation exists. The free-disease and endemic equilibrium points coexist when [Formula: see text] This means that HIV/AIDS is still spreading. A basic reproduction number below one is insufficient to constitute a free-disease condition. In order to determine essential parameters that significantly contribute to HIV/AIDS transmission, we computed sensitivity index values using a sensitivity analysis. The HIV/AIDS model and bifurcation parameter both identified the rate of HIV transmission from uneducated individuals to HIV-positive individuals who do not receive ART as the most crucial parameter. A numerical simulation supports the dynamical analysis.

When a finite group freely acts on a topological space, we can define its index and coindex. They roughly measure the size of the given action. We explore the interaction between this index theory and topological dynamics. Given a fixed-point free dynamical system, the set of p -periodic points admits a natural free action of ℤ / p ℤ for each prime number p. We are interested in the growth of its index and coindex as p → ∞. Our main result shows that there exists a fixed-point free dynamical system having the divergent coindex sequence. This solves a problem posed by M. Tsukamoto, M. Tsutaya and M. Yoshinaga, G -index, topological dynamics and marker property, preprint (2020), arXiv:2012.15372. [ABSTRACT FROM AUTHOR]

This study presents a hybrid modeling approach that integrates physics and machine learning for modeling multi-dimensional dynamics of a coupled nonlinear dynamical system. This approach leverages principles from classical mechanics, such as the Euler-Lagrange and Hamiltonian formalisms, to facilitate the process of learning from data. The hybrid model incorporates single or multiple artificial neural networks within a customized computational graph designed based on the physics of the problem. The customization minimizes the potential of violating the underlying physics and maximizes the efficiency of information flow within the model. The capabilities of this approach are investigated for various multidimensional modeling scenarios using different configurations of a coupled nonlinear dynamical system. It is demonstrated that, in addition to improving modeling criteria such as accuracy and consistency with physics, this approach provides additional modeling benefits. The hybrid model implements a physics-based architecture, enabling the direct alteration of both conservative and non-conservative components of the dynamics. This allows for an expansion in the model's input dimensionality and optimal allocation of input variable effects on conservative or non-conservative components of dynamics. [ABSTRACT FROM AUTHOR]

This article explores the examination of the widely employed zig-zag optical lattice model for cold bosonic atoms, which is commonly utilized to depict nonlinear wave in fluid mechanics and plasma physics. The focus is on obtaining soliton solutions in optics and investigating their physical properties. A wave transformation is initially applied to convert a partial differential equation (PDE) into an ordinary differential equation (ODE). Soliton solutions are subsequently obtained through the application of two distinct methods, namely the generalized logistic equation method and the Sardar sub-equation method. These solutions include bright, dark, combined dark-bright, chirped type solitons, bell-shaped, periodic, W-shape, and kink solitons. In this paper, the solutions derived from two analytical approaches were compared to enhance the understanding of the behavior of the discussed nonlinear model. The obtained solutions have significant implications across various fields such as plasma physics, fluid dynamics, optics, and communication technology. Furthermore, 3D and 2D graphs are generated to depict the physical phenomena of the derived solutions by assigning appropriate constant parameters. The qualitative evaluation of the undisturbed planar system involves the analysis of phase portraits within bifurcation theory. Subsequently, the introduction of an outward force is carried out to induce disruption, and chaotic phenomena are unveiled. The detection of chaotic trajectory in the perturbed system is achieved through 3D plots, 2D plots, time scale plots, and Lyapunov exponents. Furthermore, stability analysis of the examined model is addressed under distinct initial conditions. Finally, the sensitivity assessment of the model under consideration is carried out using the Runge–Kutta method. The results of this study are innovative and have not been previously investigated for the system under consideration. The results obtained underscore the reliability, simplicity, and effectiveness of these techniques in analyzing a variety of nonlinear models found in mathematical physics and engineering disciplines. [ABSTRACT FROM AUTHOR]

Using the Darboux transformation method, the general Lax equation is solved and a collection of new exact solutions together with one-soliton solutions, singular one-soliton solutions, periodic solutions, singular periodic solution, two-soliton solutions, singular two-soliton solutions, two-periodic solutions and singular two-periodic solutions is obtained. Using traveling wave transformation, the Lax equation is transfigured to a conservative dynamical system (CDS) of dimension four with three equilibrium points involving two parameters γ and v. The CDS has various quasi-periodic motions for fixed values of the parameters γ and v at different initial conditions. Furthermore, effects of the parameters γ and v are shown on the quasiperiodic motions of the CDS by means of phase sections and time series plots. This approach can be applied to a heterogeneity of nonlinear model equations or partial differential equations for describing their inherent nonlinear phenomena. [ABSTRACT FROM AUTHOR]

Abstract We study the dynamical assembly of weighted bipartite networks to understand the hidden mechanisms of pollination, expanding the Bianconi–Barabási model where nodes have intrinsic properties. Allowing for a non-linear interaction rate, which represents the seasonality of flowers and pollinators, our analysis reveals similarity of this extended Bianconi–Barabási model with field observations. While our current approach may not fully account for the diverse range of interaction accretion slopes observed in the real world, we regard it as an important step towards enriching theoretical models with biological realism.

In this paper, the fractional space-time nonlinear Chen-Lee-Liu equation has been considered using various methods. The investigation of the transition from periodic to quasi-periodic behavior has been conducted using a saddle-node bifurcation approach. The paper reports the conditions of multi-dimensional bifurcations of dynamical solutions. Additionally, a direct algebraic method has been used to calculate various 2D and 3D solitonic structures of the equation, and an analysis of their accuracy and effectiveness has been conducted. Furthermore, the Galilean transformation has been used to convert the equation into a planar dynamical system, which is further utilized to obtain bifurcations and chaotic structures. Chaotic structures of perturbed dynamical system are observed and detected through chaos detecting tools such as 2D-phase portrait, 3D-phase portrait, time series analysis, multistability and Lyapunov exponents over time. Further, sensitivity behavior for a range of initial conditions, both perturbed and unperturbed. The results suggest that the investigated equation exhibits a higher degree of multi-stability.

The fluid movement of an arm requires multiple spatiotemporal parameters to be set independently. Recent studies have argued that arm movements are generated by the collective dynamics of neurons in motor cortex. An untested prediction of this hypothesis is that independent parameters of movement must map to independent components of the neural dynamics. Using a task where three male monkeys made a sequence of reaching movements to randomly placed targets, we show that the spatial and temporal parameters of arm movements are independently encoded in the low-dimensional trajectories of population activity in motor cortex: each movement’s direction corresponds to a fixed neural trajectory through neural state space and its speed to how quickly that trajectory is traversed. Recurrent neural network models show that this coding allows independent control over the spatial and temporal parameters of movement by separate network parameters. Our results support a key prediction of the dynamical systems view of motor cortex, and also argue that not all parameters of movement are defined by different trajectories of population activity [ABSTRACT FROM AUTHOR]

Introduction: This study explores using Neural Ordinary Differential Equations (NODEs) to analyze hormone dynamics in the hypothalamicpituitary-adrenal (HPA) axis during Trier Social Stress Tests (TSST) to classify patients with Major Depressive Disorder (MDD). Methods: Data from TSST were used, measuring plasma ACTH and cortisol concentrations. NODE models replicated hormone changes without prior knowledge of the stressor. The derived vector fields from NODEs were input into a Convolutional Neural Network (CNN) for patient classification, validated through cross-validation (CV) procedures. Results: NODE models effectively captured system dynamics, embedding stress effects in the vector fields. The classification procedure yielded promising results, with the 1x1 CV achieving an AUROC score that correctly identified 83% of Atypical MDD patients and 53% of healthy controls. The 2x2 CV produced similar outcomes, supporting model robustness. Discussion: Our results demonstrate the potential of combining NODEs and CNNs to classify patients based on disease state, providing a preliminary step towards further research using the HPA axis stress response as an objective biomarker for MDD. [ABSTRACT FROM AUTHOR]

A new approach based on the analysis of the influence of a single perturbation is proposed as a test for the shadowing property for a broad class of dynamical systems (in particular, nonautonomous ones) under a variety of perturbations. Applications to several interesting cases are considered in detail. [ABSTRACT FROM AUTHOR]

Asymptotic and exact formulas for the growth functions of certain families of -valued coset groups are obtained, and the relationship between the theory of -valued groups and symbolic dynamics is described. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a third-order dynamical system for finding zeros of the sum of two generalized operators in a Hilbert space H . We establish the existence and uniqueness of the trajectories generated by the system under appropriate continuity conditions, and prove exponential convergence to the unique zero when the sum of the operators is strongly monotone. Additionally, we derive an explicit discretization of the dynamical system, which results in a forward–backward algorithm with double inertial effects and larger range of stepsize. We establish the linear convergence of the iterates to the unique solution using this algorithm. Furthermore, we provide convergence analysis for the class of strongly pseudo-monotone variational inequalities. We illustrate the effectiveness of our approach by applying it to structured optimization and pseudo-convex optimization problems. [ABSTRACT FROM AUTHOR]

Bifurcations of nonlinear and supernonlinear ion-acoustic waves (IAWs) are studied in an electron-ion plasmas with generalized (r, q)-distributed electrons. The IAWs are examined under the Zakharov–Kuznetsov (ZK) and modified ZK equations using the reductive perturbation technique. Transforming both the ZK and mZK equations into their corresponding dynamical systems, all possible phase spaces and potential energy functions are analyzed. The nonlinear periodic and solitary wave solutions are obtained under the ZK and mZK equations. A newly discovered supernonlinear wave, in particular, supernonlinear periodic wave under the modified ZK equation in electron-ion magnetized plasma with (r, q)-distributed electrons is reported for the first time in the literature. Nonlinear and supernonlinear wave solutions are shown under the influence of physical parameters. The proposed study contributes to new wave motions in slow solar wind streams and astrophysical plasmas. [ABSTRACT FROM AUTHOR]

In [A. Radhakrishnan, M. Belkin and C. Uhler, Overparameterized neural networks implement associative memory, Proc. Natl. Acad. Sci. USA 117(44) (2020) 27162–27170], the authors empirically show that autoencoders trained with standard SGD methods form basins of attraction around their training data. We consider network functions of width not exceeding the input dimension and prove that in this situation, such basins of attraction are bounded and their complement cannot have bounded components. Our conditions in these results are met in several experiments reported in [A. Radhakrishnan, M. Belkin and C. Uhler, Overparameterized neural networks implement associative memory, Proc. Natl. Acad. Sci. USA 117(44) (2020) 27162–27170] and we thus address a question posed therein. We also show that under some more restrictive conditions, the basins of attraction are path-connected. The necessity of the conditions in our results is demonstrated by means of examples. [ABSTRACT FROM AUTHOR]

We study the behavior of a three-dimensional dynamical system with respect to some set S given in 3-dimensional euclidean space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter a ∈ (0 , 1 / 2) , as for S it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that S is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and S for all a ∈ (0 , 1 / 2) . These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases a ∈ { 1 / 9 , 1 / 8 , 1 / 6 } of generalized Wallach spaces. [ABSTRACT FROM AUTHOR]

Modeling complex physical dynamics is a fundamental task in science and engineering. Traditional physics-based models are first-principled, explainable, and sampleefficient. However, they often rely on strong modeling assumptions and expensive numerical integration, requiring significant computational resources and domain expertise. While deep learning (DL) provides efficient alternatives for modeling complex dynamics, they require a large amount of labeled training data. Furthermore, its predictions may disobey the governing physical laws and are difficult to interpret. Physics-guided DL aims to integrate first-principled physical knowledge into datadriven methods. It has the best of both worlds and is well equipped to better solve scientific problems. Recently, this field has gained great progress and has drawn considerable interest across discipline Here, we introduce the framework of physics-guided DL with a special emphasis on learning dynamical systems. We describe the learning pipeline and categorize state-of-the-art methods under this framework. We also offer our perspectives on the open challenges and emerging opportunities. [ABSTRACT FROM AUTHOR]

We have examined collisional degenerate plasma composed of charged state of heavy positive ion and light positive as wel as negative ion. Employing the reductive perturbation method, we derived the damped Korteweg-de Vries-Burgers (dKdV-B) equation and by using its standard solution we analyze the characteristics of the solitary-shock profile under varying parameters. Furthermore, with the application of planar dynamical systems bifurcation theory, the phase portraits have been analyzed. This dynamical system analysis allowed us to extract important information on the stability of these structures as represented by the dKdV-B equation. [ABSTRACT FROM AUTHOR]

In this paper we introduce some weak dynamical properties by using subbases for the phase space. Among them, the notion of light chaos is the most significant. Several examples, which clarify the relationships between this kind of chaos and some classical notions, are given. Particular attention is also devoted to the connections between the dynamical properties of a system and the dynamical properties of the associated functional envelope. We show, among other things, that a continuous map f: X → X, where X is a metric space, is chaotic (in the sense of Devaney) if and only if the associated functional dynamical system is lightly chaotic. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive exploration of shock waves in a dusty plasma, incorporating the dynamics of inertial dust and ions with Maxwellian distributed electrons. By deriving the Korteweg-de Vries-Burgers (KdV-B) equation and by using its standard solution we analyze the characteristics of the shock front under varying parameters. Furthermore, we conduct a phase plane analysis to elucidate the system's behavior and dynamics by tuning in the parameters that provide various types of perturbation in the system. The investigation sheds light on the intricate behavior of shock waves in dusty plasmas and contributes valuable insights to the understanding of plasma dynamics. [ABSTRACT FROM AUTHOR]

This work offers a (semi-analytical) solution for a second-order nonlinear differential equation associated to the dynamical Kermack–McKendrick system. The approximate closed-form solutions are obtained by means of the Optimal Homotopy Asymptotic Method (OHAM) using only one iteration. These solutions represent the ε -approximate OHAM solutions. The advantages of this analytical procedure are reflected by comparison between the analytical solutions, numerical results, and corresponding iterative solutions (via a known iterative method). The obtained results are in a good agreement with the exact parametric solutions and corresponding numerical results, and they highlight that our procedure is effective, accurate, and useful for implementation in applications. [ABSTRACT FROM AUTHOR]

In this paper, the fractional space-time nonlinear Chen-Lee-Liu equation has been considered using various methods. The investigation of the transition from periodic to quasi-periodic behavior has been conducted using a saddle-node bifurcation approach. The paper reports the conditions of multi-dimensional bifurcations of dynamical solutions. Additionally, a direct algebraic method has been used to calculate various 2D and 3D solitonic structures of the equation, and an analysis of their accuracy and effectiveness has been conducted. Furthermore, the Galilean transformation has been used to convert the equation into a planar dynamical system, which is further utilized to obtain bifurcations and chaotic structures. Chaotic structures of perturbed dynamical system are observed and detected through chaos detecting tools such as 2D-phase portrait, 3D-phase portrait, time series analysis, multistability and Lyapunov exponents over time. Further, sensitivity behavior for a range of initial conditions, both perturbed and unperturbed. The results suggest that the investigated equation exhibits a higher degree of multi-stability. [ABSTRACT FROM AUTHOR]

Pesticides are the most common method used to eliminate pests, including aphids. Pesticides are the most common method used to eliminate pests, including aphids. Nonetheless, numerous farmers incorporate ladybugs into their pest management strategies as they serve as natural predators of aphids. By integrating these methods, farmers aim to achieve optimal outcomes in mitigating the detrimental effects of aphids on the agricultural sector. In this paper, the dynamics of interactions between aphids and ladybugs, including the impact of pesticides on aphid mortality, are represented using a system of nonlinear differential equations. This study treats the parameter representing aphid mortality caused by pesticides as a fuzzy number to account for variations in resistance levels. Additionally, the model incorporates four parameters that depict the interaction between aphids and ladybugs beyond considering the effect of pesticides. The parameters include the proportion of aphids consumed by ladybugs, the proportion of aphids capable of evading ladybugs, and the growth rates of both aphids and ladybugs. The triangular form is chosen to depict the fuzzy membership function because it reflects the resistance of aphids when pesticides are applied excessively. The dynamic model, incorporating a fuzzy parameter, is transformed into discrete-time models using the Non-Standard Finite Difference (NSFD) method for simulation purposes. The simulation outcomes align with the analysis findings, indicating a potential equilibrium between the populations of aphids and ladybugs. Various examinations on the impact of fuzzy pesticide parameters on the growth of aphids and ladybugs are provided. The findings demonstrate that pesticide application can substantially decrease the aphid population and can be tailored based on the interplay between aphids and ladybugs. Moreover, pesticide usage can be diminished with heightened ladybug growth and predation rates, thereby minimizing the occurrence of resistant aphids and enhancing the effectiveness of pesticide application. [ABSTRACT FROM AUTHOR]

This article deals with an inverse problem of determining two space-time-dependent sources in an integerfractional mobile-immobile two-region solute transport system by additional Dirichlet-Neumann data. The unique existence of a solution to the forward problem is obtained by the method of Laplace transform, and a dynamical system connecting the known data with the unknown sources is established by variational method and boundary homogenization. The dynamical system is discretized to a linear system at a given time in a homogenous polynomial space, and the sources are reconstructed by alternative iterations and Tikhonov regularization. Numerical examples are presented to illustrate the validity of the inversion algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the cosmic dynamics of varying vacuum models where the dark matter interacts with the vacuum energy. We consider the homogeneous and isotropic spacetime with spatial curvature and apply the dynamical system technique to the varying vacuum models by specifying the form of energy exchange rate (Q) between the dark energy and dark matter. Further, we utilize the cosmographic parameters and statefinder parameters in the terms of dynamical variables of the cosmological dynamical system to explore the cosmic dynamics of the universe. The Milne universe solution exists as a consequence of spatially curved geometry in the model. The models also yield radiation- matter- and dark energy-dominated phases in order and thus explain the late-time accelerated expansion of the universe. The strength of interaction terms will affect the existence of cosmological phases in the model but there will always be an attractor corresponding to the accelerating universe. We numerically solve the system to illustrate the evolution of cosmological quantities and dynamical variables in the models. The role of curvature is visible during the transition from the decelerated phase into the accelerating phase. The numerical solutions affirm that the cosmological parameters in the models are consistent with their corresponding observational values. [ABSTRACT FROM AUTHOR]

In this paper, we study some Krasnoselskii-Mann type dynamical systems in solving fixed point problems. The first one can be regarded as a continuous version of the Krasnoselskii-Mann iterations. We prove that the solution of this dynamical system converges weakly to a fixed point of the involving mapping. Next, we focus our attention on a regularized Krasnoselskii-Mann type dynamical system. Besides proving existence and uniqueness of strong global solutions, we show that the generated trajectories converge strongly to a unique solution of a variational inequality over the fixed point set. Also, we provide a convergence rate analysis for the regularized dynamical system. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of local parameter identifiability for discrete dynamical systems with discrete infinite-dimensional parameter. Our results are related to the so-called conditional local parameter identifiability. In this case, one introduces a special class P of possible perturbations of the selected parameter P 0 and finds conditions on observations of trajectories under which all parameters P ∈ P close to P 0 coincide with P 0 . We consider discrete dynamical systems for which the trajectories are observed at points k = 1 , 2 , ⋯ or at a countable set of points 0 < t 1 < t 2 < ⋯ . The case of a linearly perturbed diffeomorphism in a neighborhood of a hyperbolic set is also considered. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel algorithm to optimize the design of Reservoir Computing using Cellular Automata models for time series applications. Besides selecting the models' hyperparameters, the proposed algorithm particularly solves the open problem of Linear Cellular Automaton rule selection. The selection method pre-selects only a few promising candidate rules out of an exponentially growing rule space. When applied to relevant benchmark datasets, the selected rules achieve low errors, with the best rules being among the top 5% of the overall rule space. The algorithm was developed based on mathematical analysis of Linear Cellular Automaton properties and is backed by almost one million experiments, adding up to a computational runtime of nearly one year. Comparisons to other state-of-the-art time series models show that the proposed Reservoir Computing using Cellular Automata models have lower computational complexity and, at the same time, achieve lower errors. Hence, our approach reduces the time needed for training and hyperparameter optimization by up to several orders of magnitude. [ABSTRACT FROM AUTHOR]

In this paper, the group structure of the -adic ball and sphere are studied. The dynamical system of isometry defined on invariant sphere is investigated. We define the binary operations and on a ball and sphere, respectively, and prove that these sets are compact topological abelian group with respect to the operations. Then we show that any two balls (spheres) with positive radius are isomorphic as groups. We prove that the Haar measure introduced in is also a Haar measure on an arbitrary balls and spheres. We study the dynamical system generated by the isometry defined on a sphere and show that the trajectory of any initial point that is not a fixed point is not convergent. We study ergodicity of this -adic dynamical system with respect to normalized Haar measure reduced on the sphere. For we prove that the dynamical systems are not ergodic. But for under some conditions the dynamical system may be ergodic. [ABSTRACT FROM AUTHOR]

Action potential generation underlies some of the most consequential dynamical systems on Earth, from brains to hearts. It is therefore interesting to develop synthetic cell-free systems, based on the same molecular mechanisms, which may allow for the exploration of parameter regions and phenomena not attainable, or not apparent, in the live cell. We previously constructed such a synthetic system, based on biological components, which fires action potentials. We call it “Artificial Axon”. The system is minimal in that it relies on a single ion channel species for its dynamics. Here we characterize the Artificial Axon as a dynamical system in time, using a simplified Hodgkin-Huxley model adapted to our experimental context. We construct a phase diagram in parameter space identifying regions corresponding to different temporal behavior, such as Action Potential (AP) trains, single shot APs, or damped oscillations. The main new result is the finding that our system with a single ion channel species, with inactivation, is dynamically equivalent to the system of two channel species without inactivation (the Morris-Lecar system), which exists in nature. We discuss the transitions and bifurcations occurring crossing phase boundaries in the phase diagram, and obtain criteria for the channels’ properties necessary to obtain the desired dynamical behavior. In the second part of the paper we present new experimental results obtained with a system of two AAs connected by excitatory and/or inhibitory electronic “synapses”. We discuss the feasibility of constructing an autonomous oscillator with this system.

This study focuses on finding effective solutions to a mathematical equation known as the Lonngren wave equation. The solutions from the Lonngren wave equation can be used to evaluate electromagnetic signals in cable lines and sound waves in stochastic systems. The main equation is transformed into an ordinary differential equation by utilizing a suitable wave transformation, allowing for the exploration of mathematical models by using the modified Khater technique to detect the exact solution of a solitary wave. We use the provided method to derive the trigonometric, rational, and hyperbolic solutions. To illustrate the model’s physical behavior, we also present graphical plots of selected solutions to illustrate the physical behavior of the model. By choosing appropriate values for arbitrary factors, the visual representation enhances the understanding of the dynamical system. Furthermore, the system is transformed into a planar dynamical system, and phase portrait analysis is conducted. Additionally, the sensitivity analysis of the dynamical system confirms that slight changes in the initial conditions will have minimal impact on the stability of the solution. The existence of chaotic dynamics in the Lonngren wave equation is explored by introducing a perturbed term in the dynamical system. Two and three-dimensional phase portraits will be used to demonstrate these chaotic behaviors.