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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]

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]

By analyzing the main cable shape of the suspension bridge and the mechanical and geometric relationship between the main cable and the cable saddle, an improved algorithm is established for determining the position of the composite circular curve splay saddle. The geometric conditions of the main cable and saddle are considered to establish the equations, and the position of the composite circular curve splay saddle is obtained by solving only a one-dimensional nonlinear equation with the dichotomy method. The algorithm yields the solution interval of the dichotomy method to achieve the solution of the nonlinear equation, and the calculation process ensures convergence without any initial value. Several examples demonstrate that the improved algorithm can achieve excellent convergence. [ABSTRACT FROM AUTHOR]

This paper is concerned with the tempered exponential dichotomy for random differential systems in Banach spaces. Based on the presentation of bounded solutions for a tempered exponential dichotomous system, we give a bound under which the tempered exponentially dichotomous property of perturbed systems is preserved. Some applications of our results to stochastic partial differential equations are considered. [ABSTRACT FROM AUTHOR]

In this paper, block upper triangular systems of linear difference equations are considered, in which the coefficient matrices are not assumed invertible. The relationship between the exponential dichotomy properties of such a system and its associated block diagonal system is studied. The reason it is important to study triangular systems is that any system of linear difference equations is kinematically similar to an upper triangular system. In the bounded invertible case, it is known that for equations on the intervals J = Z + or Z − , a block upper triangular system has an exponential dichotomy if and only if the associated block diagonal system has one. However, when J = Z , only the sufficiency holds. The sufficiency extends to the noninvertible case, provided the off-diagonal matrices are bounded. However, the necessity does not hold even when J = Z + or Z − . Nevertheless, if certain conditions are added, then the necessity does hold and it is also shown that these conditions are needed since it turns out that if both the triangular and diagonal systems have dichotomies, then these extra conditions must hold. [ABSTRACT FROM AUTHOR]

In this article we study exponential dichotomies for noninvertible linear difference equations in finite dimensions. After giving the definition, we study the extent to which the projection P (k) in a dichotomy is unique. For equations on Z it is unique but for equations on Z + only its range is unique and for Z − only its nullspace. Here we strengthen Kalkbrenner's results and give a complete characterization of all possible projections. Next we study the possibility of extending the dichotomy to a larger interval. We reproduce the results of Pötzsche but also show exactly when the original projection remains unchanged. Next we prove that the roughness theorem, well known for additive perturbations, holds for multiplicative perturbations also. The proof uses ideas of Zhou, Lu and Zhang. Finally, following Ducrot, Magal and Seydi, we mention that the results by Palmer on finite time conditions on dichotomy for the invertible case can be extended to the noninvertible case. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions in the spirit of Rolewicz under which all Lyapunov exponents of a given linear cocycle are either positive or negative. As a consequence, we formulate new conditions for the existence of the so-called tempered exponential dichotomies. We consider cocycles over both maps and flows. [ABSTRACT FROM AUTHOR]

We characterize the existence of an exponential dichotomy for a nonautonomous linear delay equation via the hyperbolicity of an appropriate cocycle. An important advantage of this approach is that the base is compact under mild additional assumptions. Moreover, we give a few applications of the equivalence of the two notions of hyperbolicity. In particular, we consider the robustness and the admissibility of the equation, and we obtain stable and unstable invariant manifolds. [ABSTRACT FROM AUTHOR]

We present a theory of existence and uniqueness of μ-pseudo almost periodic solution for a delayed non-autonomous partial functional differential equation in the exponential dichotomic case, where the nonlinear operator F satisfies the ϕ-Lipschitz condition and ϕ belongs to some admissible spaces. Moreover, we prove the existence of an invariant stable manifold around the μ-pseudo almost periodic solution in that case. We give finally an application to illusrate our results. [ABSTRACT FROM AUTHOR]

This paper proposes an asymptotic approximation solution for the propagation of Love-type surface waves in a smart composite structure involving functionally graded piezoelectric material (FGPM). The considered structure is a three-layered system in which the uppermost piezoelectric (PE) layer is accomplished by an FGPM layer and piezomagnetic (PM) substrate. The distribution of material coefficients in FGPM layer is considered in two different manners: exponential and linear, along with the x-axis. The Wentzel-Kramers-Brillouin (WKB) asymptotic method is adopted to solve the differential equations. Frequency equations are derived for electrically and magnetically open and short circuit cases. Variation of phase velocity, group velocity and electromechanical coupling factor versus the wave number has been distinctly marked. Influence of different parameters (like layer width, material gradient) on phase velocity curve of different modes has been shown graphically. Comparison between phase velocity and group velocity has been also demonstrated through graphs. This study finds its application toward designing and optimization of Seismic Acoustic Wave (SAW) devices, ultrasonic transducer and Love wave sensors. [ABSTRACT FROM AUTHOR]

In this paper, we study optimal controls of a class of impulsive partial stochastic differential equations with weighted pseudo almost periodic coefficients in Hilbert spaces. Firstly, using interpolation theory, the stochastic analysis techniques and a fixed-point theorem, the existence of p-mean piecewise weighted pseudo almost periodic in distribution mild solutions for the impulsive stochastic control system is presented. Secondly, the existence of optimal pairs of system governed by impulsive partial stochastic differential equations is also obtained. Finally, application to optimal control problems of impulsive parabolic control system is considered. [ABSTRACT FROM AUTHOR]

We construct stable invariant manifolds for a large class of nonautonomous delay difference equations with infinite delay. This requires considering an appropriate class of phase spaces that are Banach spaces of sequences satisfying a certain axiom motivated by work of Hale and Kato for continuous time. We also give examples of these spaces. We consider the general cases when the linear part has a tempered exponential dichotomy and when the perturbation depends on a parameter. Moreover, we obtain the optimal regularity of the stable manifolds for Lipschitz and C 1 perturbations, jointly with respect to the base and to the parameter. [ABSTRACT FROM AUTHOR]

For a linear nonautonomous dynamics, we give a characterization of its hyperbolicity in terms of a certain cocycle, both for discrete and continuous time. The base of the cocycle is obtained taking the closure of the translations of the dynamics with respect to the topology of uniform convergence on compact sets, which in the case of discrete time corresponds to pointwise convergence. [ABSTRACT FROM AUTHOR]

Various approaches can be used to construct a model from a null distribution and a test statistic. I prove that one such approach, originating with D. R. Cox, has the property that the p-value is never greater than the Generalized Likelihood Ratio (GLR). When combined with the general result that the GLR is never greater than any Bayes factor, we conclude that, under Cox's model, the p-value is never greater than any Bayes factor. I also provide a generalization, illustrations for the canonical Normal model, and an alternative approach based on sufficiency. This result is relevant for the ongoing discussion about the evidential value of small p-values, and the movement among statisticians to "redefine statistical significance." [ABSTRACT FROM AUTHOR]

In this paper, we propose a weaker form of hyperbolicity in which the conditional stability along the stable and unstable directions is allowed to be non-uniform depending on each particular vector, instead of on each invariant space. We also give explicit examples of the notion. Moreover, we show that the size of the set of vectors for which the spoiling of the uniform exponential behaviour has a prescribed bound, measured in terms of the Lebesgue measure on the unit sphere, may grow to infinity with any given speed. [ABSTRACT FROM AUTHOR]

In this paper, using matched spaces for time scales, we introduce new types of almost periodic functions, including -almost periodic functions and -order -almost periodic functions ( -almost periodic functions). Also we introduce the definition of hull equations for homogeneous dynamic equations and obtain some existence results. Under exponential dichotomy for the corresponding homogeneous equation, we obtain the form of a -almost periodic solution with the -almost periodic affiliated function for a type of nonhomogeneous dynamic equation and we use it to study the existence of -almost periodic solutions with the -almost periodic affiliated function for new delay dynamic equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Kawahara equation [Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation, Adv Math (China).45(2016),pp.80-88] as follows: with initial data in the Sobolev space Benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488-2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on ; Acta Math Sci.37(2017),pp.385C39], first, we show that the local well-posedness is established for the initial data with ( ) and ( ) respectively. Then,using these results and conservation laws, we also prove that the IVP is globally well-posed for the initial data with ( ). Finally, benefited from ideas of [Z.Y. Zhang and J.H. Huang, Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity, Math Meth Appl Sci. 39(2016),pp.2488-2513; Z.Y. Zhang, J.H. Huang, Z.H. Liu and M.B. Sun, On the unique continuation property for the modified Kawahara equation,Adv Math (China).45(2016),pp.80-88], i.e. using complex variables technique and Paley-Wiener theorem, we prove the unique continuation property (UCP henceforth) for the IVP. [ABSTRACT FROM AUTHOR]

Criteria for the existence of limit-periodic solutions are established to nonlinear difference as well as differential systems governed by a Stepanov-like limit-periodic forcing. The restoring nonlinearities need not be globally Lipschitzean. The obtained sufficient conditions for difference systems are significantly more effective than those for differential systems. Nevertheless, in the continuous case, our theorems also extend some known results. A possibility of more explicit formulations are discussed in concluding remarks. [ABSTRACT FROM AUTHOR]

The Sacker–Sell (also called dichotomy or dynamical) spectrumis an important notion in the stability theory of nonautonomous dynamical systems. For instance, when dealing with variational equations on the (nonnegative) half line, the set Σ+determines uniform asymptotic stability or instability of a solution and more general, it is crucial to construct invariant manifolds from the stable hierarchy. Compared to the spectrum associated to dichotomies on the entire line, Σ+has stronger and more flexible perturbation features. In this paper, we study continuity properties of the Sacker–Sell spectrum by means of an operator-theoretical approach. We provide an explicit example that the generally upper-semicontinuous set Σ+can suddenly collapse under perturbation, establish continuity on the class of equations with discrete spectrum and identify system classes having a continuous spectrum. These results for instance allow to vindicate numerical approximation techniques. [ABSTRACT FROM PUBLISHER]

We construct smooth stable invariant manifolds for a class of delay equations with piecewise constant delay, for any sufficiently small perturbation of a nonuniform exponential dichotomy. We build on former work for perturbations of a uniform exponential dichotomy, also for delay equations with piecewise constant delay. These equations can be described as delay equations with an impulsive behavior of the derivative, such that at certain times the derivative changes abruptly. [ABSTRACT FROM PUBLISHER]

The present study describes a possible driving mechanism for a self-excited oscillation observed in motorcycle dynamics, often referred to as chatter. This phenomenon, affecting the performance of road racing motorcycles, has been simulated in straight running braking manoeuvres with multibody motorcycle models. It involves rear suspension bounce and driveline oscillation in the frequency range between 17 and 22 Hz. A simplified model of a motorcycle rear suspension with chain transmission is proposed and its stability in equilibrium configurations is studied via eigenvalue analysis. The sensitivity with respect to all its governing parameters is analysed by means of stability maps and the self-excitation mechanism is explained with the aid of energy balance analysis and phase diagrams. It is found that the key role for the instability onset is played by the gradient of the nonlinear characteristic slip function of the tyre. [ABSTRACT FROM PUBLISHER]

We prove the existence of some integral manifolds for the nonautonomous delayed partial differential equationwhere (A,D(A)) satisfies the Hille–Yosida condition,is a family of operators insatisfying some measurability and boundedness conditions and the nonlinear forcing termfsatisfies ∥f(t,ϕ)−f(t,ψ)∥≤φ(t)∥ϕ−ψ∥𝒞, whereφbelongs to an admissible space andϕ,ψ∈𝒞: = C([−r,0],X). To achieve our target, we resort to the properties of admissible spaces, the extrapolation theory, the characterization of exponential dichotomy and exponential trichotomy, Lyapunov–Perron method, and other technical structures. Finally, an example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

The rate-controlled constrained-equilibrium (RCCE) method is a thermodynamics-based dimension reduction method that enables representation of chemistry involvingnschemical species in terms of a smaller number,nr, of constraints, and thus reduces the computational burden imposed by detailed chemical kinetics. The accuracy in the reduced description by RCCE strongly depends on the specification of constraints. In this work, the connection between the RCCE method and the conventional partial equilibrium assumption (PEA) method is revisited. It is shown that with the constraints being specified based on rate-controlling slow reactions, the RCCE method is equivalent to PEA. A criterion based on time scale analysis is then employed for the identification of fast reactions. Then the importance index, which represents the contribution of individual species to the slow dynamics of a reactive system, is proposed for the selection of species constraints in RCCE. This makes the thermodynamics-based constrained equilibrium manifolds (CEMs) a good approximation to the actual kinetics-controlled slow invariant manifolds (SIMs) in reactive systems. The method is demonstrated in the reduced descriptions of perfectly stirred reactors (PSRs) burning stoichiometric methane/air mixtures. Sample reaction states ranging from near-equilibrium conditions to the extinction turning points on theS-curve are examined for the identification of fast reactions and species constraints. Reduced descriptions of transient PSRs by RCCE are subsequently performed with their accuracy assessed against the full descriptions, in which all the chemical species are transported. The study shows that the dynamics predicted by RCCE agrees well with that from the full description and that near-limit reaction states are important for the selection of species constraints in RCCE. [ABSTRACT FROM PUBLISHER]

In this paper, we study the structure and properties of the dichotomy spectrum for linear difference equations in Banach spaces avoiding the commonly made assumption of having a compact flow. In this situation, any compact subset of the positive-half line may occur as a dichotomy spectrum. To overcome this difficulty, we introduce the upper and lower dichotomy index by means of a measure of noncompactness. It allows to decompose the dichotomy spectrum into the upper, essential and lower dichotomy spectrum. The main result is a Spectral Theorem which describes all possible cases of the upper and lower dichotomy spectrum and yields a “nonautonomous linear algebra” appropriate also for systems in the absence of Lyapunov regularity. Furthermore, we give explicit examples of difference equations to illustrate the different cases of the Spectral Theorem and to underline that a new form of the spectrum not present in the literature can occur. Finally, we indicate applications to nonlinear difference equations and the continuous time situation. [ABSTRACT FROM AUTHOR]

In this article, we study almost automorphic solutions for semilinear stochastic differential equations driven by Lévy noise. We establish the existence and uniqueness of bounded solutions by using the Banach fixed point theorem, the exponential dichotomy property and stochastic analysis techniques. Furthermore, this unique bounded solution is almost automorphic in distribution under slightly stronger conditions. We also give two examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this article, study the almost periodic profile of solutions for nonautonomous difference equations in Banach spaces. We apply our results in population dynamics. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions ensuring the topological and strong topological equivalence of two perturbed difference systems whose linear part has a property of generalized exponential dichotomy. When the exponential dichotomy is verified, we obtain a strongly and Hölder topological equivalence. [ABSTRACT FROM PUBLISHER]

For a nonautonomous dynamics with discrete time defined by a sequence of matrices, we obtain sharp lower and upper bounds for the Lyapunov coefficient of regularity. This has the advantage of avoiding considering the adjoint dynamics (in contrast to what happens with the regularity coefficients considered by Perron and Grobman). We also show that the dynamics can always be reduced to one defined by upper triangular matrices with the additional properties that the canonical basis is normal and ordered. Moreover, we show in a simpler manner that the Lyapunov coefficient of regularity is related to the notion of nonuniform hyperbolicity, more precisely to the nonuniform part of a nonuniform exponential contraction or a nonuniform exponential dichotomy. Finally, as an application of this relation, we show that from the point of view of ergodic theory, for almost all trajectories with negative Lyapunov exponents the nonuniformity can be made arbitrarily small. [ABSTRACT FROM PUBLISHER]

We calibrate a dynamic stochastic general equilibrium model that features a transmission mechanism with different types of government spending, while the literature usually treats government spending as a homogenous compound. In this regard, we manage to distinguish between different types of government spending (namely: government investment, government wage component consumption and non-wage component consumption) where each type of spending has a varied role in the economy. The government wage increase has the largest positive effect both on private consumption and output by affecting the economy through the government production. This is a natural consequence of government production being complementary to private consumption in our model. Other two government spending types, namely government non-wage consumption and government investment, also have positive effects on output, whereas their responses on (private) consumption are mostly negative. These results provide an alternative explanation for the wide range of multipliers existing in the literature as our setup enables them to produce different effects on macroeconomic variables. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behaviour of the positive solutions of the system of two difference equationswhereare positive constants and the initial valuesare positive numbers. [ABSTRACT FROM AUTHOR]

We point out an error in the above short note and correct it under additional assumptions. [ABSTRACT FROM AUTHOR]

We develop a new version of a known theorem obtained by Van Minh, Rabiger, Schnaubelt in [N. Van Minh, F. Rabiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equ. Oper. Theory 32 (1998) pp. 332-353]. We rely completely on the classical 'test functions' method designed by Perron in 1930. The advantage of such a version is that is more readable since the classical method of Perron have been known for decades and that we do not involve a sophisticated mathematical machinery. Our approach is in contrast with the general philosophy of 'autonomization' the nonautonomous system, since we do not require to attach the evolution semigroup. Also we point out a discrete-time version of our approach extending some known results given by Li and Henry. [ABSTRACT FROM AUTHOR]

We propose a new and unified approach for the study of dichotomy of variational difference equations, establishing a link between control methods and basic techniques from interpolation theory. We obtain necessary and sufficient conditions for the existence of uniform dichotomy and, respectively, for uniform exponential dichotomy of variational difference equations in terms of the admissibility of general pairs of sequence spaces. We provide a classification of the main classes of sequence spaces where the input spaces and the output spaces may belong to, for each dichotomy property and prove that the hypotheses on the underlying sequence spaces cannot be removed. The obtained results extend the framework to the study of dichotomy of variational difference equations, hold without any requirement on the coefficients and are applicable to all systems of variational difference equations. [ABSTRACT FROM AUTHOR]

We prove that if a difference equation on Z has an exponential dichotomy on sufficiently many intervals of sufficiently large fixed length and with uniform constants and exponents, then it has an exponential dichotomy on Z. We give applications of the result to slowly varying systems, almost periodic systems, a perturbation theorem, the variational equation along pseudo orbits lying in a hyperbolic set of a diffeomorphism, transversality of homoclinic orbits of a diffeomorphism and hyperbolicity of a compact invariant set of a diffeomorphism. [ABSTRACT FROM AUTHOR]

In many ways, exponential dichotomies are an appropriate hyperbolicity notion for nonautonomous linear differential or difference equations. The corresponding dichotomy spectrum generalizes the classical set of eigenvalues or Floquet multipliers and is therefore of eminent importance in a stability theory for explicitly time-dependent systems, as well as to establish a geometric theory of nonautonomous problems with ingredients like invariant manifolds and normal forms, or to deduce continuation and bifurcation techniques. In this note, we derive some invariance and perturbation properties of the dichotomy spectrum for nonautonomous linear difference equations in Banach spaces. They easily follow from the observation that the dichotomy spectrum is strongly related to a weighted shift operator on an ambient sequence space. [ABSTRACT FROM AUTHOR]

We characterize the exponential dichotomy of difference equations with infinite delay. We apply the results to study the robustness of exponential dichotomy. This kind of dichotomy gives us relevant information about boundedness of solutions for several perturbed quasi linear systems with infinite delay. Applications to Volterra difference equations are shown. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of pseudo almost automorphic solution for some partial functional differential equations with infinite delay. We assume that the undelayed part is not necessarily densely defined and satisfies the Hille-Yosida condition. We use the variation of constant formula developed recently in 1 to get the existence and uniqueness of pseudo almost automorphic solution when the linear equation has an exponential dichotomy. We also give an application of the abstract results to a Lotka-Volterra model with diffusion. [ABSTRACT FROM AUTHOR]

Classical results of the Hartman-Wintner type lead to the asymptotic integration of certain types of linear, non-autonomous systems of differential or difference equations. Here, we show how to extend such results to linear dynamic equations on time scales having bounded graininess, using both averaged as well as pointwise dichotomy conditions. [ABSTRACT FROM AUTHOR]

In this work, we study the existence of Cn-almost periodic solutions and Cn-almost automorphic solutions (n ≥ 1), for partial neutral functional differential equations. We prove that the existence of a bounded integral solution on + implies the existence of Cn-almost periodic and Cn-almost automorphic strict solutions. When the exponential dichotomy holds for the homogeneous linear equation, we show the uniqueness of Cn-almost periodic and Cn-almost automorphic strict solutions. [ABSTRACT FROM AUTHOR]

Mantissa distributions generated by dynamical processes continue to attract much interest. In this article, it is demonstrated that one-dimensional projections of (at least) almost all orbits of many multi-dimensional nonautonomous dynamical systems exhibit a mantissa distribution that is a convex combination of a trivial point mass and Benford's Law, i.e. the mantissa distribution of the non-trivial part of the orbit is asymptotically logarithmic, typically for all bases. Both linear and power-like systems are considered, and Benford behaviour is found to be ubiquitous for either class. The results unify previously known facts and extend them to the nonautonomous setting, with many of the conclusions being best possible in general. [ABSTRACT FROM AUTHOR]

We give a new characterization of exponential dichotomy of linear difference equations in terms of -admissibility. The obtained result is a discrete version of the recent new result of Preda et al. [( L p , L q )-admissibility and exponential dichotomy of evolution processes on the half-plane, Integral Equations Operator Theory , 49 (2004), 405–418] on exponential dichotomy of evolution processes on the half-plane. Then we derive a new characterization of exponential stability of linear difference equations. This extends the recent new result of Pituk M. [A criterion for the exponential stability of linear difference equations, Applied Mathematics Letters , 17 (2004), 779–783] and of Aulbach and Nguyen Van Minh. [The concept of spectral dichotomy for linear difference equations II, Journal of Difference Equations and Applications , 2 (1996), 251–262]. [ABSTRACT FROM AUTHOR]