In this paper, we investigated some dynamical behavior of family of tent maps. Such dynamical properties include fixed points and their stability, period orbits, visualize the iterations using a kind of plot called a cobweb plot and demonstrate bifurcation diagram for r T. Furthermore, detecting the presence of chaos in the discrete dynamical system r T is investigated. Finally, we develop Matlab computer programs that reflect the results interpreting such dynamical behavior.

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Barnsley (Barnsley, M.F., 1986. Fractal functions and interpolation. Constr. Approx., 2, pp.303–329) introduced fractal interpolation function (FIF) whose graph is the attractor of an iterated function system (IFS) for describing the data that have an irregular or self-similar structure. Barnsley et al. (Barnsley, M.F., et al., 1989. Recurrent iterated function systems in fractal approximation. Constr. Approx., 5, pp.3–31) generalized FIF in the form of recurrent fractal interpolation function (RFIF) whose graph is the attractor of a recurrent iterated function system (RIFS) to fit data set which is piece-wise self-affine. The primary aim of the present research is investigating the RFIF approach and using it for fitting the piece-wise self-affine data set in ℜ2.

Lyapunov exponents play a significant part in revealing and quantifying chaos, which occurs in many areas of science and technology. The purpose of this study was to approximate the Lyapunov exponents for discrete dynamical systems and to present it as a quantifier for inferring and detecting the existence of chaos in those discrete dynamical systems. Finally, the approximation of the Lyapunov exponents for the discrete dynamical system was implemented using the Matlab code listed in the Appendix.

This research investigates the application of the QR - method for computing all the eigenvalues of the real symmetric tridiagonal matrix. The Householder method will be used for reduction of the real symmetric matrix to symmetric tridiagonal form, and then the so called QR - method with acceleration shift applies a sequence of orthogonal transformations to the symmetric tridiagonal matrix which converges to a similar matrix that is tridiagonal. This tridiagonal matrix possesses an eigenvalues similar to the eigenvalues of the symmetric tridiagonal matrix. Particular attention is paid to the shift technique that accelerates the rate of convergence. Computer algorithms for implementing the Householder's method and QR – method are presented. Computer Matlab programs for performing the Householder algorithm and the QR algorithm (with acceleration shift) are listed in the Appendix.