By Saber N. Elaydi
PREFACE FOREWORD the steadiness of One-Dimensional Maps advent Maps vs. distinction Equations Maps vs. Differential Equations Linear Maps/Difference Equations mounted (Equilibrium) issues Graphical generation and balance standards for balance Periodic issues and Their balance The Period-Doubling path to Chaos functions allure and Bifurcation advent Basin of allure of mounted issues Basin of appeal of Periodic Orbits Singer's Theorem Bifurcation Sharkovsky's Theorem The Lorenz Map Period-Doubling within the actual international Poincaré Section/Map. Read more...
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Additional info for Discrete Chaos, Second Edition : With Applications in Science and Engineering
25) where x(0) = x0 is our initial guess of the root r. 26) where fN is called Newton’s function. 4 (Taylor’s Theorem) Let f be diﬀerentiable of all orders at x0 . Then f (x) = f (x0 ) + (x − x0 )f (x0 ) + (x − x0 )2 f (x0 ) + . . 2! for all x in a small open interval containing x0 . 25) may be justiﬁed using Taylor’s Theorem. A linear approximation of f (x) is given by the equation of the tangent line to f (x) at x0 : f (x) = f (x0 ) + (x − x0 )f (x0 ). Discrete Chaos 28 The intersection of this tangent line with the x-axis produces the next point x1 in Newton’s algorithm (Fig.
12): 1. If |a| < 1, then lim |f n (x0 )| = 0 n→∞ or lim |x(n)| = 0 [see Fig. 4 (b) n→∞ and (c)]. 2. If |a| > 1, then lim |f n (x0 )| = ∞ n→∞ or lim |x(n)| = ∞ if x0 = 0 [see n→∞ Fig. 4 (a) and (d)]. 3. (a) If a = 1, then f is the identity map where every point is a ﬁxed point of f . 12) is said to be periodic of period 2. 2. Solutions of Eqs. 12) for diﬀerent values of the parameter a. Next, let us look at the aﬃne map f (x) = ax + b. By successive iteration, we get f 2 (x) = a2 x + ab + b f 3 (x) = a3 x + a2 b + ab + b ..
16 Newton’s method for g(x) = x2 − 1. 17 Cobweb diagram for Newton’s function fN when g(x) = x2 − 1. 29 Discrete Chaos 30 2. If f (x∗ ) = 0 and f (x∗ ) > 0, then x∗ is unstable. 3. If f (x∗ ) = 0 and f (x∗ ) < 0, then x∗ is asymptotically stable. PROOF 1. Assume that f (x∗ ) = 1 and f (x∗ ) = 0. Then, the curve y = f (x) is either concave upward (f (x∗ ) > 0) or concave downward (f (x∗ ) < 0), as shown in Fig. 18(a) and (b). Now, if f (x∗ ) > 0, then f (x) is increasing in a small interval containing x∗ .