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.
SIAM Journal on Applied Mathematics 09/1978; 35. DOI:10.1137/0135020 · 1.41 Impact Factor
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ABSTRACT: This book provides an introduction to modelling with both differential and difference equations. Our approach to mathematical modelling is to emphasize what is involved by looking at specific examples from a variety of disciplines. From each discipline enough background is provided to enable students to understand both the assumptions and the predictions of the models. Exercises have been included at the end of each section. They are intended to provide a balanced development of some of the main skills used in mathematical modelling, and hence they are an essential part of the book. The book is divided into parts, each of which corresponds to the areas in which the problems arise and to the types of mathematical equations to which they lead. In part I we model some basic problems in mechanics modelled by differential equations. We introduce Newton’s laws and set them against the background of the set of postulates for mechanics due to the ancient Greek philosopher Aristotle. We also refer to the results discovered by Galileo and Kepler, which showed the inadequacy of the model proposed by Aristotle and thereby set the stage for Newton. In part II we turn to some models described by difference equations. In particular, we introduce the idea of a difference equation via a problem involving rabbit populations. Basic ideas regarding the solutions of these equations are then explained. In parts III (the growth of large populations in which breeding is not restricted to specific seasons; the absorption of drugs into the body tissues; the decay of radioactive substances), IV (drag force and buoyant force in viscous fluids), and V (biological models with linear interaction; in particular, the glucose-insulin homeostasis in the bloodstream and the mother-fetus exchange of nutrients via the placenta) we consider the models which lead to progressively more advanced types of differential equations.