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.

In this paper, we prove some new reverse dynamic inequalities of Renaud- and Bennett-type on time scales. The results are established using the time scales Fubini theorem, the reverse Hölder inequality and a time scales chain rule.

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$. Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.

We show that the solution of the dynamic boundary value problem $y^{{\rm\Delta}{\rm\Delta}}=f(t,y,y^{{\rm\Delta}})$, $y(t_{1})=y_{1}$, $y(t_{2})=y_{2}$, on a general time scale, may be delta-differentiated with respect to $y_{1},~y_{2},~t_{1}$ and $t_{2}$. By utilising an analogue of a theorem of Peano, we show that the delta derivative of the solution solves the boundary value problem consisting of either the variational equation or its dynamic analogue along with interesting boundary conditions.

The settlement record of the Neolithic of the northern Alpine foreland is used to address the question of what difference having high-resolution chronology — in this case principally provided by dendrochronology — makes to the kinds of narrative we seek to write about the Neolithic. In a search for detailed histories, three kinds of scale are examined. The longer-term development of cultural patterns and boundaries is found to correlate very imprecisely with the character and architecture of settlements. Individual houses and settlements were generally short-lived, suggesting considerable fluidity in social relations at the local level. Greater continuity can be found in the landscape, perhaps involving more than individual communities. We argue that the particular history of the northern Alpine foreland is best understood by interweaving multiple temporal scales, an approach that will need to be extended to other case studies.

Using Krasnoselskii’s fixed point theorem, we deduce the existence of periodic solutions of nonlinear system of integro-dynamic equations on periodic time scales. These equations are studied under a set of assumptions on the functions involved in the equations. The equations will be called almost linear when these assumptions hold. The results of this paper are new for the continuous and discrete time scales.

In this work we review the aggregation of variables method for discrete dynamicalsystems. These methods consist of describing the asymptotic behaviour of a complex systeminvolving many coupled variables through the asymptotic behaviour of a reduced systemformulated in terms of a few global variables. We consider population dynamics modelsincluding two processes acting at different time scales. Each process has associated a mapdescribing its effect along its specific time unit. The discrete system encompassing bothprocesses is expressed in the slow time scale composing the map associated to the slow oneand the k-th iterate of the map associated to the fast one. In the linear case a result isstated showing the relationship between the corresponding asymptotic elements of bothsystems, initial and reduced. In the nonlinear case, the reduction result establishes theexistence, stability and basins of attraction of steady states and periodic solutions ofthe original system with the help of the same elements of the corresponding reducedsystem. Several models looking over the main applications of the method to populationsdynamics are collected to illustrate the general results.

We introduce the concept of ‘shift operators’ in order to establish sufficient conditions for the existence of the resolvent for the Volterra integral equation on time scales. The paper will serve as the foundation for future research on the qualitative analysis of solutions of Volterra integral equations on time scales, using the notion of the resolvent.

Approximate aggregation techniques consist of introducing certain approximations thatallow one to reduce a complex system involving many coupled variables obtaining a simplerʽʽaggregated systemʼʼ governed by a few variables. Moreover, they give results that allowone to extract information about the complex original system in terms of the behavior ofthe reduced one. Often, the feature that allows one to carry out such a reduction is thepresence of different time scales in the system under consideration. In this work we dealwith aggregation techniques in stochastic discrete time models and their application tothe study of multiregional models, i.e., of models for an age structured populationdistributed amongst different spatial patches and in which migration between the patchesis usually fast with respect to the demography (reproduction-survival) in each patch.Stochasticity in population models can be of two kinds: environmental and demographic. Wereview the formulation and the main properties of the dynamics of the different models forpopulations evolving in discrete time and subjected to the effects of environmental anddemographic stochasticity. Then we present different stochastic multiregional models withtwo time scales in which migration is fast with respect to demography and we review themain relationships between the dynamics of the original complex system and the aggregatedsimpler one. Finally, and within the context of models with environmental stochasticity inwhich the environmental variation is Markovian, we make use these techniques to analyzequalitatively the behavior of two multiregional models in which the original complexsystem is intractable. In particular we study conditions under which the population goesextinct or grows exponentially.