Modeling Technical and Mathematical Tasks of Applied Knowledge Areas on Computers

The paper considers modeling of technical problems and problems of applied mathematics, their algorithms and programming. The characteristics of the numerical modeling of technical problems and applied mathematics are given: physical and technical experiments, energy, ballistic and seismic methods of I.V. Kurchatov, starting with mathematical methods of the 17-20th centuries, the first computers and computers. The analysis of the first technical problems and problems of applied mathematics, their modeling, algorithmization and programming using the A.A. Lyapunov graph-schematic language, address language and programming languages is given. Numerical methods are presented, implemented under the guidance of A.A. Dorodnitsyn, A.A. Samarsky, O.M. Belotserkovsky and other scientists on modern supercomputers. Examples of mathematical modeling of the biological problem of eye treatment and the subject of «Computational geometry» on the Internet are given.

Smale flows on

In this paper, we use abstract Lyapunov graphs as a combinatorial tool to obtain a complete classification of Smale flows on$\mathbb{S}^{2}\times \mathbb{S}^{1}$. This classification gives necessary and sufficient conditions that must be satisfied by an (abstract) Lyapunov graph in order for it to be associated to a Smale flow on$\mathbb{S}^{2}\times \mathbb{S}^{1}$.

Gradient-like flows on high-dimensional manifolds

The main purpose of this paper is to study the implications that the homology index of critical sets of smooth flows on closed manifolds M have on both the homology of level sets of M and the homology of M itself. The bookkeeping of the data containing the critical set information of the flow and topological information of M is done through the use of Lyapunov graphs. Our main result characterizes the necessary conditions that a Lyapunov graph must possess in order to be associated to a Morse–Smale flow. With additional restrictions on an abstract Lyapunov graph L we determine sufficient conditions for L to be associated to a flow on M.

Lyapunov Graph for Two-Parameters Map: Application to the Circle Map

In a Lyapunov graph the Lyapunov exponent, λ, is represented by a color in the parameter space. The color shade varies from black to white as λ goes from -∞ to 0. Some of the main aspects of the complex dynamics of the circle map (θn+1=θn+Ω+(1/2π)K sin (2πθn)( mod 1)), can be obtained by analyzing its Lyapunov graph. For K>1 the map develops one maximum and one minimum and may exhibit bistability that corresponds to the intersection of topological structures (stability arms) in the Lyapunov graph. In the bistability region, there is a strong sensitivity to the initial condition. Using the fact that each of the coexisting stable solution is associated to one of the extrema of the map, we construct a function that allows to obtain the boundary separating the set of initial conditions converging to one stable solution, from the set of initial conditions converging to the other coexisting stable solution.

Gradient-like flows on 3-manifolds

In this paper, we determine properties that a Lyapunov graph must satisfy for it to be associated with a gradient-like flow on a closed orientable three-manifold. We also address the question of the realization of abstract Lyapunov graphs as gradient-like flows on three-manifolds and as a byproduct we prove a partial converse to the theorem which states the Morse inequalities for closed orientable three-manifolds. We also present cancellation theorems of non-degenerate critical points for flows which arise as realizations of canonical abstract Lyapunov graphs.