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In this paper, we establish Hartman-Grobman theorem for mean hyperbolic systems on the whole axis $ \mathbb{R} $ and half axis $ \mathbb{R^+} $. Mean hyperbolic systems possess fixed average contraction and expansion rates measured at sufficiently long evolution time. The coexistence of expansion and contraction behaviors in generalized stable (unstable) subspace leads to non-hyperbolic phenomena during evolution process. For mean hyperbolic systems, the Hölder regularity and continuous dependence on perturbation for $ h(t,x) $ are discussed. Particularly, the damped wave equation with variable speed of propagation can be applied to illustrate our main result. [ABSTRACT FROM AUTHOR]

We study the existence of an inertial manifold for the fully non-autonomous evolution equation of the form du dt + A (t) u (t) = f (t , u) , t ∈ R , in certain admissible spaces. We prove the existence of such an inertial manifold in the cases that the family of linear partial differential operators (A (t)) t ∈ R generates an evolution family (U (t , s)) t ≥ s satisfying certain dichotomy estimates, and the nonlinear forcing term f(t, x) satisfies the φ -Lipschitz condition, i.e., f (t , x 1) - f (t , x 2) ⩽ φ (t) A (t) θ (x 1 - x 2) , where φ (·) belongs to some admissible function space such that certain dichotomy gap condition holds. This dichotomy gap condition, on the one hand, extends the spectral gap condition known in the case of autonomous equations, on the other hand, provides a chance to come over the restricted spectral gap condition. [ABSTRACT FROM AUTHOR]

In this paper, we provide a necessary and sufficient condition ensuring the property of exponential dichotomy for periodic linear systems of generalized differential equations. This condition allows us to revisit a recent result of admissibility, obtaining an alternative formulation with a particular simplicity. [ABSTRACT FROM AUTHOR]

We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain nonautonomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to the state variable is uniformly negative in a neighborhood of the prescribed set. The results are applicable to important classes of model equations including the logistic equation, whose conditional shadowing has recently been studied. Several examples are constructed showing that the obtained conditions are optimal. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and stability of traveling waves in a three-variable reaction-diffusion-mechanics system, which was derived by Matt Holzer, Arjen Doelman, and Tasso J. Kaper [15]. This system consists of a modified FitzHugh-Nagumo equation coupled with two elasticity equations. We provide the proof of existence of traveling pulses by using geometric singular perturbation theory and an exchange lemma. Then, we analyze the spectrum of linearized operators about traveling pulses by combining geometric singular perturbation theory and the Lin-Sandstede method, and we prove that the traveling pulse is spectrally stable. Precisely, we show that there is at most a nontrivial eigenvalue near the origin, which turns out to be negative. [ABSTRACT FROM AUTHOR]

In this article, we first showed that conditions given by Hale and Lin, Steinlein and Walther, and Sander, which ensured the presence of chaotic dynamics near a homoclinic orbit of a non-invertible map, are equivalent to the exponential dichotomy of the variational equation along the homoclinic orbit. Next, we studied the notion of generalized exponential dichotomy, which arose from Steinlein and Walther's notion of hyperbolicity. Finally, we corrected a slight mistake in our article 'Exponential dichotomy for noninvertible linear difference equations', which appeared in volume 27 of the Journal of Difference Equations and Applications. [ABSTRACT FROM AUTHOR]

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ 1 , μ 2) -exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ 1 , μ 2) -exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research. [ABSTRACT FROM AUTHOR]

We consider the existence of invariant manifolds to evolution equations u'(t)=Au(t), A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A(0)=0 under the assumption that its proto-derivative \partial A(x) exists and is continuous in x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U and V in a Banach space \mathbb {X} is defined as d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \|, where U_\mu and V_\mu are the Yosida approximations of U and V, respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new. [ABSTRACT FROM AUTHOR]

This study deals with the Ulam stability of nonautonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two‐dimensional nonautonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on (−∞,∞)$(-\infty,\infty)$, but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for nonautonomous systems other than periodic systems. [ABSTRACT FROM AUTHOR]

Starting from a characterization of polynomial dichotomy by means of admissibility, recently proved in [Dragicevic, D.; Sasu, A. L.; Sasu, B. Admissibility and polynomial dichotomy of discrete nonautonomous systems. Carpath. J. Math. 38 (2022), 737-762.], the aim of this paper is to explore the roughness of polynomial dichotomy in the presence of perturbations and to obtain a new robustness criterion. We show that the polynomial dichotomy is robust when subjected to linear additive perturbations which are bounded by a well-chosen sequence. We emphasize that the new bounds imposed to the perturbation family improve and extend the previous approaches. Furthermore, we mention that the main result applies to discrete nonautonomous systems in Banach spaces with the only requirement that their propagators exhibit a polynomial growth. [ABSTRACT FROM AUTHOR]

We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analyzing the complexity for restricted patterns and restricted hosts. Specifically, we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H) ⋅ |G|O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes Q as our central objects of study and establish the following crisp dichotomies as consequences of the exponential time hypothesis: (1) Counting k-matchings in a graph G ∈ Q is fixed-parameter tractable if and only if Q is nowhere dense. (2) Counting k-independent sets in a graph G ∈ Q is fixed-parameter tractable if and only if Q is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if Q is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colorable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19). At the same time, our proofs are much simpler: using structural characterizations of somewhere dense graphs, we show that a colorful version of a recent breakthrough technique for analyzing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of Eberlein weakly almost periodic solutions for differential equations of the form u ′ = A u + f (t) and u ′ = A (t) u + f (t) . In the first scenario, when A generates a strongly asymptotically semigroup, we establish the existence of Eberlein-weakly almost periodic solutions, thereby extending and improving a previous result in Zaidman(Ann Univ Ferrara 14(1): 29–34, 1969). In the second case, we consider a more general situation where A(t) is a (possibly nonlinear) operator satisfying a monotony condition. Unlike most existing works in the literature, our approach does not rely on tools of exponential dichotomy and Lipschitz nonlinearity. Lastly, we illustrate the practical relevance of our findings by presenting real-world models, including a hematopoiesis model, that exemplify the key findings. A numerical simulation is also provided. [ABSTRACT FROM AUTHOR]

In this paper, the normal forms of nonautonomous nonlinear systems with discrete time are investigated. We first employ the nonuniform kinematic similarity to prove the nonuniform dichotomy spectrum theorem, which is not based on linear integral manifolds in most of the previous works. Using this spectral theorem, we extend smooth normal forms from autonomous difference systems to nonautonomous ones. [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]

The aim of this paper is to study the robustness of the family of pullback attractors associated with a non-autonomous coupled system of strongly damped wave equations, which is a modified version of the well known Klein–Gordon–Zakharov system. Under appropriate hyperbolicity conditions, we establish the gradient-like structure of the limit pullback attractor associated with this evolution system, and we prove the continuity of the family of pullback attractors at zero. [ABSTRACT FROM AUTHOR]

In this review paper, we present some basic notions and properties of quaternionic exponentially dichotomous operators. Some perturbation results of quaternionic exponentially dichotomous operators are illustrated which will help to consider the exponential dichotomous solutions to quaternionic evolution equations through semigroup theory. [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]

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]

We show that the tempered spectrum of a sufficiently small perturbation of a sequence of matrices varies little, in the sense that it is contained in a small open neighbourhood of the tempered spectrum of the original sequence. In addition, we show that for perturbations that decay exponentially, all the Lyapunov exponents of the perturbation belong to the tempered spectrum of the original sequence. [ABSTRACT FROM AUTHOR]

We prove that smoothness of nonautonomous linearization is of class C 2. Our approach admits the existence of stable and unstable manifolds determined by a family of nonautonomous hyperbolicities, including the non uniform exponential case, while for the classic exponential dichotomy we obtain the same class of differentiability except for a zero Lebesgue measure set. Moreover, our goal is reached without spectral conditions. [ABSTRACT FROM AUTHOR]

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 investigate the problem of the exponential stability of a stationary solution for a hyperbolic system with nonlocal characteristic velocities and measurement error. The formulation of the initial boundary value problem of boundary control for the specified hyperbolic system is given. A difference scheme is constructed for the numerical solution of the considered initial boundary value problem. The definition of the exponential stability of the numerical solution in ℓ 2 -norm with respect to a discrete perturbation of the equilibrium state of the initial boundary value difference problem is given. A discrete Lyapunov function for a numerical solution is constructed, and a theorem on the exponential stability of a stationary solution of the initial boundary value difference problem in ℓ 2 -norm with respect to a discrete perturbation is proved. [ABSTRACT FROM AUTHOR]

Low-fidelity analytic and computational wave drag prediction methods assume linear aerodynamics and small perturbations to the flow. Hence, these methods are typically accurate for only very slender geometries. The present work assesses the accuracy of these methods relative to high-fidelity Euler, compressible computational-fluid-dynamics solutions for a set of axisymmetric geometries with varying radius-to-length ratios (R / L) . Grid-resolution studies are included for all computational results to ensure grid-resolved results. Results show that the low-fidelity analytic and computational methods match the Euler CFD predictions to around a single drag count (∼ 1.0 × 10 − 4 ) for geometries with R / L ≤ 0.05 and Mach numbers from 1.1 to 2.0. The difference in predicted wave drag rapidly increases, to over 30 drag counts in some cases, for geometries approaching R / L ≈ 0.1 , indicating that the slender-body assumption of linear supersonic theory is violated for larger radius-to-length ratios. All three methods considered predict that the wave drag coefficient is nearly independent of Mach number for the geometries included in this study. Results of the study can be used to validate other numerical models and estimate the error in low-fidelity analytic and computational methods for predicting wave drag of axisymmetric geometries, depending on radius-to-length ratios. [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]

We consider conservation laws with nonlocal velocity and show, for nonlocal weights of exponential type, that the unique solutions converge in a weak or strong sense (dependent on the regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal weight approaches a Dirac distribution. To this end, we first establish a uniform total variation bound on the nonlocal velocity, which can be used to pass to the limit in the weak solution. For the required entropy admissibility, we use a tailored entropy-flux pair and take advantage of a well-known result that a single strictly convex entropy-flux pair is sufficient for uniqueness, given some additional constraints on the velocity. For general weights, we show that the monotonicity of the initial datum is preserved over time, which enables us to prove convergence to the local entropy solution for rather general kernels if the initial datum is monotone. This case covers the archetypes of local conservation laws: shock waves and rarefactions. These results suggest that a "nonlocal in the velocity" approximation might be better suited to approximating local conservation laws than a nonlocal in the solution approximation, in which such monotonicity only holds for specific velocities. [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]

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]

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]

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]

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 work we study the response for a class of Lotka-Volterra preypredator systems with diffusion and time-dependent parameters to a large class of oscillatory type functions, namely the pseudo almost automorphic type oscillations. To this end, using the exponential dichotomy approach and a fixed point argument, we propose to analyze a class of nonautonomous semilinear abstract evolution equation of the form (*) z'(h) = A(h)z(h)+g(h, z(h)), h ∈ R, where A(h), h ∈ R is a family of closed linear operators acting in a Banach space T, the nonlinear term g is µ-pseudo-almost automorphic in a weak sense (Stepanov sense) with respect to h and Lipschitzian in T with respect to the second variable. Therefore, according to the results obtained for equation (*) we establish the existence and uniqueness of µ-pseudo-almost automorphic solutions in the strong sense (Bohr sense) to a nonautonomous system of reactiondiffusion equations describing a Lotka-Volterra prey-predator model with diffusion and time-dependent parameters in a generalized almost automorphic environment. [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]

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]

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 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]

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]