Family Of Two-dimensional Maps Research Articles

Entropy formula and continuity of entropy for piecewise expanding maps

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.

A family of dissipative two-dimensional mappings: Chaotic, regular and steady state dynamics investigation

Some dynamical properties for a family of two dimensional mappings controlled by three parameters are considered in this paper. For null dissipation and depending on the other control parameters, diffusion in phase space is observed. A connection with the standard mapping is made to determine the range of control parameters leading to unlimited diffusion. In the dissipative case, for low nonlinearity, we obtain the fixed points and an expression to find any periodic orbit with any period, if it exists. For large nonlinearity we investigate the Lyapunov exponents and characterise the steady state of the system for a long time.

COMMON DYNAMICAL FEATURES OF PERIODICALLY DRIVEN STRICTLY DISSIPATIVE OSCILLATORS

Periodically driven strictly dissipative nonlinear oscillators in general possess a recurring bifurcation structure in parameter space. It consists of slightly modified versions of a basic pattern of bifurcation curves that was found to be essentially the same for many different oscillators. The periodic orbits involved in these bifurcation scenarios also possess common topological properties characterized in terms of their torsion numbers and the way they are connected when parameters are varied. In this paper, this typical bifurcation structure of periodically driven strictly dissipative oscillators will be presented and discussed in terms of examples from Duffing’s equation. Furthermore a family of two-dimensional maps is given that models (strictly) dissipative oscillators and shows essential features of the bifurcation pattern found.

ON MODELING DRIVEN OSCILLATORS BY MAPS

A family of two-dimensional maps showing essential features of the bifurcation set of some periodically driven, strictly dissipative oscillators is constructed.