Volume 3 (2013)

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1.  Solving Nonlinear Equations Using Two-Step Optimal Methods Full text (PDF)

M.A. Hafiz, M. S. M. Bahgat

Pages: 1-11

Abstract:  Finding iterative methods for solving nonlinear equations is an important area of research in numerical analysis at it has interesting applications in several branches of pure and applied science can be studied in the general framework of the nonlinear equations f(x)=0. Due to their importance, several numerical methods have been suggested and analyzed under certain condition. These numerical methods have been constructed using different techniques such as Taylor series, homotopy perturbation method and its variant forms, quadrature formula, variational iteration method, and decomposition method ( see [1-9]). In this study we describe new iterative free from second derivative to find a simple root r of a nonlinear equation. In the implementation of the method of Noor et al. [10], one has to evaluate the second derivative of the function, which is a serious drawback of these methods. To overcome these drawbacks, we modify the predictor-corrector Halley method by replacing the second derivatives of the function by its suitable finite difference scheme.

2. A Dynamical Analysis of an Autocatalytic Model Full text (PDF)

Lakshmi Burra, Uma Maheswari

Pages: 12-22

Abstract:  In this paper we study an autocatalytic reaction and we derive the differential equations arising from this reaction. We analyze these equations using phase-space analysis. We next use Center Manifold theory to derive a stable Center Manifold for this system. Since this system is a polynomial differential system, we study the orbits of the system which go or come from infinity using the Poincaré compactification. Interestingly these equations would also represent a population model. All these concepts are illustrated graphically.

3. Chaos in the Planar Two-Body Coulomb Problem with a Uniform Magnetic Field  Full text (PDF)

Vladimir Zhdankins, J. C. Sprott  

Pages: 23-33

Abstract:  The dynamics of the classical two-body Coulomb problem in a uniform magnetic field are explored numerically in order to determine when chaos can occur. The analysis is restricted to the configuration of planar particles with an orthogonal magnetic field, for which there is a four-dimensional phase space. Parameters of mass and charge are chosen to represent physically motivated systems. To check for chaos, the largest Lyapunov exponent and Poincaré section are determined for each case. We find chaotic solutions when particles have equal signs of charge. We find cases with opposite signs of charge to be numerically unstable, but a Poincaré section shows that chaos occurs in at least one case.

4 On a Conjecture of Trichotomy and Bifurcation in a Third Order Rational Difference Equation Full text (PDF)

 Xianyi  Li, Cheng Wang

Pages: 34-44

Abstract:  In this paper it is first investigated for a conjecture of trichotomy of period two for a third order rational difference equation, and then the bifurcation of this equation is further considered. The results obtained partially verify a conjecture in the known literature.

5.  Book Review: Lozi Mappings: Theory and Applications, by Zeroualia Elhadj, CRC Press, 2013  Full text (PDF)

René Lozi   

Pages: 45-48

Abstract:  When, more than two years ago, Prof. Zeroualia Elhadj informed me of his willing to write a book on what is known as Lozi map since the Misiurewicz's communication in the congress organized by the New York Academy of Science, 17-21 December 1979, I warned the task was not straightforward because hundreds of articles were published on this topic in thirty years. These papers were scattered in various fields of research, not only in mathematics (dynamical systems), but in physics, computer science, electronics, chemistry, control science and engineering, etc..