Vol. 6, No. 2 (2017): Regular Issue

An oscillator which impacts against a rigid barrier is taken into account. A cycle with zero impact velocity is discussed. The main result of this article concerns stability of the grazing cycle. A significant attention to a model with the variable coefficient of restitution depending on velocity is paid. The mechanical reasons for that are provided as well as new theoretical advantages have been discovered for the investigation of dynamics near a grazing cycle. The W-map which reduces the system with variable moments of impacts to that with fixed moments and simplifies the analysis, is defined. A new type of linearization system with two compounds is applied to investigate the stability of the grazing cycle whose existence is easily examined. A new approach to suppress a singularity, caused by the tangency, in linearization has been developed. Simulations are provided to visualize the stability of the grazing cycle.

In this paper, we generalize the work of Bender and co-workers to derive new partially-integrable hierarchies of various ℘ T -symmetric, nonlinear partial differential equations. The possible integrable members are identified employing the Painlevé Test, a necessary but not sufficient, integrability condition, and are indexed by the integer n, corresponding to the negative of the order of the dominant pole in the singular part of the Painlevé expansion for the solution. For the ℘ T -symmetric Korteweg-de Vries (KdV) equation, as with some other hierarchies, the first or n = 1 equation fails the test, the n = 2 member corresponds to the regular KdV equation, while the remainder form an entirely new, possibly integrable, hierarchy. Bäcklund Transformations and analytic solutions of the n = 3 and n = 4 members are derived. The solutions, or solitary waves, prove to be algebraic in form. The ℘ T -symmetric Burgers’ equation fails the Painlevé Test for its n = 2 case, but special solutions are nonetheless obtained. Also, a ℘ T - Symmetric hierarchy of the (2+1) Burgers’ equation is analyzed. The Painlevé Test and invariant Painlevé analysis in (2+1) dimensions are utilized, and BTs and special solutions are found for those cases that pass the Painlevé Test.

We introduce a new class of quadratic stochastic operators corresponding to cyclic groups. We study the set of fixed points and prove that almost all (w.r.t. Lebesgue measure) trajectories of such operators converge to the center of the simplex. For the cyclic groups of order 2n we show that for any subgroup corresponding quadratic stochastic operator is a regular operator.

We find the Lie point symmetries of the generalized two-component Hunter-Saxton system. Then we show that it is nonlinearly self-adjoint and establish the corresponding conservation laws using a recent theorem of Nail Ibragimov which enables one to determine conservation laws for problems without variational structure. Finally we obtain some invariant solutions.

We study the existence and uniqueness of a mild solution of fractional impulsive differential equations with nonlocal conditions. Here we consider fractional derivative in the non-instantaneous impulsive conditions. We use fixed point techniques and resolvent operators to prove our existence results.

The Hodgkin and Huxley model describes the electrophysiology of the membrane of the giant squid’s axon. This model was developed from measurements of passive and active electrical behavior of the nerve cell. The four-coupled nonlinear ordinary differential equations, which describe the model, are based on the behavior of sodium and potassium channels. The aim of this work is to study the dynamic behavior of a neuronal physiological system described by the Hodgkin-Huxley (H-H) model through the analysis and interpretation of numerical simulations.

One of the most prominent feature of research in Russia and the former Soviet Union is so-called scientific schools. It is a collaboration of researchers with a common scientific background working, as a rule, together in a specific city or even at an institution. The genealogical tree of scientific schools on nonlinear dynamics in Russia and the former Soviet Union is grown. We use these terminology in a broad sense including theory of dynamical systems and chaos and its applications in nonlinear physics. In most cases we connect two persons if one was an advisor of the Doctoral thesis of another one. It is an analogue of the Candidate of Science thesis in Russia. If the person had no official advisor or we don’t know exactly who was an advisor, we fix that person who was known to be an informal teacher and has influenced on him/her very much.

Discontinuous systems of nonlinear non-autonomous differential equations with impulsive effects are the main object of investigation in the paper. These systems consist of two basic parts: (i) A set of non-linear nonautonomous systems of ordinary differential equations that define the continuous parts of the solutions. The right-hand sides of the systems are elements of the set of functions f = { f1, f2, ...} ; (ii) The conditions which consistently determine “the switching moments”. The structural change (discontinuity) of the right-hand side and impulsive perturbations take place at the moments of switching. In these moments, the trajectory meets the “switching sets”. They are parts of the hyperplanes, situated in the phase space of the system considered. Sufficient conditions are found so that the nonzero solutions of the studied discontinuous system with impulsive effects are asymptotically stable.