Antonios Valaristos

B-spline method for solving Bratu's problem

In this paper, we propose a B-spline method for solving the one-dimensional Bratus problem. The numerical approximations to the exact solution are computed and then compared with other existing methods. The effectiveness and accuracy of the B-spline method is verified for different values of the parameter, below its critical value, where two solutions occur.

An automated acquisition setup for the evaluation of intermittency statistics

The design and realization of an automated acquisition setup, dedicated to experimental evaluation of intermittent chaotic phenomena and the related statistics, is presented. The setup was implemented in National Instruments LabView environment and it was structured in such a way that it is not dependent of the signal-registering devices used. The circuit evaluation is achieved by registering only one signal. An experimental intermittency example confirms the systems effectiveness.

Period doubling patterns of interval maps

Abstract We prove the existence and demonstrate the construction of period doubling patterns centered at periodic orbits of continuous maps on the interval. In particular we prove that f ∈ C 0 ( I , I ) exhibits a period doubling pattern centered at a fixed point of f if and only if the set of periodic points is not closed. Furthermore, we prove that if f has a periodic orbit of period n >1, which is not a power of two, and n ≺ m in Sarkovskii’s ordering, then f exhibits a period doubling pattern centered at a periodic orbit of period m . An analytic configuration of such period doubling patterns is exhibited.

A characterization of the dynamics of Newton’s derivative

In the present report the dynamic behaviour of the one dimensional family of maps \( F_{a,b,c} \left( x \right) = c\left[ {\left( {1 - a} \right)x - b} \right]^{\frac{1} {{1 - a}}} \) is examined, for different ranges of the control parametres a, b and c. These maps are of special interest, since they are solutions of N f ′ (x) = a, where N f ′ is the Newton’s method derivative. In literature only the case N f ′ (x) = 2 has been completely examined. Simultaneously, they may be viewed as solutions of normal forms of second order homogeneous equations, F″(x)+p(x)F(x) = 0, with immense applications in mechanics and electronics. The reccurent form of these maps, \( x_n = c\left[ {\left( {1 - a} \right)x_{n - 1} - b} \right]^{\frac{1} {{1 - a}}} \) , after excessive iterations, shows an oscillatory behaviour with amplitudes undergoing the period doubling route to chaos. This behaviour was confirmed by calculating the corresponding Lyapunov exponents.