The relaxed Newton method derivative: Its dynamics and non-linear properties
ABSTRACT The dynamic behaviour of the one-dimensional family of maps f(x)=c2[(a−1)x+c1]−λ/(α−1) is examined, for representative values of the control parameters a,c1, c2 and λ. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a. The maps f(x) are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an xn versus λ plot, an initial exponential decay followed by a bifurcation. The value of λ at which this bifurcation takes place depends on the values of the parameters a,c1 and c2. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x) undergoing a period doubling. For values of a higher than 1 and at higher values of λ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter c1 between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.
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ABSTRACT: The emerging discipline known as "chaos theory" is a relatively new field of study with a diverse range of applications (i.e., economics, biology, meteorology, etc.). Despite this, there is not as yet a universally accepted definition for "chaos" as it applies to general dynamical systems. Various approaches range from topological methods of a qualitative description; to physical notions of randomness, information, and entropy in ergodic theory; to the development of computational definitions and algorithms designed to obtain quantitative information. This thesis develops some of the current definitions and discusses several quantitative measures of chaos. It is intended to stimulate the interest of undergraduate and graduate students and is accessible to those with a knowledge of advanced calculus and ordinary differential equations. In covering chaos for continuous systems, it serves as a complement to the work done by Philip Beaver, which details chaotic dynamics for discrete systems.11/1992;
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ABSTRACT: Second edition of the first edition in (1986; Zbl 0632.58005). From the author’s preface: “We have added new material on the orbit diagram and a new section on the Mandelbrot set. Apart from this, the only other major changes from the first edition include a revised treatment of elementary bifurcation theory and Sarkovskij’s theorem.”The Mathematical Gazette 06/1990; 74(468).
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ABSTRACT: In , Gilbert Strang explored the chaotic behavior of Newton's method applied to x 2 + 1 when the iteration starts from only real initial conditions. We embed Strang's problem in the one-parameter family of iteration methods known as relaxed Newton's method. By extending this analysis to include a discussion of invariant measures, probability densities, and Lyapunov exponents, we prove that in the quadratic case, relaxed Newton's method exhibits chaotic dynamics as well. 1 Introduction This article makes a contribution to an old problem rst considered by Sir Arthur Cayley in 1879, the convergence of Newton's method for quadratic polynomials. Gilbert Strang later used the divergence of Newton's method to illustrate the idea of chaos in , since most calculus textbooks cover this familiar root nding algorithm. Newton's method is dened here for any function f(x) xn+1 = xn f(xn )=f 0 (x n ): (1) Generating xn+1 from xn is called an iteration step. Repeating the iteration proces...09/1999;