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## Linear switching systems with slow growth of trajectories

We prove the existence of positive linear switching systems (continuous time), whose trajectories grow to infinity, but slower than a given increasing function. This implies that, unlike the situation with linear ODE, the maximal growth of trajectories of linear systems may be arbitrarily slow. For systems generated by a finite set of matrices, this phenomenon is proved to be impossible in dimension 2, while in all bigger dimensions the sublinear growth may occur. The corresponding examples are provided and several open problems are formulated.

We summarize some of the recent works, devoted to the study of one-dimensional (pseudo)group actions and codimension one foliations. We state a conjectural alternative for such actions (generalizing the already obtained results) and describe the properties in both alternative cases. We also discuss the generalizations for holomorphic one-dimensional actions. Finally, we state some open questions that seem to be already within the reach.

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has two purely imaginary eigenvalues, while the other eigenvalues lie outside the imaginary axis. We study the reducibility of such systems to pseudonormal form. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

Nonlinear differential dynamic model of the relation between the branches of production was proposed. Mathematically, this model is expressed as a system of first-order ODE. Dynamic variables of the model – the value of the output of each branch of production. Each differential equation of the system includes independent growth and diminution of finished goods; growth and decline of production related to the production of allied industries. Two models were proposed: a model with Malthusian products growth (model with no restrictions on the amount of product), the model with the Verhulst limiting of the growth of output. The equilibrium points of dynamical systems, system stability were determined as well as the qualitative analysis of dynamic systems was made.

We present an approach to study degenerate ODE with periodic nonlinearities; for resonant higher order nonlinear equations L(p)x=f(x)+b(t), p=d/dt, with 2pi-periodic forcing b and periodic f we give multiplicity results, in particular, conditions of existence of infinite and unbounded sets of 2pi-periodic solutions.

The volume is dedicated to Stephen Smale on the occasion of his 80th birthday. Besides his startling 1960 result of the proof of the Poincaré conjecture for all dimensions greater than or equal to five, Smale’s ground breaking contributions in various fields in Mathematics have marked the second part of the 20th century and beyond. Stephen Smale has done pioneering work in differential topology, global analysis, dynamical systems, nonlinear functional analysis, numerical analysis, theory of computation and machine learning as well as applications in the physical and biological sciences and economics. In sum, Stephen Smale has manifestly broken the barriers among the different fields of mathematics and dispelled some remaining prejudices. He is indeed a universal mathematician. Smale has been honored with several prizes and honorary degrees including, among others, the Fields Medal(1966), The Veblen Prize (1966), the National Medal of Science (1996) and theWolf Prize (2006/2007).

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.