(2) ^ © = A' X(t) + X\t) A + B, te(0,co) dt for continuous systems. Here A is a given stable matrix, i.e. all its eigenvalues a, satisfy |a,| < i or Re a,< 0, respectively, and B is a given symmetric matrix. This difference/differential equation is to be solved with a given initial condition X(0). Of special interest is a limit for / -» co; it is a solution of the stationary Lyapunov equation (...

Second order linear non-autonomous differential equations with negative stiffness are considered. Using Chetaev-like (Lyapunov-like) functions, necessary (sufficient) conditions are found for the solutions to be bounded for all initial conditions if any one of the coefficients is constant. Conclusions are then extended to include systems where both coefficients are time-varying.

Background:Leukemia is cancer of blood and bone marrow cells. In general, there are four types of leukemia: chronic myelogenous leukemia (CML), acute myeloid leukemia (AML), B-cell chronic lymphocytic leukemia (CLL) and acute lymphoblastic leukemia (ALL).  Fractal geometry can be introduced as one of the effective ways to detect this type of cancer. In this study, with introduc...

Fractal analysis of earthquakes, lineages and waterways are considered as one of the practical tools to evaluate tectonic activity. With the help of fractal analysis, structural maturity and tectonic dynamics can be evaluated. In this research, using fractal analysis and box counting method and fild studies, the measured fractal dimensions for the designed network of Alborz province were studie...

Abstract In this paper, the design and development of dual-band triple-band fractal microstrip patch antennas with enhanced gain are presented. The structure is based on Sierpinski carpet fractal, where multiband functionality achieved by applying iteration technique. antenna characteristics along analysis reflection coefficient radiation patterns for each (DBFMPA) operating at 4.9 5.3 GHz (TBF...

Chaotic dynamics occur in deterministic systems which display extreme sensitivity on initial conditions. These systems often have attractors which are geometric figures exhibiting affine self-similarity that have non-integer dimension, otherwise known as fractals. We investigated the link between chaos and the eventual fate of a ball on a frictionless elliptical billiards table with one pocket....

Stability investigation of hereditary systems is connected often with construction of Lyapunov functionals. One general method of Lyapunov functionals construction was proposed and developed in [1-9] both for differential equations with aftereffect and for difference equations with discrete time. Here, some modification of Lyapunov-type stability theorem is proposed, which allows one to use thi...

Let G be a connected Lie group with Lie algebra g , expG : g −→ G the exponential map and E(G) its range. E(G) will denote the set of all n -fold products of elements of E(G). G is called exponential if E(G) = E(G) = G . Since most real (or complex) connected Lie groups are not exponential, it is of interest to know that the weaker conclusion E(G) = G is always true (Theorem 5.6). This result w...

The increasing development of the electric power grid, the largest engineered system ever, to an even more complicated and larger system requires a new generation of stability assessment methods that are computationally tractable and feasible in real-time. In this paper we first extend the recently introduced Lyapunov Functions Family (LFF) transient stability assessment approach, that has pote...

A new phase of temporal evolution of the one-dimensional selfgravitating system is numerically discovered. Fractal structure is dynamically created from non-fractal initial conditions. Implication to astrophysics and mathematical physics is discussed.