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Discretizing a differential equation may change the qualitative behaviour drastically, even if the stepsize is small. We illustrate this by looking at the discretization of a piecewise continuous differential equation that models a population of agents playing the Rock-Paper-Scissors game. The globally asymptotically stable equilibrium of the differential equation turns, after discretization, into a repeller surrounded by an annulus shaped attracting region. In this region, more and more periodic orbits emerge as the discretization step approaches zero. [ABSTRACT FROM AUTHOR]

The present paper dealt with the study of solitons of Lorentzian Kähler space-time manifolds. In this paper, we have discussed different conditions for solitons to be steady, expanding, or shrinking in terms of isotropic pressure, the cosmological constant, energy density, nonlinear equations, and gravitational constant in pseudo-quasi-conformally flat and pseudo-W8 flat Lorentzian Kahler space-time manifolds. [ABSTRACT FROM AUTHOR]

This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research. [ABSTRACT FROM AUTHOR]

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point 0 ≤ c < d and fixed points at the end of the interval [0, d]. For i = 0, 1, the boundary conditions are as follows: ai, bi > 0, a0p(c) = -b0p(d), a1p'(c) = -b1p'(d). Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixedpoint theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved. [ABSTRACT FROM AUTHOR]

In this paper, a new class of extended (2+1)-dimensional Ito equations is investigated for its group invariant solutions. The Lie symmetry method is employed to transform the nonlinear Ito equation into an ordinary differential equation. The general solution of the solvable linear differential equation with different parameters is obtained, and the plot of the solvable linear differential equation is given. A power series solution for the equation is then derived. Furthermore, a conservation law for the equation is constructed by utilizing a new Ibragimov conservation theorem. [ABSTRACT FROM AUTHOR]

The generalized Drinfeld-Sokolov system is a widely-used model that describes wave phenomena in various contexts. Many properties of this system, such as Hamiltonian formulations and integrability, have been extensively studied and exact solutions have been derived for specific cases. In this paper we applied the direct method of multipliers to obtain all low-order local conservation laws of the system. These conservation laws correspond to physical quantities that remain constant over time, such as energy and momentum, and we provided a physical interpretation for each of them. Additionally, we investigated the Lie point symmetries and first-order symmetries of the system. Through the point symmetries and constructing the optimal systems of one-dimensional subalgebras, we were able to reduce the system of partial differential equations to ordinary differential systems and obtain new solutions for the system. [ABSTRACT FROM AUTHOR]

This paper introduces a novel numerical scheme, the conformable finite difference method (CFDM), for solving time-fractional gas dynamics equations. The method was developed by integrating the finite difference method with conformable derivatives, offering a unique approach to tackle the challenges posed by time-fractional gas dynamics models. The study explores the significance of such equations in capturing physical phenomena like explosions, detonation, condensation in a moving flow, and combustion. The numerical stability of the proposed scheme is rigorously investigated, revealing its conditional stability under certain constraints. A comparative analysis is conducted by benchmarking the CFDM against existing methodologies, including the quadratic B-spline Galerkin and the trigonometric B-spline functions methods. The comparisons are performed using L2 and L∞ norms to assess the accuracy and efficiency of the proposed method. To demonstrate the effectiveness of the CFDM, several illustrative examples are solved, and the results are presented graphically. Through these examples, the paper showcases the capability of the proposed methodology to accurately capture the behavior of time-fractional gas dynamics equations. The findings underscore the versatility and computational efficiency of the CFDM in addressing complex phenomena. In conclusion, the study affirms that the conformable finite difference method is well-suited for solving differential equations with time-fractional derivatives arising in the physical model. [ABSTRACT FROM AUTHOR]

This paper employed the well-known Riccati transformation method to deduce a Kneser-type oscillation criterion for second-order dynamic equations. These results are considered an extension and improvement of the known Kneser results for second-order differential equations and are new for other time scales. We have included examples to highlight the significance of the results we achieved. [ABSTRACT FROM AUTHOR]

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $ \beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi = \alpha $ are studied, where $ \phi\in \mathbb S^{1} $ and $ (\alpha,\beta,\gamma)\in \mathbb R^{3} $. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear. [ABSTRACT FROM AUTHOR]

In this paper, we investigated n-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in 3-dimensional space at the end of the paper. [ABSTRACT FROM AUTHOR]

This paper presents novel criteria for investigating the oscillatory behavior of even-order neutral differential equations. By employing a comparative approach, we established the oscillation properties of the studied equation through comparisons with well-understood first-order equations with known oscillatory behavior. The findings of this study introduce fresh perspectives and enrich the existing body of oscillation criteria found in the literature. To illustrate the practical application of our results, we provide an illustrative example. [ABSTRACT FROM AUTHOR]

We consider the two-point boundary value problems for a nonlinear one-dimensional second-order differential equation with involution in the second derivative and in lower terms. The questions of existence and uniqueness of the classical solution of two-point boundary value problems are studied. The definition of the Green's function is generalized for the case of boundary value problems for the second-order linear differential equation with involution, indicating the points of discontinuities and the magnitude of discontinuities of the first derivative. Uniform estimates for the Green's function of the linear part of boundary value problems are established. Using the contraction mapping principle and the Schauder fixed point theorem, theorems on the existence and uniqueness of solutions to the boundary value problems are proved. The results obtained in this paper cover the boundary value problems for one-dimensional differential equations with and without involution in the lower terms. [ABSTRACT FROM AUTHOR]

In this paper, a classical risk model with liquid reserves and proportional investment is considered, and the expected total discounted dividend before ruin of insurance companies under the threshold dividend strategy is studied. First, the integral differential equations of the expected total discounted dividend before ruin satisfying certain boundary conditions is derived. Second, since the explicit solutions of the equations cannot be obtained, the numerical approximation solutions are obtained by the sinc approximation method. Finally, we discuss the effects of parameters such as risk capital ratio and liquid reserve on the expected total discounted dividend before ruin by some examples. [ABSTRACT FROM AUTHOR]

This paper explored the existence and uniqueness of a neutral fractional impulsive dynamic equation over time scales that included nonlocal initial conditions and employed the Caputo-nabla derivative (C∇D). The establishment of existence and uniqueness relies on the fine fixed point theorem. Furthermore, a comparison was conducted between the fractional order C∇D and the Riemann-Liouville nabla derivative (RL∇D) over time scales. Theoretical findings were substantiated through a numerical methodology, and an illustrative graph using MATLAB was presented for the provided example. [ABSTRACT FROM AUTHOR]

This paper studies a discontinuous Sturm-Liouville problem in which the spectral parameter appears not only in the differential equation but also in the transmission conditions. By constructing an appropriate Hilbert space and inner product, the eigenvalue and eigenfunction problems of the Sturm-Liouville problem are transformed into an eigenvalue problem of a certain self-adjoint operator. Next, the eigenfunctions of the problem and some properties of the eigenvalues are given via construction of the basic solution. The Green's function for the Sturm-Liouville problem is also given. Finally, the continuity of the eigenvalues and eigenfunctions of the problem is discussed. Especially, the differential expressions of the eigenvalues for some parameters have been obtained, including the parameters in the eigenparameter-dependent transmission conditions. [ABSTRACT FROM AUTHOR]

In this paper we study a complete second order differential equation of Sturm-Liouville type under Dirichlet boundary condition and where the variable coefficients are allowed to be sign changing. Through critical point theory, we obtain the existence of two nontrivial generalized solutions by requiring a specific growth on the nonlinearity. Moreover, the solutions turn out to be nonnegative and with opposite energy sign. [ABSTRACT FROM AUTHOR]

This paper discusses model order reduction of large sparse secondorder index-3 differential algebraic equations (DAEs) by applying Iterative Rational Krylov Algorithm (IRKA). In general, such DAEs arise in constraint mechanics, multibody dynamics, mechatronics and many other branches of sciences and technologies. By deecting the algebraic equations the secondorder index-3 system can be altered into an equivalent standard second-order system. This can be done by projecting the system onto the null space of the constraint matrix. However, creating the projector is computationally expensive and it yields huge bottleneck during the implementation. This paper shows how to find a reduce order model without projecting the system onto the null space of the constraint matrix explicitly. To show the efficiency of the theoretical works we apply them to several data of second-order index-3 models and experimental resultants are discussed in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a multi-order -Hilfer fractional pantograph implicit differential equation on unbounded domains . The existence and uniqueness of solution are established for a such problem by utilizing the Banach fixed point theorem in an applicable Banach space. In addition, stability of the types Ulam-Hyers ( ), Ulam-Hyers-Rassias ( ) and semi-Ulam-Hyers-Rassias (s- ) are discussed by using nonlinear analysis topics. Finally, a concrete example includes some particular cases is enhanced to illustrate rightful of our results. [ABSTRACT FROM AUTHOR]

In this paper, the solutions of some typical nonlinear fractional differential equations are discussed, and the implicit analytical solutions are obtained. The fractional derivative concerned here is the Caputo-Fabrizio form, which has a nonsingular kernel. The calculation results of different fractional orders are compared through images. In addition, by comparing the results obtained in this paper with those under Caputo fractional derivative, it is found that the solutions change relatively gently under Caputo-Fabrizio fractional derivative. It can be concluded that the selection of appropriate fractional derivatives and appropriate fractional order is very important in the modeling process. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the existence of unique almost periodic solutions to fuzzy shunting inhibitory cellular neural networks (FSICNN) with several delays. Further, by means of Halanay inequality we analyze the global exponential stability of these solutions and obtain corresponding convergence rate. The results of this paper are new, and they are concluded with numerical simulations confirming them. [ABSTRACT FROM AUTHOR]

This research paper focused on the solution of systems of fractional integro-differential equations (FIDEs) of the Volterra type with variable coefficients. The proposed approach combined the tau method and shifted Gegenbauer polynomials in a matrix form. The investigation of the existence and uniqueness of solutions for these systems was carried out using Krasnoselskii's fixed point theorem. The equations employed Caputo-style derivative operators, and to minimize computational operations involving derivatives and multiplications, integral and product operational matrices were derived. By introducing suitable polynomial approximations and employing the tau spectral method, the original system of FIDE was transformed into an algebraic system. Solving this algebraic system provided approximate solutions to the main system. Error bounds were computed in the Gegenbauer- weighted Sobolev space. The proposed algorithm was implemented and tested on two systems of integro-fractional differential equations to demonstrate its efficiency and simplicity. By varying the parameter a in the Gegenbauer polynomials, the impact of this variation on the approximate solutions can be observed. A comparison with another method utilizing the block-by-block approach was also presented. [ABSTRACT FROM AUTHOR]

In this paper, I study a planar m-piecewise discontinuous polynomial differential system ẋ = y, ẏ = -x - ε(f (x, y) + gm(x, y)h(x)), which has a linear center in each zone partitioned by those switching lines, where f (x, y) = ∑i+j=0n aijxiyj, h(x) = ∑j=0l bjxj, aij, bj ∈ ℝ, n, l ∈ ℕ, and gm(x, y) with the positive even number m as the union of m/2 different straight lines passing through the origin of coordinates dividing the plane into sectors of angle 2π/m. Using the averaging theory, I provide the lower bound Lm(n, l) for the maximun number of limit cycles, which bifurcates which bifurcating from the annulus of the origin of this system. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable. [ABSTRACT FROM AUTHOR]

In this paper, we establish a quadratic integral inequality involving the second order derivative of functions in the following form: for all f ∈ D , ∫ a b r | f ″ | 2 + p | f ′ | 2 + q | f | 2 ≥ μ 0 ∫ a b | f | 2. Here r , p , q are real- valued coefficient functions on the compact interval [ a , b ] with r (x) > 0 . D is a linear manifold in the Hilbert function space L 2 (a , b) such that all integrals of the above inequality are finite and μ 0 is a real number that can be determined in terms of the spectrum of a uniquely determined self adjoint differential operator in L 2 (a , b) . The inequality is the best possible, i.e., the number μ 0 cannot be increased. f is a complex-valued function in D. [ABSTRACT FROM AUTHOR]

In this paper, we propose a two-grid algorithm for nonlinear time fractional parabolic equations by H 1 -Galerkin mixed finite element discreitzation. First, we use linear finite elements and Raviart-Thomas mixed finite elements for spatial discretization, and L 1 scheme on graded mesh for temporal discretization to construct a fully discrete approximation scheme. Second, we derive the stability and error estimates of the discrete scheme. Third, we present a two-grid method to linearize the nonlinear system and discuss its stability and convergence. Finally, we confirm our theoretical results by some numerical examples. [ABSTRACT FROM AUTHOR]

The 2-variable modified partially degenerate Hermite (MPDH) polynomials are the subject of our study in this paper. We found basic properties of these polynomials and obtained several types of differential equations related to MPDH polynomials. Based on the MPDH polynomials, we looked at the structures of the approximation roots for a particular polynomial and checked the values of the approximate roots. Further, we presented some conjectures for MPDH polynomials. [ABSTRACT FROM AUTHOR]

This paper established sufficient conditions for the stability of Abel's differential equation of the first kind. These conditions explicate the impact of the asymptotic behaviors exhibited by the time-varying coefficients on the overall stability of the system. More precisely, we studied the positivity, the continuation and the boundedness of solutions. Additionally, we investigated the attractivity, the asymptotic stability, the uniform stability and the instability of the system. The results were scrutinized by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the existence of multiple nontrivial T-periodic solutions of the nonlinear second ordinary differential equation x + Vx(t; x) = 0 in N(≥ 1) dimensions. Using homological linking and morse theory, we get at least two critical points of the functional corresponding to our problem. And, we also prove that two critical points are different by critical groups. Then, we obtain there are at least two nontrivial T-periodic solutions of the problem. [ABSTRACT FROM AUTHOR]

The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on the properties of the differential subordination and superordination one of the newest techniques used in this field, we obtain some differential subordination and superordination results for multivalent functions defined by differintegral operator with j -derivatives ℑ p (ν , ρ ; ℓ) f (z) for ℓ > 0 , ν , ρ ∈ R , such that (ρ − j) ≥ 0 , ν > − ℓ p (p ∈ N) in the open unit disk U . Differential sandwich result is also obtained. Also, the results are followed by some special cases and counter examples. The results from this paper are related to the geometric function theory. In order to obtain them, we use the technique based on the properties of the differential subordination and superordination one of the newest techniques used in this field, we obtain some differential subordination and superordination results for multivalent functions defined by differintegral operator with -derivatives for such that in the open unit disk . Differential sandwich result is also obtained. Also, the results are followed by some special cases and counter examples. [ABSTRACT FROM AUTHOR]

In this paper, by using generating functions for the Chebyshev polynomials, we have obtained the convolution formulas involving the bi-periodic Fibonacci and Lucas polynomials. [ABSTRACT FROM AUTHOR]

This paper deals with the following quasilinear attraction-repulsion chemotaxis system with homogeneous Neumann boundary conditions in a bounded, smooth domain where and It is proved that if the nonlinear exponents of the system satisfy then the system has globally bounded classical solutions. Furthermore, assume that if one of the following conditions holds: then the classical solutions of the system would be globally bounded. The global boundedness criteria generalize the results established by previous researchers. This paper deals with the following quasilinear attraction-repulsion chemotaxis system with homogeneous Neumann boundary conditions in a bounded, smooth domain where and It is proved that if the nonlinear exponents of the system satisfy then the system has globally bounded classical solutions. Furthermore, assume that if one of the following conditions holds: then the classical solutions of the system would be globally bounded. The global boundedness criteria generalize the results established by previous researchers. [ABSTRACT FROM AUTHOR]

Stability theory has significant applications in technology, especially in control systems. On the other hand, the newly defined generalized mean-square stochastic fractional (GMSF) operators are particularly interesting in control theory and systems due to their various controllable parameters. Thus, the combined study of stability theory and GMSF operators becomes crucial. In this research work, we construct a new class of GMSF differential equations and provide a rigorous proof of the existence of their solutions. Furthermore, we investigate the stability of these solutions using the generalized Ulam-Hyers-Rassias stability criterion. Some examples are also provided to demonstrate the effectiveness of the proposed approach in solving fractional differential equations (FDEs) and evaluating their stability. The paper concludes by discussing potential applications of the proposed results in technology and outlining avenues for future research. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to find the explicit forms for entire solutions of two certain types of Fermat-type q-difference differential equations. Some previous results are generalized and examples are constructed to show that the results are accurate. [ABSTRACT FROM AUTHOR]

In this paper, for a regularized fuzzy system, a generalization of the direct Lyapunov method is adapted on the base of matrix-valued Lyapunov-like functions. First, the new concept of a regularization scheme for fuzzy systems is discussed and the matrix-valued Lyapunov function technique is introduced. Then, sufficient conditions are established for the boundedness and stability of the equilibrium set of solutions of the regularized fuzzy system of differential equations. Scalar and vector Lyapunov-type functions are used based on an auxiliary matrix-valued function. Finally, a discussion is offered for the future directions of the proposed approach. Since the strategies for the analysis of the stability of fuzzy models are very important in numerous aspects, we expect that our results will inspire researchers to develop the introduced concept. [ABSTRACT FROM AUTHOR]

In this paper we provide a novel approach to the analysis of kinetic models for label switching, which are used for particle systems that can randomly switch between gradient flows in different energy landscapes. Besides problems in biology and physics, we also demonstrate that stochastic gradient descent, the most popular technique in machine learning, can be understood in this setting, when considering a time-continuous variant.Our analysis is focusing on the case of evolution in a collection of external potentials, for which we provide analytical and numerical results about the evolution as well as the stationary problem. [ABSTRACT FROM AUTHOR]

In this paper, we study a time-delayed free boundary of tumor growth with Gibbs-Thomson relation in the presence of inhibitors. The model consists of two reaction diffusion equations and an ordinary differential equation. The reaction diffusion equations describe the nutrient and inhibitor diffusion within tumors and take into account the Gibbs-Thomson relation at the outer boundary of the tumor. The tumor radius evolution is described by the ordinary differential equation. It is assumed that the regulatory apoptosis process takes longer than the natural apoptosis and proliferation processes. We first show the existence and uniqueness of the solution to the model. Next, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is a constant. Finally, we further demonstrate the existence of the stationary solutions and the asymptotic behavior of the stationary solutions when the blood vessel density is bounded. The result implies that, under certain conditions, the tumor will probably become dormant or will finally disappear. The conclusions are illustrated by numerical computations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-sided jumps risk model with proportional investments and random observation periods. The downward jumps represent the claim while the upward jumps represent the random returns. Suppose an insurance company invests all of their surplus in risk-free and risky investments in proportion. In real life, corporate boards regularly review their accounts rather than continuously monitoring them. Therefore, we assume that insurers regularly observe surplus levels to determine whether they will ruin and that the random observation periods are exponentially distributed. Our goal is to study the Gerber-Shiu function (i.e., the expected discounted penalty function) of the two-sided jumps risk model under random observation. First, we derive the integral differential equations (IDEs) satisfied by the Gerber-Shiu function. Due to the difficulty in obtaining explicit solutions for the IDEs, we utilize the sinc approximation method to obtain the approximate solution. Second, we analyze the error between the approximate and explicit solutions and find the upper bound of the error. Finally, we discuss examples of sensitivity analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multiplicity of solutions for the following three-point boundary value problem of second-order p-Laplacian differential equations with instantaneous and non-instantaneous impulses: and f j 2 C((s j; t j+1]-R;R), I j 2 C(R;R). ≥ 2 (0; +1), - 2 R are two parameters. -(t) ≥ 1, 1 ≤ g(t) ≤ c for t 2 (s j; t j+1], -(t); g(t) 2 Lp[0; 1], and c is a positive constant. By using variational methods and the critical points theorems of Bonanno-Marano and Ricceri, the existence of at least three classical solutions is obtained. In addition, several examples are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, based on the E. Study map, clear terms are offered for the differential equations of one-parameter Lorentzian spatial kinematics that are coordinate system-independent. This cancels the request of demanding coordinate transformations that are typically required in the determination of the canonical systems. With the suggested technique, new proofs of the Disteli formulae of a spacelike line trajectory are instantly gained and their spatial equivalents are studied in detail. As a consequence, we address the kinematic geometry of a point trajectory for the one-parameter Lorentzian spherical and planar movements. [ABSTRACT FROM AUTHOR]

In the value distribution theory of complex analysis, Petrenko's deviation is to describe more precisely the quantitative relationship between T(r, f) and log M(r, f) when the modulus of variable |z| = r is sufficiently large. In this paper we introduce Petrenko's deviations to the coefficients of three types of complex equations, which include difference equations, differential equations and differential-difference equations. Under different assumptions we study the lower bound of limiting directions of Julia sets of solutions of these equations, where Julia set is an important concept in complex dynamical systems. The results of this article show that the lower bound of limiting directions mentioned above is closely related to Petrenko's deviation, and our conclusions improve some known results. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytical solutions of two-dimensional fractional-order linear system Dαt X(t) = AX(t) described by fractional differential equations, where D is the fractional derivative in the Caputo-Fabrizio sense and A = (ai j)2x2 is nonsingular coefficient matrix with aij ∈ R. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system. [ABSTRACT FROM AUTHOR]

This paper is concerned with parameter estimation for partially observed stochastic differential equations driven by fractional Brownian motion. Firstly, the state estimation equation is given and the parameter estimator is derived. Then, the strong consistency and asymptotic normality of the maximum likelihood estimator are derived by applying the strong law of large numbers for continuous martingales and the central limit theorem for stochastic integrals with respect to Gaussian martingales. Finally, an example is provided to verify the results. [ABSTRACT FROM AUTHOR]

Since the Marcus-Wyse (MW-, for brevity) topological spaces play important roles in the fields of pure and applied topology (see Remark 2.2), the paper initially proves that the MWtopological space satisfies the semi-T3-separation axiom. To do this work more efficiently, we first propose several techniques discriminating between the semi-openness or the semi-closedness of a set in the MW-topological space. Using this approach, we suggest the condition for simple MW-paths to be semi-closed, which confirms that while every MW-path P with j P j 2 is semi-open, it may not be semi-closed. Besides, for each point p 2 Z2 the smallest open neighborhood of the point p is proved to be a regular open set so that it is semi-closed. Note that the MW-topological space is proved to satisfy the semi-T3-separation axiom, i.e., it is proved to be a semi-T3-space so that we can confirm that it also satisfies an s-T3-separation axiom. Finally, we prove that the semi-T3-separation axiom is a semi-topological property. [ABSTRACT FROM AUTHOR]

In this paper, we study a food chain chemostat model with Michaelis-Menten function response and double delays. Applying the stability theory of functional differential equations, we discuss the conditions for the stability of three equilibria, respectively. Furthermore, we analyze the sufficient conditions for the Hopf bifurcation of the system at the positive equilibrium. Finally, we present some numerical examples to verify the correctness of the theoretical analysis and give some valuable conclusions and further discussions at the end of the paper. [ABSTRACT FROM AUTHOR]

Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider linear-quadratic (LQ) leader-follower Stackelberg differential games for mean-field type stochastic systems with jump diffusions, where the system includes mean-field variables, i.e., the expected value of state and control variables. We first solve the LQ mean-field type control problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the coupled integro-Riccati differential equations (CIRDEs) via the Four-Step Scheme. Next, we solve the problem of the leader, which is the LQ control problem subject to the mean-field type forward-backward stochastic system with jump diffusions, where the constraint characterizes the rational behavior of the follower. Using the variational approach, we obtain the (mean-field type) stochastic maximum principle. However, to obtain the state-feedback representation of the open-loop optimal solution of the leader, there is a technical challenge due to the jump process. We consider two different cases, in which the state-feedback type control in terms of the CIRDEs can be characterized by generalizing the Four-Step Scheme. We finally show that the state-feedback type controls of the open-loop optimal solutions for the leader and the follower constitute the Stackelberg equilibrium. [ABSTRACT FROM AUTHOR]

The article offers information on the conference "LlaveFest: A Broad Perspective on Finite and Infinite Dimensional Dynamical Systems," that was held at University of Barcelona from June 12-16, 2017. It mentions that the conference was supported by the Institute of Mathematics of the University of Barcelona (IMUB). It focuses on the aspects of the dynamical systems and partial differential equations.

This paper focuses on the key issues of mosquito population control, particularly exploring the impact of periodic releases of sterile males in the population model with a stage structure. We construct and analyze a model that includes only sexually active sterile mosquitoes in the dynamic interaction system. We focus on the system's dynamical behaviors under two scenarios: when the sexual lifespan T ¯ equals the release period T of sterile mosquitoes, and when T ¯ is less than T. In the first scenario, we explore the existence and stability of equilibria, identifying a pivotal threshold m ∗ that determines the requisite release amount. In the second scenario, we convert the problem into an impulsive switched system and derive sufficient conditions for the local asymptotic stability of the extinction equilibrium. We also establish the existence of positive periodic solutions using the geometric method of differential equations and the fixed point theorem. Our conclusions show that the relationship between the sexual lifespan and release period of sterile mosquitoes significantly impacts the stability of the mosquito population. Additionally, our numerical simulations not only corroborate but they also complement our theoretical findings. [ABSTRACT FROM AUTHOR]