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In this paper, we consider some free boundary problems that arise in the mathematical modeling of cancer and its treatment. In these models, a parameter η is introduced to represent either the killing rate of the cancer cells by the T cells or a drug called checkpoint inhibitor. We study the long-time behavior of the tumor as the parameter η varies. [ABSTRACT FROM AUTHOR]

We derive, at leading order in the thin-shell parameter, a consistent set of nonlinear governing equations for the dynamics of flows confined to a zonal cloud band such as those on Jupiter, in a thin layer near the top of the planetary troposphere. Some exact solutions are provided in the material (Lagrangian) framework. The explicit specification of the individual particle paths enables a detailed study of these flows that model oscillations superimposed on a mean current. This approach is applied to Jupiter's Great Red Spot and to the filamentary zonal flow at its southern boundary. Significance Statement: We propose a new approach to the study of some flow patterns visible in zonal cloud bands on Jupiter. Motivated by observations showing that the dominant motions in the cloud bands on Jupiter are zonal and rotational, we provide some exact solutions to the governing equations for the leading-order dynamics. These solutions model rotating particle paths interacting with a straight-line flow. The approach offers detailed insight into basic features of the flow, highlighting the interplay between density variations and wind forcing. The exact solutions presented here are a useful starting point for a perturbation analysis. Current advances in computing methods enhance the feasibility of numerical simulations of perturbed flows that are designed to capture a wider range of effects, whose relevance can be ascertained by comparison with the exact solutions presented in this paper. [ABSTRACT FROM AUTHOR]

Non-autonomous differential equations exhibit highly intricate dynamics, and various concepts have been introduced to describe their qualitative behavior. In general, it is rare to obtain time dependent invariant compact attracting sets when time goes to plus infinity. Moreover, there are only a few papers in the literature that explore the geometric structure of such sets. In this paper we investigate the long time behaviour of cooperative $ n $-dimensional non-autonomous Lotka–Volterra systems in population dynamics. We provide sufficient conditions for the existence of a globally stable (forward in time) entire solution in which one species becomes extinct, or where all species except one become extinct. Furthermore, we obtain the precise geometrical structure of the non-autonomous forward attractor in one, two, and three dimensions by establishing heteroclinic connections between the globally stable solution and the semi-stable solutions in cases of species permanence and extinction. We believe that understanding time-dependent forward attractors paves the way for a comprehensive analysis of both transient and long-term behavior in non-autonomous phenomena. [ABSTRACT FROM AUTHOR]

The article explores geodesics on polyhedra, particularly focusing on simple closed geodesics and their properties. It discusses the classification of geodesics on regular polyhedra, the relationship between geodesics and billiards, and the uniqueness of disphenoids in having arbitrarily long geodesics. The text also addresses the existence of simple closed geodesics on tetrahedra and the properties of geodesics on nonconvex polyhedra, including the construction of long closed geodesics using seven cubes. Open questions are posed regarding geodesics on nonconvex polyhedra in different spaces and the minimal number of vertices needed for long geodesics. The author acknowledges the contributions of an anonymous referee for their feedback. [Extracted from the article]

The article in the Journal of Fluid Mechanics explores the prediction of thermoacoustic instabilities using van der Pol oscillators derived from physics-informed neural networks (PINNs). The study aims to understand and control oscillations in combustion systems to prevent overheating and engine failures. By integrating PINNs with a modelled Rijke tube system, the study demonstrates an effective approach to predict nonlinear characteristics and develop control strategies. The findings offer valuable insights into the dynamics of thermoacoustic instabilities and potential control methods, providing a comprehensive analysis of fluid mechanics and combustion instability control. [Extracted from the article]

The article explores the dynamics of heavy inertial particles in non-axisymmetric vortices, focusing on the clustering and chaotic motion of particles in flow fields generated by elliptical vortices like the Kirchhoff and Kida vortices. The study reveals unexpected behaviors, such as particle clustering around co-rotating attractors and chaotic tracer transport in the presence of external straining. By analyzing the interplay between particle inertia and straining, the research uncovers mechanisms that can suppress chaotic transport in certain conditions. The document provides valuable insights into the complex behavior of particles in non-axisymmetric vortices, shedding light on their dispersion and capture rates in coherent structures. [Extracted from the article]

Continuing from Paper I [Ohsawa and Yagasaki, J. Math. Phys. 65, 102706 (2024)], we study semiclassical perturbations of single-degree-of-freedom analytic Hamiltonian systems and provide a sufficient condition for its meromorphic nonintegrability such that the first integrals depend on the small parameter meromorphically. Our approach is based on a generalization due to Ayoul and Zung of the Morales-Ramis theory, which enables us to show the meromorphic nonintegrability of dynamical systems by using the differential Galois theory. We remark that standard systems of Hagedorn and Heller for the semiclassical Gaussian wave packet dynamics are analytically integrable as well as the corresponding classical systems. We illustrate our theory for a bounded potential. [ABSTRACT FROM AUTHOR]

This article is a preface to a special issue of the Journal of Difference Equations and Applications dedicated to Saber Elaydi on his 80th birthday. Elaydi is recognized for his significant contributions to the field of difference equations and discrete dynamical systems. He has had a successful academic career, serving as a professor and chair at Trinity University in Texas. Elaydi has also been a valuable teacher and mentor, inspiring students and attracting disciples from around the world. He has written influential books on difference equations and has been involved in organizing conferences and founding journals in the field. Elaydi's research interests cover a wide range of topics, including mathematical biology, neuroscience, and economics. He has made substantial contributions to these areas and has been recognized with prestigious awards. The article expresses appreciation for Elaydi's achievements and wishes him continued success in his mathematical and personal life. [Extracted from the article]

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

The adaptive spectral Koopman (ASK) method was introduced to numerically solve autonomous dynamical systems that laid the foundation for numerous applications across different fields in science and engineering. Although ASK achieves high accuracy, it is computationally more expensive for multidimensional systems compared with conventional time integration schemes like Runge--Kutta. In this work, we combine the sparse grid and ASK to accelerate the computation for multidimensional systems. This sparse-grid-based ASK (SASK) method uses the Smolyak structure to construct multidimensional collocation points as well as associated polynomials that are used to approximate eigenfunctions of the Koopman operator of the system. In this way, the number of collocation points is reduced compared with using the tensor product rule. We demonstrate that SASK can be used to solve ordinary differential equations (ODEs) and partial differential equations (PDEs) based on their semidiscrete forms. Numerical experiments are illustrated to compare the performance of SASK and state-of-the-art ODE solvers. [ABSTRACT FROM AUTHOR]

The detection of limit cycles of differential equations poses a challenge due to the type of the nonlinear system, the regime of interest, and the broader context of applicable models. Consequently, attempts to solve Hilbert's sixteenth problem on the maximum number of limit cycles of polynomial differential equations have been uniformly unsuccessful due to failing results and their lack of consistency. Here, the answer to this problem is finally obtained through information geometry, in which the Riemannian metrical structure of the parameter space of differential equations is investigated with the aid of the Fisher information metric and its scalar curvature R. We find that the total number of divergences of | R | to infinity provides the maximum number of limit cycles of differential equations. Additionally, we demonstrate that real polynomial systems of degree n ≥ 2 have the maximum number of 2 (n − 1) (4 (n − 1) − 2) limit cycles. The research findings highlight the effectiveness of geometric methods in analyzing complex systems and offer valuable insights across information theory, applied mathematics, and nonlinear dynamics. These insights may pave the way for advancements in differential equations, presenting exciting opportunities for future developments. [ABSTRACT FROM AUTHOR]

This paper deals with a two-person zero-sum differential game for a dynamical system described by a Caputo fractional differential equation of order α ∈ (0 , 1) and a Bolza cost functional. The differential game is associated to the Cauchy problem for the path-dependent Hamilton–Jacobi–Bellman–Isaacs equation with so-called fractional coinvariant derivatives of order α and the corresponding right-end boundary condition. A notion of a viscosity solution of the Cauchy problem is introduced, and the value functional of the differential game is characterized as a unique viscosity solution of this problem. [ABSTRACT FROM AUTHOR]

This paper presents a control strategy synthesis method for dynamical systems with differential constraints, emphasizing the prioritization of specific rules. Special attention is given to scenarios where not all rules can be simultaneously satisfied to complete a given task, necessitating decisions on the extent to which each rule is satisfied, including which rules must be upheld or disregarded. We propose a learning-based Model Predictive Control (MPC) method designed to address these challenges. Our approach integrates a learning method with a traditional control scheme, enabling the controller to emulate human expert behavior. Rules are represented as Signal Temporal Logic (STL) formulas. A robustness margin, quantifying the degree of rule satisfaction, is learned from expert demonstrations using a Conditional Variational Autoencoder (CVAE). This learned margin is then applied in the MPC process to guide the prioritization or exclusion of rules. In a track driving simulation, our method demonstrates the ability to generate behavior resembling that of human experts and effectively manage rule-based dilemmas. [ABSTRACT FROM AUTHOR]

Due to the noise interference and the weak characterization ability of the fault vibration signal of rotation vector (RV) reducer crankshaft bearing, it is difficult to obtain satisfactory results for the available fault diagnosis methods. For that, this paper proposes a variable‐speed‐condition fault diagnosis method with WSO‐VMD and ResNet‐SWIN. A signal reconstruction method with WSO‐VMD was carried out, Firstly, the performance of VMD algorithm is improved by using war strategy optimization algorithm to select parameters adaptively. Then the signal is reconstructed considering the fault characteristic frequency, so as to realize the noise reduction of the signal. By using the residual network module and attention mechanism to replace the first stage of the original SWIN model, a novel ResNet‐SWIN fault diagnosis model is established to enhance the feature extraction ability for the weak signal. The experiments with the constant‐operating‐condition and the variable‐operating‐condition are carried out to verify the effectiveness of the proposed method. The results show that, whether at variable‐speed or constant‐speed conditions, WSO algorithm has been proven to be the fastest convergence speed compared with WOA, SSA, and NGO optimization algorithms, and by the signal reconstruction with WSO‐VMD, the variance evaluation indicator of the reconstructed signal has 36%, 21%, 46%, and 40%, respectively. ResNet‐SWIN model has achieved the optimal diagnosis accuracy compared with SWIN, VIT, and CNN‐SVM models in both variable‐speed and constant‐speed conditions. [ABSTRACT FROM AUTHOR]

We use the inverse spectral problem method for integrating the nonlinear complex modified Korteweg-de Vries equation (cmKdV) with additional terms in the class of periodic infinite-gap functions. Also, we deduce the evolution of the spectral data of the periodic Dirac operator whose coefficient is a solution to cmKdV. We prove that the Cauchy problem is solvable for an infinite system of Dubrovin differential equations in the class of six times continuously differentiable periodic infinite-gap functions. Moreover, we establish the solvability of the Cauchy problem for cmKdV with additional terms in the class of six times continuously differentiable periodic infinite-gap functions. [ABSTRACT FROM AUTHOR]

Majorization–minimization (MM) is a versatile optimization technique that operates on surrogate functions satisfying tangency and domination conditions. Our focus is on differentiable optimization using inexact MM with quadratic surrogates, which amounts to approximately solving a sequence of symmetric positive definite systems. We begin by investigating the convergence properties of this process, from subconvergence to R-linear convergence, with emphasis on tame objectives. Then we provide a numerically stable implementation based on truncated conjugate gradient. Applications to multidimensional scaling and regularized inversion are discussed and illustrated through numerical experiments on graph layout and X-ray tomography. In the end, quadratic MM not only offers solid guarantees of convergence and stability, but is robust to the choice of its control parameters. [ABSTRACT FROM AUTHOR]

In this paper, we present an example of piecewise linear systems with infinitely many crossing limit cycles defined in two zones separated by a piecewise linear curve with countable corners. Then we prove that under piecewise linear perturbations, the perturbed system can have infinitely many limit cycles, or exactly l limit cycles for any given nonnegative integer l. [ABSTRACT FROM AUTHOR]

The slow–fast Leslie–Gower model with piecewise-smooth Holling type-I functional response and weak Allee effect is studied in this paper. It is shown that the model undergoes singular Hopf bifurcation and nonsmooth Hopf bifurcation as the parameters vary. The theoretical analysis implies that the predator's food quality and Allee effect play an important role and lead to richer dynamical phenomena such as the globally stable equilibria, canard explosion phenomenon, a hyperbolically stable relaxation oscillation cycle enclosing almost two canard cycles with different stabilities and so on. Moreover, the predator and prey will coexist as multiple steady states or periodic oscillations for different positive initial populations and positive parameter values. Finally, we present some numerical simulations to illustrate the theoretical analysis such as the existence of one, two or three limit cycles. [ABSTRACT FROM AUTHOR]

Integrating logical reasoning and machine learning by approximating logical inference with differentiable operators is a widely used technique in the field of Neuro-Symbolic Learning. However, some differentiable operators could introduce significant biases during backpropagation, which can degrade the performance of Neuro-Symbolic systems. In this paper, we demonstrate that the loss functions derived from fuzzy logic operators commonly exhibit a bias, referred to as Implication Bias. To mitigate this bias, we propose a simple yet efficient method to transform the biased loss functions into Reduced Implication-bias Logic Loss (RILL). Empirical studies demonstrate that RILL outperforms the biased logic loss functions, especially when the knowledge base is incomplete or the supervised training data is insufficient. [ABSTRACT FROM AUTHOR]

We use a multiscale symbolic approach to study the complex dynamics of temporal lobe refractory epilepsy employing high-resolution intracranial electroencephalogram (iEEG). We consider the basal and preictal phases and meticulously analyze the dynamics across frequency bands, focusing on high-frequency oscillations up to 240 Hz. Our results reveal significant periodicities and critical time scales within neural dynamics across frequency bands. By bandpass filtering neural signals into delta, theta, alpha, beta, gamma, and ripple high-frequency bands (HFO), each associated with specific neural processes, we examine the distinct nonlinear dynamics. Our method introduces a reliable approach to pinpoint intrinsic time lag scales τ within frequency bands of the basal and preictal signals, which are crucial for the study of refractory epilepsy. Using metrics such as permutation entropy (H), Fisher information (F), and complexity (C), we explore nonlinear patterns within iEEG signals. We reveal the intrinsic τ max that maximize complexity within each frequency band, unveiling the nonlinear subtle patterns of the temporal structures within the basal and preictal signal. Examining the H × F and C × F values allows us to identify differences in the delta band and a band between 200 and 220 Hz (HFO 6) when comparing basal and preictal signals. Differences in Fisher information in the delta and HFO 6 bands before seizures highlight their role in capturing important system dynamics. This offers new perspectives on the intricate relationship between delta oscillations and HFO waves in patients with focal epilepsy, highlighting the importance of these patterns and their potential as biomarkers. [ABSTRACT FROM AUTHOR]

In this paper we deal with the problem of strong stabilisation for a class of distributed semilinear systems by means of a nonlinear feedback control. The systems under consideration evolve in a Hilbert space and present periodic time-varying state delay. We first give a proof for the existence and uniqueness of the solution for the considered systems. Then, under a null controllability condition, we establish the stabilisation result and provide an explicit optimal decay rate estimate. The particular case of such systems for which the linear part generates a differentiable semigroup is also investigated. Some illustrating applications to hyperbolic and parabolic equations are displayed. Finally, a conclusion and some perspectives are given. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the dynamic bifurcation of the general Brusselator model when the order of reaction is $ p \in (1,\infty) $. We verify that the Turing instability occurs above the critical control number and obtain a rigorous formula for the bifurcated stable patterns. We define a constant $ s_N $ that gives a criterion for the continuous transition. We obtain continuous transitions for $ s_N>0 $, but jump transitions for $ s_N<0 $. By using this criterion, we prove mathematically that higher-molecular reactions are rarely observed. We also provide some numerical results that illustrate the main results. [ABSTRACT FROM AUTHOR]

It is well-known that any dynamical system governed by a differential equation containing time derivatives higher than second order unavoidably holds unbounded energy solutions, dubbed ghosts that appear in the Hamiltonian. They correspond to instabilities displayed at the classical level. In this study, we show first that it is possible to construct in classical mechanics, characterized by non-standard Lagrangians and nonlocal-in-time kinetic energy, higher-order derivative theories that avoid the Ostrogradsky ghost. Second, we show that in the realm of quantum mechanics, higher-order discretized energies emerge in the theory which may lead to extended quantum mechanics formalism. The problem of quantum wells has been treated where we showed that negative energy and negative action, complexified energies and complexified actions may emerge in our formalism. We have also discussed the quantum motion of a neutron in the Earth's gravitational field. It was observed that within the realm of higher-order non-standard Lagrangians, the quantum energies of the neutrons are higher than the energy levels obtained in the basic formalism. [ABSTRACT FROM AUTHOR]

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for discrete or continuous dynamical systems globally defined in R 3 . We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of R 3 that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of S 3 which arise naturally when considering toroidal sets. [ABSTRACT FROM AUTHOR]

We study periodic trajectories of nonlinear dynamical systems considered as models of the simplest molecular repressilator. In the phase portraits of these systems, we find hidden attractors and nonlocal oscillations. The cases of nonuniqueness of cycles in these portraits are described as well. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics (including shadowing property, expansiveness, topological stability and entropy) of several types of upper semi-continuous set-valued maps are mainly considered from differentiable dynamical systems points of view. It is shown that (1) if f is a hyperbolic endomor-phism then for each ε> 0 there exists a C1-neighborhood U of f such that the induced set-valued map F f , U has the ε-shadowing property, and moreover, if f is an expanding endomorphism then there exists a C1-neighborhood U of f such that the induced set-valued map F f , U has the Lipschitz shadowing property; (2) when a set-valued map F is generated by finite expanding endomorphisms, it has the shadowing property, and moreover, if the collection of the generators has no coincidence point then F is expansive and hence is topologically stable; (3) if f is an expanding endomorphism then for each ε> 0 there exists a C1-neighborhood U of f such that h (F f , U , ε) = h (f) (4) when F is generated by finite expanding endomorphisms with no coincidence point, the entropy formula of F is given. Furthermore, the dynamics of the set-valued maps based on discontinuous maps on the interval are also considered. [ABSTRACT FROM AUTHOR]

In this study, we extended the applicability of a derivative-free algorithm to encompass the solution of operators that may be either differentiable or non-differentiable. Conditions weaker than the ones in earlier studies are employed for the convergence analysis. The earlier results considered assumptions up to the existence of the ninth order derivative of the main operator, even though there are no derivatives in the algorithm, and the Taylor series on the finite Euclidian space restricts the applicability of the algorithm. Moreover, the previous results could not be used for non-differentiable equations, although the algorithm could converge. The new local result used only conditions on the divided difference in the algorithm to show the convergence. Moreover, the more challenging semi-local convergence that had not previously been studied was considered using majorizing sequences. The paper included results on the upper bounds of the error estimates and domains where there was only one solution for the equation. The methodology of this paper is applicable to other algorithms using inverses and in the setting of a Banach space. Numerical examples further validate our approach. [ABSTRACT FROM AUTHOR]

The problem of a stabilisation system synthesis for the motion of a control object along a given spatial trajectory is considered. The complexity of the problem is that the preset trajectory is defined in the state subspace and not in time. This paper describes a stabilisation system synthesis for motion along a trajectory specified in time and along a trajectory specified in the form of a manifold in a state space. In order to construct a stabilisation system, it is necessary to determine a distance between an object and the given trajectory at each moment in time. For trajectories that are not given in time, the determination of this distance can be ambiguous. An object may be exactly on a trajectory but at a different time. This paper proposes some approaches to solve the problem. One of the approaches is to transform a given trajectory in a state subspace into a trajectory given in time. A description of a universal method to perform this transformation is presented. In order to solve the synthesis problem automatically, without having to analyse the mathematical model of the control object, it is suggested that machine learning control by symbolic regression is used. In computational experiments, examples of stabilisation system syntheses for quadcopter motion along a given spatial trajectory are presented. [ABSTRACT FROM AUTHOR]

In order to reveal the general escape mechanism of non-hyperbolic chaotic invariant sets under both weak noise limit and finite noise intensity, the simplest example of Hénon map, which represents the stretching and folding of the phase space, is taken to study the escape mechanism under two kinds of global bifurcation: the fractal boundary crisis and the attractor contact crisis. In this paper, we revealed the general exit mechanism by analyzing the escape paths derived from the shooting method and Monte Carlo simulation. Finally, to further demonstrate universality, an example of high-dimensional differential dynamical system, namely the transient chaos Duffing oscillator, is examined, which underscores the main idea of this paper analyzing specific deterministic structures not only enhances the comprehensibility of the escape process, but also allows for predictions of general escape behavior under weak noise intensity. [ABSTRACT FROM AUTHOR]

This paper presents experimental switching between two attractors in the swinging bell. In the considered yoke–bell–clapper system, two coexisting solutions appear. In the first one, we observe a single impact between the bell and the clapper per one period of motion, and in the second solution, no impacts occur—no sound is produced. Based on the time-dependent stability margin method, we numerically detect parts of the trajectories where the system is most prone to perturbations. Using this knowledge, we experimentally investigate switching between attractors by applying the perturbation to the clapper. We show that we can easily enforce the change of attractor by properly timing the perturbation. The results prove that, based on the results from the time-dependent stability margin numerical method, we are able to effectively alter the wrong operation of the bell (lack of impact) to the correct operation (solution with impact). The analysis is conducted on the real-world mechanical system rather than paradigmatic examples. Therefore, it contributes to the subject of multistability and nonlinearity in engineering design. Novel, recently developed methods for analyzing multistable systems are successfully employed during the investigation. The paper shows that a complex phenomenon of multistability observed in the system, which is considered simple and undemanding from an engineering design point of view. [ABSTRACT FROM AUTHOR]

The present paper describes, in a theoretical fashion, a variational approach to formulate fourth-order dynamical systems on differentiable manifolds on the basis of the Hamilton–d'Alembert principle of analytic mechanics. The discussed approach relies on the introduction of a Lagrangian function that depends on the kinetic energy and the covariant acceleration energy, as well as a potential energy function that accounts for conservative forces. In addition, the present paper introduces the notion of Rayleigh differential form to account for non-conservative forces. The corresponding fourth-order equation of motion is derived, and an interpretation of the obtained terms is provided from a system and control theoretic viewpoint. A specific form of the Rayleigh differential form is introduced, which yields non-conservative forcing terms assimilable to linear friction and jerk-type friction. The general theoretical discussion is complemented by a brief excursus about the numerical simulation of the introduced differential model. [ABSTRACT FROM AUTHOR]

The main goal of the current work is to study dynamical behavior and dispersive optical solitons for the fractional Ginzburg–Landau equation in optical fibers. Starting with the traveling wave transformations, the fractional Ginzburg–Landau model is converted into an equivalent ordinary differential traveling wave system. Then, the Hamiltonian function and orbits phase portraits of this system are found. Here, we derived explicit fractional periodic wave solutions, bell-shaped solitary wave solutions and kink-shaped solitary wave solutions through the bifurcation theory of differential dynamical system. In addition to, some other traveling wave solutions are obtained by using the polynomial complete discriminant method and symbolic computation. Most notably, we give the classification of all single traveling wave solutions of fractional Ginzburg–Landau equation at the same time. The obtained optical soliton solutions in this work may substantially improve or complement the corresponding results in the known references. Finally, we give the comparison between our solutions and other's results. [ABSTRACT FROM AUTHOR]

We investigate dynamics of a class of abstract Timoshenko model with damping and source terms. We obtain a abstract Kirchhoff model as a singular limit of the abstract Timoshenko model when a certain coupling parameter $ a $ tends to infinity. Smooth global attractors with finite fractal dimension and exponential attractors are obtained using the recent quasi-stability theory. We conclude the work comparing the abstract Timoshenko model with abstract Kirchhoff model, in the sense of the upper-semicontinuity of their attractors as $ a $ tends to infinite. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mass-critical nonlinear Schrödinger equation in one dimension. Ogawa-Tsutsumi [Proc. Amer. Math. Soc. 111 (1991), no. 2, 487-496] proved a blow-up result for negative energy solution by using a scaling argument for initial data. In general, a equation with a linear potential does not have a scale invariant, so the method by Ogawa-Tsutsumi cannot be used directly to that. In this paper, we prove a blow-up result for the equation with the linear potential by modifying the argument of Ogawa-Tsutsumi. [ABSTRACT FROM AUTHOR]

One-dimensional maps showing chaotic intermittency with discontinuous reinjection probability density functions are studied. For these maps, the reinjection mechanism possesses two different processes. The M function methodology is applied to analyze the complete reinjection mechanism and to determine the discontinuous reinjection probability density function. In these maps the function M(x) is continuous and non-differentiable. Theoretical equations are found for the function M(x) and for the reinjection probability density function. Finally, the theoretical results are compared with numerical data finding high accuracy. [ABSTRACT FROM AUTHOR]

Through this research, an extension for a differential form of Hilbert's integral inequality for three variables is provided, and the form of the reverse of the main conclusion will also be given. [ABSTRACT FROM AUTHOR]

In this paper, we present a new fixed-point result to draw conclusions about the existence and uniqueness of the solution for a nonlinear Fredholm integral equation of the second kind with non-differentiable Nemytskii operator. To do this, we will transform the problem of locating a fixed point for an integral operator into the problem of locating a solution of an integral equation. Thus, assuming conditions on the Nemytskii operator, we will obtain a global convergence domain for the solution of the considered integral equation, taking for this a uniparametric family of derivativefree iterative processes with quadratic convergence. This result provides us a new fixed-point result for the integral operator considered. [ABSTRACT FROM AUTHOR]

In this paper, we establish some sufficient conditions for the existence of a random exponential attractor for a random dynamical system in a Banach space. As an application, we consider a stochastic reaction-diffusion equation with multiplicative noise. We show that the random dynamical system ϕ (t , ω) generated by this stochastic reaction-diffusion equation is uniformly Fréchet differentiable on a positively invariant random set in L 2 p (D) and satisfies the conditions of the abstract result, then we obtain the existence of a random exponential attractor in L 2 p (D) , where p is the growth of the nonlinearity satisfying 1 < p ≤ 3 . [ABSTRACT FROM AUTHOR]

Kinematic control of redundant serial manipulators has been carried out for the past half century based primarily on a generalized inverse velocity formulation that is known to have mathematical deficiencies. A recently developed inverse kinematic configuration mapping is employed in an operational configuration space differentiable manifold formulation for redundant-manipulator kinematic control with obstacle avoidance. This formulation is shown to resolve deficiencies in the generalized inverse velocity formulation, especially for high-degree-of-redundancy manipulators. Tracking a specified output trajectory while avoiding obstacles for four- and twenty-degree-of-redundancy manipulators is carried out to demonstrate the effectiveness of the differentiable manifold approach for applications with a high degree of redundancy and to show that it indeed resolves deficiencies of the conventional generalized inverse velocity formulation in challenging applications. [ABSTRACT FROM AUTHOR]

We study the problem of existence of Poisson stable (in particular, almost periodic, almost automorphic, recurrent) solutions to the semi-linear differential equation x' = (A0 + A(t))x + F(t, x) with unbounded closed linear operator A0, bounded operators A(t) and Poisson stable functions A(t) and F(t, x). Under some conditions we prove that there exists a unique (at least one) solution which possesses the same recurrence property as the coefficients. [ABSTRACT FROM AUTHOR]

In this paper, the inverse spectral problem method is used to integrate a modified Korteweg-de Vries-Liouville (mKdV-L) equation with an additional term in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator is introduced, and the coefficient of the Dirac operator is a solution for a modified Korteweg-de Vries-Liouville equation with an additional term. A simple algorithm for deriving the Dubrovin system of differential equations is proposed. The solvability of the Cauchy problem for a Dubrovin infinite system of differential equations in the class of six times continuously differentiable periodic infinite-gap functions is proven. It is proven that there is a global solution of the Cauchy problem for a modified Korteweg-de Vries-Liouville equation with an additional term for sufficiently smooth initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, the inverse spectral problem method is used to integrate the nonlinear mKdV–L equation in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of times continuously differentiable periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent function series constructed by solving the system of Dubrovin equations and by using the first trace formula satisfies the mKdV–L equations. Moreover, we prove that if the initial function is a real-valued -periodic analytic function, then the solution of the Cauchy problem for the mKdV–L equation is a real-valued analytic function in the variable as well; and if the number is a period (respectively, antiperiod) of the initial function, then the number is the period (respectively, antiperiod) in the variable of the solution of the Cauchy problem for the mKdV–L equations. [ABSTRACT FROM AUTHOR]

This investigation discusses the problems of fixed-/predefined-time stabilization and synchronization of memristor chaotic circuits (MCCs). Specially, all of the proposed control schemes are differentiable, namely smooth, which are superior to the previous finite-/fixed-time control techniques, because the discontinuous signum and absolute functions are not contained anymore. Comparing with the traditional fast convergence of chaotic systems, the upper-bound estimation of convergence time in this investigation is not only irrelevant to the initial values of MCCs, but also concise and explicit. Moreover, according to the Lyapunov stability theory, the sufficient criteria are established successively for ensuring the fixed-/predefined-time stabilization and synchronization of MCCs. Finally, the numerical simulations are placed to validate the effectiveness and feasibility of obtained results, in which the comparison is made and the effect of controlling parameters on the convergence speed is further explored. [ABSTRACT FROM AUTHOR]

The theory of convexity pertaining to fractional calculus is a well-established concept that has attracted significant attention in mathematics and various scientific disciplines for over a century. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we establish new fractional identities. Employing these identities, some extensions of the fractional H-H type inequality via generalized preinvexities are explored. Finally, we discuss some applications to the q-digamma and Bessel functions via the established results. We believe that the methodologies and approaches presented in this work will intrigue and spark the researcher's interest even more. [ABSTRACT FROM AUTHOR]

In Morocco, with the growing wind installed capacity it becomes crucial to perform further the operation and maintenance (O&M) process, the wind assets are exposed to several constraints and raindom of maintenances for incidents. The target of this work is to optimize the gross and net production while simultaneously adhering to the minimum turbine unavailability and minimizing all contractual power curtailments due to unforeseeable factors. The proposed system is modeled using technical and mathematical formulas composed of the main function and the associated constraints. The project's modularization ends with a complex mathematical system that is non-linear, non-differentiable and requires the genetic algorithm for optimal resolution. Using MATLAB to determine the optimized solution that represents the number of operational turbines per day, allowing for maximum production, minimizing curtailment, and reducing the unavailability of turbines. The 365-vector containing the numbers of turbines per day will opimally define the long-term O&M strategy. [ABSTRACT FROM AUTHOR]

The exploration of stochastic partial differential equations in noisy perturbations of dynamical systems remains a major challenge at this stage. The study analyzes the effective dynamical system combining degenerate additive noise-driven stochastic partial differential equations, firstly in the first class of stochastic partial differential equations, the terms in the non-nuclear space formed by nonlinear interactions are overcome by effectively replacing the elements in the non-nuclear space through the ItÔ formulation, and thus the final effective dynamical system is obtained. The effective dynamical system is then obtained in the second type of stochastic partial differential equation using the O-U process similar to the terms in the non-nuclear space. At noise disturbance amplitudes of 5%, 10%, 15% and 20% AC voltage maxima in that order, the effective dynamical systems for the first type of stochastic partial differential equation and the second type of stochastic partial differential equation are more stable compared to the other types of partial differential equation dynamical systems, with the maximum range of error rate improvement for the sampling points 0–239 voltage rms and voltage initial phase value being 3.62% and 26.85% and 2.13% and 19.86% for sampling points 240–360, respectively. The effective dynamic system and stochastic partial differential equation obtained by the research have very high approximation effect, and can be applied to mechanical devices such as thermal power machines. [ABSTRACT FROM AUTHOR]