V. V. Fedorenko’s research while affiliated with National Academy of Sciences of Ukraine and other places

The coexistence of different types of homoclinic and periodic trajectories for dynamical systems generated by 1D continuous maps and a special class of n-dimensional continuous maps is investigated.

We consider dynamical systems defined by continuous maps of an interval I of the real axis into itself. We prove that if an interval J in I contains the preimage of a periodic point of period p of a map f ∈ C 0(I, I), then the sequence of intervals f 2pn (J), n= 0, 1, 2,…, is convergent.

We consider dynamical systems generated by continuous mappings of an interval I into itself. We prove that the trajectory of an interval J ⊂ I is asymptotically periodic if and only if J contains an asymptotically periodic point.

The necessity of considering trajectories just of sets rather than of points occurs both in dynamical systems theory by itself and in many evolutionary problems reducible to dynamical systems. In the case of one-dimensional dynamical systems, we present a number of conditions for the trajectory of an interval to be asymptotically periodic. The obtained results find significant applications in the theory of continuous time difference equations and some classes of boundary value problems for partial differential equations.

Many effects of the real turbulence can be observed in infinite-dimensional dynamical systems given by certain classes of boundary value problems for linear partial differential equations. The possibility of using one-dimensional maps under investigation of such infinite-dimensional systems allows to understand the mathematical mechanisms of development of complex structures in the solutions of these boundary value problems. We describe the bifurcations in infinite-dimensional systems resulting from the bifurcations in the corresponding one-dimensional maps, namely, the period-doubling bifurcations and the tangent bifurcations, the "period-adding" bifurcations and the bifurcations subordinate to Farey's rule, and also universal phenomena connected with these bifurcations.

Difference equations with piecewise continuous nonlinearities and their singular perturbations, first order neutral type delay differential equations with small parameters, are considered. Solutions of the difference equations are shown to be asymptotically periodic with period-adding bifurcations and bifurcations determined by Farey's rule taking place for periods and types of solutions. Solutions of the singularly perturbed delay differential equations are considered and compared with solutions of the difference equations within finite time intervals. The comparison is based on a continuous dependence of solutions on the singular parameter.

An info you can find there - http://books.google.de/books?hl=uk&lr=&id=_52wnl2symIC&oi=fnd&pg=PR7&dq=Dynamics+of+one-dimensional+maps&ots=yhhoKOD3-1&sig=j2e67nasp2fE5btIGu-ehRuCOlw#v=onepage&q=Dynamics%20of%20one-dimensional%20maps&f=false

Symbolic dynamics is a part of the general theory of dynamical systems dealing with cascades generated by shifts in various spaces of sequences $\Theta = \left( {...{\theta _{ - 2}},{\theta _{ - 1}},{\theta _0},{\theta _1},{\theta _2},...} \right)\quad or\quad \Theta = \left( {{\theta _0},{\theta _1},{\theta _2},...} \right),$ where θn are letters of an alphabet A = {θ1, θ2, ..., θm } The methods of symbolic dynamics are now widely applied to the investigation of various types of dynamical systems.

Dynamical systems generated by continuous maps of an interval into itself are characterized by the following important property: The data on the relative location of points of a single trajectory on the interval I may contain much information about the dynamical system as a whole. Clearly, this is explained by the fact that the phase space (the interval I) is onedimensional. The points of a trajectory define a decomposition of the phase space, and information on the mutual location of these points often enables one to apply the methods of symbolic dynamics. These ideas are especially useful for the investigation of periodic trajectories.