Rigorous numerical study of the density of periodic windows for the logistic map.

Numerical study of periodic windows for the logistic map is carried out. Accurate rigorous bounds for periodic windows' end points are computed using interval arithmetic based tools. An efficient method to find the periodic window with the smallest period lying between two other periodic windows is proposed.

Rigorous numerical study of the Colpitts oscillator with an exponential nonlinearity.

The dynamics of the Colpitts oscillator with an exponential nonlinearity is investigated using rigorous interval arithmetic based tools. The existence of various types of periodic attractors is proved using the interval Newton method. The main results involve the chaotic case for which a trapping region for the associated return map is constructed and a rigorous lower bound for the value of the topological entropy is computed, thus proving that the system is chaotic in the topological sense.

Systematic search for wide periodic windows and bounds for the set of regular parameters for the quadratic map.

An efficient method to find positions of periodic windows for the quadratic map f(x)=ax(1-x) and a heuristic algorithm to locate the majority of wide periodic windows are proposed. Accurate rigorous bounds of positions of all periodic windows with periods below 37 and the majority of wide periodic windows with longer periods are found. Based on these results, we prove that the measure of the set of regular parameters in the interval [3,4] is above 0.

Is the Hénon attractor chaotic?

By performing a systematic study of the Hénon map, we find low-period sinks for parameter values extremely close to the classical ones. This raises the question whether or not the well-known Hénon attractor-the attractor of the Hénon map existing for the classical parameter values-is a strange attractor, or simply a stable periodic orbit. Using results from our study, we conclude that even if the latter were true, it would be practically impossible to establish this by computing trajectories of the map.

On the structure of existence regions for sinks of the Hénon map.

An extensive search for stable periodic orbits (sinks) for the Hénon map in a small neighborhood of the classical parameter values is carried out. Several parameter values which generate a sink are found and verified by rigorous numerical computations. Each found parameter value is extended to a larger region of existence using a simplex continuation method.

Influence of system non-uniformity on dynamic phenomena in arrays of coupled nonlinear networks.

In this paper we investigate the influence of system non-uniformity on the existence and stability of synchronous motion in an array of bi-directionally coupled electronic circuits. In computer simulations we find the level of non-uniformity for which synchronous behavior is sustained. We also present several examples of attractors, which appear when the synchronous motions is no longer stable.