The Brockett problem in the theory of nonstationary stabilization of linear differential equations

In the present work a review of algorithms for nonstationary linear stabilization is given. In many cases these algorithms, together with the criterion of nonstabilizing, allow us to obtain a solution of the Brockett problem

The time-varying stabilization of linear discrete control systems

The Brockett stabilization problem for linear discrete control systems is considered. The method of synthesis of time-varying feedback for stabilization is described

Longtime dynamics in adaptive gain control systems

We study the longtime dynamics of a nonlinear adaptive control system introduced by Mareels et al. [10] to control the behavior of a plant which can be described by a finite dimensional SISO linear time invariant system stabilizable by a high gain output feedback. We apply frequency domain methods to derive conditions for global stability, to approximate the region containing the global attractor and to estimate its Hausdorff dimension

Pendulum with positive and negative dry friction. Continuum of homoclinic orbits

A two-order differential equation of pendulum with dry friction is considered. The existence of a continuum of homoclinic orbits with various homotopic properties on the cylinder is proven

A method of constructing of dynamical systems with bounded nonperiodic trajectories

A fifth-order system is considered for which the existence of a set of bounded trajectories that are neither periodic nor almost periodic is proven by means of analytical methods. The set is situated in the region of dissipation and has a positive Lebesgue measure

Spreadsheets and the discovery of new knowledge

The paper introduces a new class of polynomials discovered as an extended inquiry into a two-parametric difference equation using a spreadsheet. These polynomials possess a number of interesting properties connected to the notion of a generalized golden ratio and can be used as a background for a spreadsheet-enhanced teaching of mathematics. The paper reflects on activities designed for a technology-rich mathematics education course for prospective teachers of secondary mathematics

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered

The Brockett problem in the theory of nonstationary stabilization of linear differential equations

In the present work a review of algorithms for nonstationary linear stabilization is given. In many cases these algorithms, together with the criterion of nonstabilizing, allow us to obtain a solution of the Brockett problem