Discrete Dynamical Systems

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Discrete Dynamical Systems with an Introduction to Discrete Optimization Problems Arild Wikan Download free books at Arild Wikan Discrete Dynamical Systems with an Introduction to Discrete Optimization Problems Download free eBooks at bookboon.com Discrete Dynamical Systems with an Introduction to Discrete Optimization Problems 1st edition © 2013 Arild Wikan & bookboon.com ISBN 978-87-403-0327-8 Download free eBooks at bookboon.com Discrete Dynamical Systems with an Introduction to Discrete Optimization Contents Contents Acknowledgements Introduction Part One-dimensional maps f :R→R x → f (x) 11 1.1 Preliminaries and deinitions 12 1.2 One-parameter family of maps 16 1.3 Fixed points and periodic points of the quadratic map 19 1.4 Stability 24 1.5 Bifurcations 30 1.6 he lip bifurcation sequence 35 1.7 Period implies chaos Sarkovskii’s theorem 38 1.8 he Schwarzian derivative 42 1.9 Symbolic dynamics I 45 1.10 Symbolic dynamics II 50 1.11 Chaos 60 1.12 Superstable orbits and a summary of the dynamics of the quadratic map 64 www.sylvania.com We not reinvent the wheel we reinvent light Fascinating lighting offers an ininite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges An environment in which your expertise is in high demand Enjoy the supportive working atmosphere within our global group and beneit from international career paths Implement sustainable ideas in close cooperation with other specialists and contribute to inluencing our future Come and join us in reinventing light every day Light is OSRAM Download free eBooks at bookboon.com Click on the ad to read more Discrete Dynamical Systems with an Introduction to Discrete Optimization Contents Part II n-dimensional maps f : Rn → Rn x → f (x) 68 2.1 Higher order diference equations 69 2.2 Systems of linear diference equations Linear maps from R to R 2.3 he Leslie matrix 2.4 Fixed points and stability of nonlinear systems 106 2.5 he Hopf bifurcation 115 2.6 Symbolic dynamics III (he Horseshoe map) 132 2.7 he center manifold theorem 138 2.8 Beyond the Hopf bifurcation, possible routes to chaos 147 2.9 Diference-Delay equations 173 Part III Discrete Time Optimization Problems 187 3.1 he fundamental equation of discrete dynamic programming 188 3.2 he maximum principle (Discrete version) 198 3.3 Ininite horizon problems 206 3.4 Discrete stochastic optimization problems 218 Appendix (Parameter Estimation) 234 References 247 n n 86 97 360° thinking Discover the truth at www.deloitte.ca/careers Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities Click on the ad to read more Discrete Dynamical Systems with an Introduction to Discrete Optimization Acknowledgements Acknowledgements My special thanks goes to Einar Mjølhus who introduced me to the fascinating world of discrete dynamical systems Responses from B Davidsen, A Eide, O Flaaten, A Seierstad, A StrØm, and K Sydsæter are also gratefully acknowledged I also want to thank Liv Larssen for her excellent typing of this manuscript and Ø Kristensen for his assistance regarding the igures Financial support from Harstad University College is also gratefully acknowledged Finally I would like to thank my family for bearing over with me throughout the writing process Autumn 2012 Arild Wikan Download free eBooks at bookboon.com Discrete Dynamical Systems with an Introduction to Discrete Optimization Introduction Introduction In most textbooks on dynamical systems, focus is on continuous systems which leads to the study of diferential equations rather than on discrete systems which results in the study of maps or diference equations his fact has in many respects an obvious historical explanation If we go back to the time of Newton (1642–1727), Leibniz (1646–1716), and some years later to Euler (1709–1783), many important aspects of the theory of continuous dynamical systems were established Newton was interested in problems within celestial mechanics, in particular problems concerning the computations of planet motions, and the study of such kind of problems lead to diferential equations which he solved mainly by use of power series method Leibniz discovered in 1691 how to solve separable diferential equations, and three years later he established a solution method for irst order linear equations as well Euler (1739) showed how to solve higher order diferential equations with constant coeicients Later on, in ields such as luid mechanics, relativity, quantum mechanics, but also in other scientiic branches like ecology, biology and economy, it became clear that important problems could be formulated in an elegant and oten simple way in terms of diferential equations However, to solve these (nonlinear) equations proved to be very diicult herefore, throughout the years, a rich and vast literature on continuous dynamical systems has been established Regarding discrete systems (maps or diference equations), the pioneers made important contributions here too Indeed, Newton designed a numerical algorithm, known as Newton’s method, for computing zeros of equations and Euler developed a discrete method, Euler’s method (which oten is referred to as a irst order Runge–Kutta method), which was applied in order to solve diferential equations numerically Modern dynamical system theory (both continuous and discrete) is not that old It began in the last part of the nineteenth century, mainly due to the work of Poincaré who (among lots of other topics) introduced the Poincaré return map as a powerful tool in his qualitative approach towards the study of diferential equations Later in the twentieth century Birkhof (1927) too made important contributions to the ield by showing how discrete maps could be used in order to understand the global behaviour of diferential equation systems Julia considered complex maps and the outstanding works of Russian mathematicians like Andronov, Liapunov and Arnold really developed the modern theory further In this book we shall concentrate on discrete dynamical systems here are several reasons for such a choice As already metioned, there is a rich and vast literature on continuous dynamical systems, but there are only a few textbooks which treat discrete systems exclusively Download free eBooks at bookboon.com Discrete Dynamical Systems with an Introduction to Discrete Optimization Introduction Secondly, while many textbooks take examples from physics, we shall here illustrate large parts of the theory we present by problems from biology and ecology, in fact, most examples are taken from problems which arise in population dynamical studies Regarding such studies, there is a growing understanding in biological and ecological communities that species which exhibit birth pulse fertilities (species that reproduce in a short time interval during a year) should be modelled by use of diference equations (or maps) rather than diferential equations, cf the discussion in Cushing (1998) and Caswell (2001) herefore, such studies provide an excellent ground for illuminating important ideas and concepts from discrete dynamical system theory Another important aspect which we also want to stress is the fact that in case of “low-dimensional problems” (problems with only one or two state variables) the possible dynamics found in nonlinear discrete models is much richer than in their continuous counterparts Indeed, let us briely illustrate this aspect through the following example: Let N = N(t) be the size of a population at time t In 1837 Verhulst suggested that the change of N could be described by the diferential equation (later known as the Verhulst equation) N˙ = rN 1− N K (I1) where the parameter r ( r > ) is the intrinsic growth rate at low densities and K is the carrying capacity Now, deine x = N/K hen (I1) may be rewritten as (I2) x˙ = rx(1 − x) which (as (I1) too) is nothing but a separable equation Hence, it is straightforward to show that its solution becomes x(t) = 1− (I3) x0 −1 −rt e x0 where we also have used the initial condition x(0) = x0 > From (I3) we conclude that x(t) → as t → ∞ which means that x∗ = is a stable ixed point of (I2) Moreover, as is true for (I1) we have proved that the population N will settle at its carrying capacity K Next, let us turn to the discrete analogue of (I2) From (I2) it follows that xt+1 − xt ≈ rxt (1 − xt ) ∆t (I4) Download free eBooks at bookboon.com Discrete Dynamical Systems with an Introduction to Discrete Optimization Introduction which implies xt+1 = xt + r∆txt − r∆tx2t = (1 + r∆t)xt − r∆t xt + r∆t (I5) and through the deinition y = r∆t(1 + r∆t)−1 x we easily obtain (I6) yt+1 = µyt (1 − yt ) where µ = + r∆t he “sweet and innocent-looking” equation (I6) is oten referred to as the quadratic or the logistic map Its possible dynamical outcomes were presented by Sir Robert May in an inluential review article called “Simple mathematical models with very complicated dynamics” in Nature (1976) here, he showed, depending on the value of the parameter µ , that the asymptotic behaviour of (I6) could be a stable ixed point (just as in (I2)), but also periodic solutions of both even and odd periods as well as chaotic behaviour hus the dynamic outcome of (I6) is richer and much more complicated than the behaviour of the continuous counterpart (I2) Hence, instead of considering continuous systems where the number of state variables is at least (the minimum number of state variables for a continuous system to exhibit chaotic behaviour), we ind it much more convenient to concentrate on discrete systems so that we can introduce and discuss important deinitions, ideas and concepts without having to consider more complicated (continuous) models than necessary — he book is divided into three parts In Part I, we will develop the necessary qualitative theory which will enable us to understand the complex nature of one-dimensional maps Deinitions, theorems and proofs shall be given in a general context, but most examples are taken from biology and ecology Equation (I6) will on many occasions serve as a running example throughout the text In Part II the theory will be extended to n-dimensional maps (or systems of diference equations) A couple of sections where we present various solution methods of higher order and systems of linear diference equations are also included As in Part I, the theory will be illustrated and exempliied by use of population models from biology and ecology In particular, Leslie matrix models and their relatives, stage structured models shall frequently serve as examples In Part III we focus on various aspects of discrete time optimization problems which include both dynamic programming as well as discrete time control theory Solution methods of inite and ininite horizon problems are presented and the problems at hand may be of both deterministic and stochastic nature We have also included an Appendix where we briely discuss how parameters in models like those presented in Part I and Part II may be estimated by use of time series he motivation for this is that several of our population models may or have been applied on concrete species which brings forward the question of estimation Hence, instead of referring to the literature we supply the necessary material here — Download free eBooks at bookboon.com ...Arild Wikan Discrete Dynamical Systems with an Introduction to Discrete Optimization Problems Download free eBooks at bookboon.com Discrete Dynamical Systems with an Introduction to Discrete Optimization... at bookboon.com Discrete Dynamical Systems with an Introduction to Discrete Optimization Introduction Introduction In most textbooks on dynamical systems, focus is on continuous systems which leads... are only a few textbooks which treat discrete systems exclusively Download free eBooks at bookboon.com Discrete Dynamical Systems with an Introduction to Discrete Optimization Introduction Secondly,