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In this paper, we investigate a class of competitive and cooperative Nicholson's blowfly equations. By applying the Lyapunov functional and analysis technique, the conditions for the existence and exponential convergence of a positive almost periodic solution are derived. Moreover, an example and numerical simulation for justifying the theoretical analysis are also provided.

In this paper we study the unique weighted measurable shadowing property for weighted pseudo-orbits of random systems on Banach spaces with the property that linear part of random system admits a tempered exponential dichotomy. We also prove for linear random systems that the tempered exponential dichotomy is necessary for the unique weighted measurable shadowing property to hold. [ABSTRACT FROM AUTHOR]

In A. Czornik et al. [Spectra based on Bohl exponents and Bohl dichotomy for non-autonomous difference equations, J. Dynam. Differ. Equ. (2023)] the concept of Bohl dichotomy is introduced which is a notion of hyperbolicity for linear non-autonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set $ \mathrm {BD} $ BD of systems with Bohl dichotomy and the set $ \mathrm {ED} $ ED of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that $ \mathrm {ED} \subseteq \mathrm {BD} $ ED ⊆ BD . Using a counterexample we show that the closure of $ \mathrm {ED} $ ED is not contained in $ \mathrm {BD} $ BD . The main result of this paper is the characterization $ \operatorname {int}\mathrm {BD} = \mathrm {ED} $ int ⁡ BD = ED . The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of non-autonomous difference equations as a universal tool. [ABSTRACT FROM AUTHOR]

In this paper, we apply Lin's method to study the existence of heteroclinic orbits near the degenerate heteroclinic chain under m$$ m $$‐dimensional periodic perturbations. The heteroclinic chain consists of two degenerate heteroclinic orbits γ1$$ {\gamma}_1 $$ and γ2$$ {\gamma}_2 $$ connected by three hyperbolic saddle points q1,q2,q3$$ {q}_1,{q}_2,{q}_3 $$. Assume that the degeneracy of the unperturbed heteroclinic orbit γi$$ {\gamma}_i $$ is ni$$ {n}_i $$, the splitting index is δi$$ {\delta}_i $$. By applying Lin's method, we construct heteroclinic orbits connected q1$$ {q}_1 $$ and q3$$ {q}_3 $$ near the unperturbed heteroclinic chain. The existence of these orbits is equivalent to finding zeros of the corresponding bifurcation function. The lower order terms of the bifurcation function is the map from ℝn1+n2+m$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+m} $$ to ℝn1+n2+δ1+δ2$$ {\mathrm{\mathbb{R}}}^{n_1+{n}_2+{\delta}_1+{\delta}_2} $$. Using the contraction mapping principle, we provide a detailed analysis on how zeros can exist based on different cases of splitting indices δ1$$ {\delta}_1 $$, δ2$$ {\delta}_2 $$ and then obtain the existence of the heteroclinic orbits which backward asymptotic to q1$$ {q}_1 $$ and forward asymptotic to q3$$ {q}_3 $$. [ABSTRACT FROM AUTHOR]

We consider a non-autonomous coupled system (x , y) , whose x coordinate satisfies a semilinear parabolic equation, and y coordinate satisfies a differential equation whose solutions do not converge too rapidly. Our aim is to study the persistence of dynamical behavior for such coupled system under a small random perturbation driven by stationary multipilcative noise. We show that for the perturbed system there exists a C 1 invariant manifold S (t , ω) = { (x , y) ∈ X × Y | x = σ (t , y , ω) } under the condition that the linear part of x equation satisfies the exponential dichotomy. Meanwhile, we observe that typically if the linear part of x equation is uniformly attracting or uniformly expanding, then the corresponding invariant manifold shares the same qualitative properties. In the end, as the perturbation tends to 0, we show that the invariant manifold and its derivative in y are approaching to those of the original system, suggesting that such structure is persistent under the small random perturbation. [ABSTRACT FROM AUTHOR]

This paper studies two nonlocal diffusion problems with a free boundary and a fixed boundary in heterogeneous environment. The main goal is to understand how the evolution of the two species is affected by the heterogeneous environment. We first prove the existence and uniqueness of a global solution for such systems. Then, for models with Lotka–Volterra type competition or predator–prey growth terms, we establish the spreading‐vanishing dichotomy. Sharp criteria of spreading and vanishing are also obtained. Furthermore, we show that accelerated spreading occurs if and only if the kernel function violates the threshold condition. [ABSTRACT FROM AUTHOR]

For linear differential systems, the Sacker–Sell spectrum (dichotomy spectrum) and the contractible set are the same. However, we claim that this is not true for the linear difference equations. A counterexample is given. For the convenience of research, we study the relations between the dichotomy spectrum and the contractible set under the framework on time scales. In fact, by a counterexample, we show that the contractible set could be different from dichotomy spectrum on time scales established by Siegmund [J. Comput. Appl. Math., 2002]. Furthermore, we find that there is no bijection between them. In particular, for the linear difference equations, the contractible set is not equal to the dichotomy spectrum. To counter this mismatch, we propose a new notion called generalized contractible set and we prove that the generalized contractible set is exactly the dichotomy spectrum. Our approach is based on roughness theory and Perron's transformation. In this paper, a new method for roughness theory on time scales is provided. Moreover, we provide a time-scaled version of the Perron's transformation. However, the standard argument is invalid for Perron's transformation. Thus, some novel techniques should be employed to deal with this problem. Finally, an example is given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we investigate the homogeneous linear differential equation v'(t) = A(t)v(t) and the semi-linear differential equation v'(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A: R → L(X) is a strongly continuous function, g: R × X → X is continuous and satisfies φ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E1), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class C¹. [ABSTRACT FROM AUTHOR]

This work deals with the existence and uniqueness of μ -pseudo almost periodic solutions to some transport processes along the edges of a finite network with inhomogeneous conditions in the vertices. For that, the strategy consists of seeing these systems as a particular case of the semilinear boundary evolution equations (S H B E) du dt = A m u (t) + f (t , u (t)) , t ∈ R , L u (t) = g (t , u (t)) , t ∈ R , where A : = A m | k e r L generates a C 0 -semigroup admitting an exponential dichotomy on a Banach space. Assuming that the forcing terms taking values in a state space and in a boundary space respectively are only μ -pseudo almost periodic in the sense of Stepanov, we show that (SHBE) has a unique μ -pseudo almost periodic solution which satisfies a variation of constant formula. Then we apply the previous result to obtain the existence and uniqueness of μ -pseudo almost periodic solution to our model of network. [ABSTRACT FROM AUTHOR]

We consider a nonlocal diffusive Fisher-KPP model with free boun-daries in time periodic environment. When the growth term is Logistic type, Zhang et al. [DCDS-B 2021] proved that this model admits a unique global solution and its long time behavior is governed by a spreading-vanishing dichotomy. However, when spreading happens, the spreading speed estimates for such free boundary problems remain unsolved. In this paper, we answer this question. By solving a corresponding time-periodic semi-wave problem, we obtain a threshold condition on the kernel function such that the spreading grows linearly in time, and provide a sharp estimate for the spreading speed; when the threshold condition is not satisfied, we observe an accelerating spreading phenomenon. [ABSTRACT FROM AUTHOR]

We investigate the roughness of nonuniform exponential dichotomies in Banach spaces subject to a new class of small linear time variable perturbations that satisfy an integral inequality which can benefit from a smallness integrability condition. We establish the continuous dependence of constants in terms of a dichotomy notion. Our proofs introduce a new development based on integral inequalities. Notably, we do not require the notion of admissibility for bounded nonlinear perturbations. Furthermore, we derive related roughness results for nonuniform exponential contractions and expansions. Our results are also new to uniform exponential dichotomy. We construct the evolution operator and projections directly, without the need for admissibility. [ABSTRACT FROM AUTHOR]

In this work, we prove some new results concerning the class of Eberlein weakly almost periodic functions in Stepanov’s sense. We prove that if the forcing term of a partial functional differential equation with infinite delay is Eberlein-weakly almost periodic in Stepanov’s sense, then the solution is even Eberlein-weakly almost periodic. This shows that a less regular almost periodic behavior in the forcing term yields a more regular almost periodic behavior in the solution. The theoretical results are illustrated in the Lotka–Volterra model. [ABSTRACT FROM AUTHOR]

Based on the estimation of the Cauchy matrix of linear impulsive differential equation, by using Banach fixed point theorem and exponential dichotomy, sufficient conditions for the existence of almost periodic solutions in shifts δ± of some nonlinear dynamic equations with impulses on time scales are established. Finally, two impulsive ecosystems defined on some specific time scales are studied to illustrate the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

We consider the Nernst-Planck-Navier-Stokes system describing the electrodiffusion of ions in a viscous Newtonian fluid. We prove the exponential nonlinear stability of constant steady states in the case of periodic boundary conditions in any dimension of space without constraints on the number of species, valences and diffusivities. We consider also the case of two spatial dimensions, and we prove the exponential stability from arbitrary large data. [ABSTRACT FROM AUTHOR]

We introduce a novel option pricing model that features stochastic interest rates along with an underlying price process driven by stochastic string shocks combined with pure jump Lévy processes. Substituting the Brownian motion in the Black–Scholes model with a stochastic string leads to a class of option pricing models with expiration-dependent volatility. Further extending this Generalized Black–Scholes (GBS) model by adding Lévy jumps to the returns generating processes results in a new framework generalizing all exponential Lévy models. We derive four distinct versions of the model, with each case featuring a different jump process: the finite activity lognormal and double–exponential jump diffusions, as well as the infinite activity CGMY process and generalized hyperbolic Lévy motion. In each case, we obtain closed or semi-closed form expressions for European call option prices which generalize the results obtained for the original models. Empirically, we evaluate the performance of our model against the skews of S&P 500 call options, considering three distinct volatility regimes. Our findings indicate that: (a) model performance is enhanced with the inclusion of jumps; (b) the GBS plus jumps model outperform the alternative models with the same jumps; (c) the GBS-CGMY jump model offers the best fit across volatility regimes. [ABSTRACT FROM AUTHOR]

We consider linear differential equations of the first order with delayed arguments in a Banach space. We establish conditions for the operator coefficients necessary for the existence of exponential dichotomy on the real axis. It is proved that the analyzed differential equation is equivalent to a difference equation in a certain space. It is shown that, under the conditions of existence and uniqueness of a solution bounded on the entire real axis, the condition of exponential dichotomy is also satisfied for any known bounded function. We also deduce the explicit formula for projectors, which form this dichotomy in the case of a single delay. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction-diffusion system with mechanical deformation of medium. This system consists of an excitable system bidirectionally coupled with an elasticity equation. The main content consists of two parts. First, for γ > 0 sufficiently small, simple pulse of homoclinic type exists. We prove that the traveling pulse is linearly stable. Specifically, there is at most one nontrivial eigenvalue near the origin and it is negative. Second, for γ > 0 large, we show existence of double twisted front-back wave loop, indicating bifurcations of various complicated traveling waves, including N -front, N -back and wave train. Then we prove that N -fronts are linearly stable. Our arguments are mainly based on geometric singular perturbation theory, exponential dichotomy, heteroclinic bifurcation and the Melnikov method. [ABSTRACT FROM AUTHOR]

For linear nonautonomous differential equations we introduce a new family of spectrums defined with general nonuniform dichotomies: for a given growth rate μ in a large family of growth rates, we consider a notion of spectrum, named nonuniform μ -dichotomy spectrum. This family of spectrums contain the nonuniform dichotomy spectrum as the very particular case of exponential growth rates. For each growth rate μ , we describe all possible forms of the nonuniform μ -dichotomy spectrum, relate its connected components with adapted notions of Lyapunov exponents, and use it to obtain a reducibility result for nonautonomous linear differential equations. We also give illustrative examples where the spectrum is obtained, including a situation where a normal form is obtained for polynomial behavior. [ABSTRACT FROM AUTHOR]

This document is an author index for Volume 23 of the journal Stochastics & Dynamics. It provides a list of authors and their respective articles published in this volume. The articles cover a wide range of topics in stochastic dynamics, including stochastic differential equations, ergodic optimization, homogenization, and more. The index is a valuable resource for library patrons conducting research in these areas. [Extracted from the article]

This paper presents an approach for designing angular and linear velocity observers for rigid bodies using two inertial vectors and one landmark measurements directly. Compared with the classical velocity estimation algorithms, the proposed observer does not need to reconstruct the pose information from measurements nor be coupled with velocity sensors. The linear‐time varying dynamics of the estimation error equation are analyzed, rigorously showing the uniform local exponential stability of the observer. Simulations are conducted to illustrate the significant performance of the proposed method. [ABSTRACT FROM AUTHOR]

Palmer's linearization theorem states that a hyperbolic linear system is topologically conjugated to its bounded perturbation. Recently, Huerta (DCDS 2020 [8]), Castañeda and Robledo (DCDS 2018 [3]) and Lin (NA 2007 [13]) generalized Palmer's theorem to the linearization with unbounded perturbation (continuous or discrete) by assuming that the linear part of the system is contractive or nonuniformly contractive. However, these previous works sacrifice the hyperbolicity of the linear part. Is it possible to study the linearization with unbounded perturbations in the hyperbolic case? In this paper, we improve the previous works [3,8,13] to the hyperbolic unbounded systems. For the contraction, each trajectory crosses its respective unit sphere exactly once. However, for the hyperbolic system, either the trajectory does not cross the unit sphere, or the trajectory cross it twice. Thus, the standard method used in the previous works for the contractive case is not valid for the hyperbolic case yet. We develop a method to overcome the difficulty based on two 'cylinders'. Furthermore, quantitative results for the parameters are provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of impulsive quaternion-valued neural networks with mixed delays. By using the General Lipschitz condition, the contraction mapping principle, the exponential dichotomy of linear dynamic equations and the generalized Gronwall–Bellman inequality technique, we obtain the conditions for the existence, uniqueness and global exponential stability of asymptotically almost automorphic solutions of the system. Finally, two examples are given to illustrate the efficiency of our theoretical results. [ABSTRACT FROM AUTHOR]

Control of complex turbulent dynamical systems involving strong nonlinearity and high degrees of internal instability is an important topic in practice. Different from traditional methods for controlling individual trajectories, controlling the statistical features of a turbulent system offers a more robust and efficient approach. Crude first-order linear response approximations were typically employed in previous works for statistical control with small initial perturbations. This paper aims to develop two new statistical control strategies for scenarios with more significant initial perturbations and stronger nonlinear responses, allowing the statistical control framework to be applied to a much wider range of problems. First, higher-order methods, incorporating the second-order terms, are developed to resolve the full control-forcing relation. The corresponding changes to recovering the forcing perturbation effectively improve the performance of the statistical control strategy. Second, a mean closure model for the mean response is developed, which is based on the explicit mean dynamics given by the underlying turbulent dynamical system. The dependence of the mean dynamics on higher-order moments is closed using linear response theory but for the response of the second-order moments to the forcing perturbation rather than the mean response directly. The performance of these methods is evaluated extensively on prototype nonlinear test models, which exhibit crucial turbulent features, including non-Gaussian statistics and regime switching with large initial perturbations. The numerical results illustrate the feasibility of different approaches due to their physical and statistical structures and provide detailed guidelines for choosing the most suitable method based on the model properties. [ABSTRACT FROM AUTHOR]

In the present paper, we first introduce the concept of double measure r$$ r $$‐mean piecewise pseudo almost periodic for stochastic processes for r≥2$$ r\ge 2 $$. Next, we make extensive use of the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory to obtain the existence of doubly measure r$$ r $$‐mean piecewise pseudo almost periodic mild solutions for a class of impulsive non‐autonomous partial stochastic evolution equations in Hilbert spaces. In addition, we study the exponential stability of r$$ r $$‐mean piecewise pseudo almost periodic mild solutions. Finally, we give an example to confirm the reliability and feasibility of our findings. [ABSTRACT FROM AUTHOR]

It is known that a tempered exponential dichotomy (which is the version of nonuniform exponential dichotomy for random dynamical systems) can be described by admissibility of a pair of function classes defined in terms of Lyapunov norms. Moreover, for MET-systems (i.e. random dynamical systems which satisfy the assumptions of the Multiplicative Ergodic Theorem) over an ergodic base system, tempered exponential dichotomy can be described by admissibility of a pair of function spaces which are defined in terms of the original norm. In present paper, we find an admissibility description of tempered exponential dichotomies for MET-systems in terms of new output and input classes. These classes can be regarded as a critical case of admissibility classes in the known result. Finally, we use our new pair of admissible spaces to prove the roughness of tempered exponential dichotomies for parametric MET-systems and give a Hölder continuous dependence of the associated projections on the parameter. [ABSTRACT FROM AUTHOR]

For strictly entropic Riemann shock solutions of strictly hyperbolic systems of balance laws, we prove that exponential spectral stability implies large-time asymptotic orbital stability. As a preparation, we also prove similar results for constant solutions of initial value and initial boundary value problems, which seem to be new in this generality. Main key technical ingredients include the design of a nonlinear change of variables providing a hypocoercive Kawashima-type structure with dissipative boundary conditions in the high-frequency regime and the explicit identification of most singular parts of the linearized evolution, both being deduced from the mere spectral assumption. [ABSTRACT FROM AUTHOR]

In this paper, we aim to present the concept of S v -asymptotically T-periodic functions taking values in a Banach space and investigate some of their properties. Also, we establish conditions under which semi linear evolution equations in a Banach space have a unique global mild S v -asymptotically T-periodic solution. Then, we apply the results obtained to prove the existence and uniqueness of S v -asymptotically ω -periodic mild solutions to a nonautonomous semi linear differential equations. [ABSTRACT FROM AUTHOR]

Let |$G=\textrm {GL}_{2n}({\mathbb {R}})$| or |$G=\textrm {GL}_n({\mathbb {H}})$| and |$H=\textrm {GL}_n({\mathbb {C}})$| regarded as a subgroup of |$G$|⁠. Here, |${\mathbb {H}}$| is the quaternion division algebra over |${\mathbb {R}}$|⁠. For a character |$\chi $| on |${\mathbb {C}}^\times $|⁠ , we say that an irreducible smooth admissible moderate growth representation |$\pi $| of |$G$| is |$\chi _H$| -distinguished if |$\operatorname {Hom}_H(\pi , \chi \circ \det _H)\neq 0$|⁠. We compute the root number of a |$\chi _H$| -distinguished representation |$\pi $| twisted by the representation induced from |$\chi $|⁠. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math. 2011). The proof is based on the analysis of the contribution of |$H$| -orbits in a flag manifold of |$G$| to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology |$H_\ast (H, \pi \otimes \chi)$| is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair |$(G, H)$| and a finite-dimensional representation |$\chi $| of |$H$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a concept of pseudo S -asymptotically ω-periodic (PSAPω) solutions, we establish some new composition theorems. By using the roughness theory of exponential dichotomies and the contraction mapping, some sufficient conditions are obtained for the existence and uniqueness of PSAPω solutions (abbreviated in EPCA) for some difference and differential equations with piecewise constant argument. [ABSTRACT FROM AUTHOR]

Many previous works used the admissibility of function classes to characterize nonuniform exponential dichotomy (for short, NEDs) by employing Lyapunov norms. Recently, L. Zhou and W. Zhang [J. Funct. Anal. 271 (2016), pp. 1087–1129] characterized NEDs without Lyapunov norms. They utilized two admissible pairs of function classes and an assumption on certain subspaces to describe NEDs under the prerequisite of bounded growth, which plays an essential role in their arguments. However, in this paper, we remove the condition of bounded growth when characterizing NEDs. Neither bounded growth nor Lyapunov norms are used to describe NEDs in this paper. [ABSTRACT FROM AUTHOR]

For a µ;n dichotomic systems generalization of Palmer's lemma was proved. Necessary and sufficient conditions of the existence of bounded on the whole axis solutions and quasisolutions that minimize the residual norm were obtained. Index of the corresponding operator was found. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some Barbashin type characterizations for the nonuniform h-dichotomy of reversible evolution families in Banach spaces. Two necessary and sufficient conditions for the nonuniform h-dichotomy are pointed out using some important sets of growth functions. Additionally, as particular cases, we obtain a Barbashin type characterization for nonuniform exponential dichotomy and a necessary and sufficient condition for the nonuniform polynomial dichotomy. [ABSTRACT FROM AUTHOR]

Citizenship is a contested concept that can be interpreted from different – and also conflicting – perspectives. Inspired by inclusive mathematics education, we try to illustrate what it could mean to work toward inclusive citizenship. We illustrate the idea of inclusive mathematics education, showing how deaf and hearing students can learn together. We interpret inclusive mathematics education as 'meetings amongst differences'. This interpretation inspires us to see meetings amongst differences in a broader perspective. Such meetings can be considered fundamental to the process of establishing inclusive citizenship. We elaborate on this idea by exemplifying how meetings across national, economic, cultural and existential differences can be established in mathematics education. We conclude that, to work toward inclusive citizenship, it is important to establish dialogues across distances; to make students interested in and able to see situations and problems from perspectives different from their own; to construct tolerance as an expression of shared concerns; and finally, to challenge any form of We–Other dichotomy. [ABSTRACT FROM AUTHOR]

We study mean-square invariant manifolds for stochastic wave equations with nonlinear infinite-dimensional noise and weak damping, where all nonlinear terms satisfy the nonhomogeneous Lipschitz conditions. First, we consider the existence of a mean-square random dynamical system generated from the mild solution. Second, we give a careful analysis for the spectrum of the wave operator and show that the wave operator satisfies the pseudo exponential dichotomy. Finally, by using the Lyapunov-Perron method and analyzing the conditional expectation solution, we show the existence of a mean-square random invariant unstable manifold as well as a mean-square random invariant stable set. [ABSTRACT FROM AUTHOR]

Given a norm ν on ℝ², the set of ν-Dirichlet improvable numbers DIν was defined and studied in the papers (Andersen and Duke, Acta Arith. 198 (2021) 37-75 and Kleinbock and Rao, Internat. Math. Res. Notices 2022 (2022) 5617-5657). When ν is the supremumnorm, DIν = BA ∪ ℚ, where BA is the set of badly approximable numbers. Each of the sets DIν, like BA, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, BA ∩ DIν is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either BA ⊂ DIν or else BA \ DIν has full Hausdorff dimension. We give several examples for each of the two cases. The dichotomy is based on whether the critical locus of ν intersects a precompact gt-orbit, where {gt} is the one-parameter diagonal subgroup of SL2(ℝ) acting on the space X of unimodular lattices in ℝ². Thus, the aforementioned dichotomy follows from the following dynamical statement: for a lattice Λ ∈ X, either gℝΛ is unbounded (and then any precompact gℝ>0-orbit must eventually avoid a neighborhood of Λ), or not, in which case the set of lattices in X whose gℝ>0 -trajectories are precompact and contain Λ in their closure has full Hausdorff dimension. [ABSTRACT FROM AUTHOR]

This paper is concerned with piecewise pseudo almost periodic solutions of a class of interval general BAM neural networks with mixed time-varying delays and impulsive perturbations. By adopting the exponential dichotomy of linear differential equations and the fixed point theory of contraction mapping. The sufficient conditions for the existence of piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations are obtained. By adopting differential inequality techniques and mathematical methods of induction, the global exponential stability for the piecewise pseudo almost periodic solutions of the interval general BAM neural networks with mixed time-varying delays and impulsive perturbations is discussed. An example is given to illustrate the effectiveness of the results obtained in the paper. [ABSTRACT FROM AUTHOR]

For finite-dimensional linear differential systems with bounded coefficients, we prove that their exponential dichotomy on $\mathbb{R}$ is equivalent to their Ulam–Hyers stability on $\mathbb{R}$ with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam–Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant.

For a discrete time multitype supercritical Galton–Watson process (Z n) n ∈ N and corresponding genealogical tree T , we associate a new discrete time process (Z n Φ) n ∈ N such that, for each n ∈ N , the contribution of each individual u ∈ T to Z n Φ is determined by a (random) characteristic Φ evaluated at the age of u at time n. In other words, Z n Φ is obtained by summing over all u ∈ T the corresponding contributions Φ u , where (Φ u) u ∈ T are i.i.d. copies of Φ. Such processes are known in the literature under the name of Crump–Mode–Jagers (CMJ) processes counted with characteristic Φ. We derive a LLN and a CLT for the process (Z n Φ) n ∈ N in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten and Stigum (1966). [ABSTRACT FROM AUTHOR]

In this article, we experimentally investigate the nonlinear behaviour of a viscoplastic film flow down an inclined plane. We focus on the nonlinear instabilities that appear as roll waves. Roll waves are generated by perturbing a permanent flow of Herschel–Bulkley fluid (Carbopol 980) at low frequencies. To determine the local thickness of the film, we used a laser sensor and a camera to globally capture the transverse shape of the waves. For a regular forcing, the results show the existence of different regimes. First, we observe primary instabilities below the cut-off frequency at the entrance of the channel. After the exponential growth of the wave in the linear regime, we recognise the nonlinear dynamics with the existence of finite amplitude waves. This finite amplitude depends on the frequency, the Reynolds number and the inclination angle. The results show that this instability is supercritical. At moderate Reynolds numbers, the finite 2-D waves become sensitive to transverse perturbations, due to a secondary instability, and become 3-D waves. The experimental results illustrate a phenomenology of viscoplastic film flows similar to Newtonian fluids, except for the capillary waves. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish some sufficient conditions ensuring the existence and uniqueness of positive (Bohr) almost periodic solutions to a class of semilinear evolution equations of the form: u′(t)=A(t)u(t)+f(t,u(t)),t∈ℝ$$ {u}^{\prime }(t)=A(t)u(t)+f\left(t,u(t)\right),t\in \mathbb{R} $$. We assume that the family of closed linear operators (A(t))t∈ℝ$$ {\left(A(t)\right)}_{t\in \mathbb{R}} $$ on a Banach lattice X$$ X $$ satisfies the "Acquistapace–Terreni" conditions, so that the associated evolution family is positive and has an exponential dichotomy on ℝ$$ \mathbb{R} $$. The nonlinear term f$$ f $$, acting on certain real interpolation spaces, is assumed to be almost periodic only in a weaker sense (i.e., in Stepanov's sense) with respect to t$$ t $$, and Lipschitzian in bounded sets with respect to the second variable. Moreover, we prove a new composition result for Stepanov almost periodic functions by assuming only continuity of f$$ f $$ with respect to the second variable (see the condition Lemma 1‐(ii)). Finally, we provide an application to a system of Lotka–Volterra predator–prey type model with diffusion and time–dependent parameters in a generalized almost periodic environment. [ABSTRACT FROM AUTHOR]

We study linear and semi-linear differential-algebraic equations (DAEs) on the half-line ℝ + . In our strategy, we first show a characterization for the existence of exponential dichotomy for linear DAEs based on the Lyapunov-Perron method. Then, we prove the existence of invariant stable manifolds for semi-linear DAEs in the case that the evolution family corresponding to a linear DAE admits an exponential dichotomy and the non-linear forcing function fulfills the non-uniform φ-Lipschitz condition where the Lipschitz function φ belongs to wide classes of admissible function spaces such as Lp, 1 ≤ p ≤ ∞ , and Lp,q. [ABSTRACT FROM AUTHOR]

The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. Two general theoretical results on this topic are established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic lattice reaction-diffusion equation driven by a time-dependent nonlinear noise. A time-average argument and an extended Krylov-Bogolyubov method due to Da Prato and Röckner [ Seminar on stochastic analysis, random fields and applications V , Birkhäuser, Basel, 2008] are employed to prove the existence of evolution systems of probability measures. A mild condition on the time-dependent diffusion function is used to prove that the limit of every evolution system of probability measures must be an evolution system of probability measures of the limiting equation. The theoretical results are expected to be applied to various stochastic lattice systems/ODEs/PDEs in the future. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of extremals for the Trudinger–Moser functional under the weighted Sobolev norm involved with the trapping potential. Since the trapping potential is non-radial, possible extremals of the Trudinger–Moser functional do not have to be radially symmetric. As a result, one can't expect to apply the symmetrization argument to obtain the existence of extremal functions for these inequalities. Thus, the standard blow-up analysis relying on the symmetrization argument no longer works. In this paper, we will develop a new symmetrization-free approach to establish the existence of an extremal (Theorem 1.1) We first show that if the maximizing sequence does not converge strongly to an extremal function, then it must be a concentration or vanishing sequence through excluding the dichotomy phenomenon (see Lemma 2.9). Since the vanishing or concentration sequence may not be radially symmetric, this causes much challenge to exclude the vanishing or concentration phenomenon of the maximizing sequence. Then the elimination of the vanishing phenomenon can be done by comparing the supremums of the Trudinger–Moser functional under the constraint of the classical Sobolev norm (i.e., S ∞ (α) ) and weighted Sobolev norm associated with the potential (i.e., S (V , α) ) respectively. For the elimination of the concentration phenomenon of the maximizing sequence, we will develop the blow-up analysis procedure for the non-radial maximizing sequence. Furthermore, we also study the existence of extremals for the perturbed Trudinger–Moser inequality in R 2 involved with trapping potential (see Theorem 1.3). While an extremal function of the classical Trudinger–Moser and the perturbed Trudinger–Moser inequality does not exist in some situations, surprisingly, the supremum of the Trudinger–Moser and the perturbed Trudinger–Moser functional involved with the trapping potential can always be achieved. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic dynamics of a time-periodic parasitic–mutualistic model of mistletoes and birds in heterogeneous environment with the especial concerns over the spreading–vanishing scenarios, in which the Stefan class free boundary is introduced as the spreading frontier. By defining the ecological reproduction number and generalizing it as the spatial-temporal risk index, a considerably universal spreading–vanishing dichotomy and the sharp criteria are first established in birds world in the absence of mistletoes, and some estimates of the asymptotic spreading speed of the free boundary provided that spreading occurs are also obtained. Furthermore, the comprehensive considerations containing the spreading frontiers, asymptotic profiles and estimates of the asymptotic spreading speed are exhibited in mistletoes-birds world by the monotone iteration technique with the proper upper and lower solutions. The results suggest that even for the spreading case, the mistletoes population will eventually persist in long term provided that its own risk index is larger than 1, otherwise it may be eradicated. [ABSTRACT FROM AUTHOR]

In this paper, we study a quasilinear parabolic equation with free boundary. A priori estimates of the H¨older norms are established. On the basic of a priori estimates, the existence and uniqueness of the solution is proved. Furthermore, we consider spreading-vanishing dichotomy result with advection term, specifically the species either successfully spreads to the infinity as t → ∞, or it fails to establish and does out in the long run. [ABSTRACT FROM AUTHOR]

We add relevant references about which we learned after the completion of the initial work. We mainly show how the concept of exponential trichotomy can successfully replace the one of exponential dichotomy in some results from the paper in the title.