Vol. 2, No. 3 (2013): Regular Issue

It is well-known for vibro-impact systems that the existence of a periodic solution with a low-velocity impact (so-called grazing) may yield complex behavior of the solutions. In this paper we show that unstable periodic motions which pass near the delimiter without touching it may give birth to chaotic behavior of nearby solutions. We demonstrate that the number of impacts over a period of forcing varies in a small neighborhood of such periodic motions. This allows us to use the technique of symbolic dynamics. It is shown that chaos may be observed in a two-sided neighborhood of grazing and this bifurcation manifests at least two distinct ways to a complex behavior. In the second part of the paper we study the robustness of this phenomenon. Models of impact Particularly, we show that the same effect can be observed in “soft” models of impacts.

In this paper, we investigate the pulse phenomena for a class of impulsive dynamical systems. Some sufficient conditions that guarantee the absence or presence of pulse phenomena are obtained, without the boundeness requirement on impulse surfaces. Besides, we also utilize methods of the flow theory, focus on the dynamical behavior in the normal direction to the switching boundary and generalize several known results to apply to an important example. Then, we study the stability of a nontrivial solution in a class of functional differential equations with pulse phenomena by using the concept of quasistability and method of comparison.

We introduce a new notion-the spectrum of dimensions of escape time- and study its properties. The escape time was defined for an initial point of the trajectory according to its ability to reach a hole in the phase space. In the article we generalize this notion onto ”spots” of initial points making the escape time to be a function of a set (spot). Then we apply the Caratheodory-Pesin machinary of fractal dimensions to define the spectrum. For dynamical systems generated by maps of the interval or the circle we obtain explicit formulas in the case where an element of Markov partition is chosen as a hole.

In this paper, analytical bifurcation trees of period-1 motions to chaos in the periodically forced, quadratic nonlinear oscillator are discussed from the generalized harmonic balance method. The analytical solutions for stable and unstable periodic motions in such quadratic nonlinear oscillator are achieved, and the corresponding stability and bifurcation were discussed. The analytical bifurcation trees from period-1 motions to period-4 motions in such quadratic oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out by the numerical and analytical solutions. This investigation provides a comprehensive picture of complex periodic motion in the periodically excited, quadratic nonlinear oscillator.

Discrete time scale dependent random walks on a graph model of a relational database can be generated by a variety of equivalence relations imposed between walks (i.e.composite functions, inheritance, property relations, ancestor-descendant relations, data queries, address allocation and assignment polices). The Green function of diffusion process induce by the random walks allows to define scale dependent geometry. Geometric relations on databases can guide the datainterpretation. In particular, first passage times in a urban spatial network help to evaluate the tax assessment value of land.