Discrete Dynamical Systems, Bifurcations and Chaos in by Wei-Bin Zhang

By Wei-Bin Zhang

This e-book is a distinct mix of distinction equations conception and its intriguing functions to economics. It offers with not just idea of linear (and linearized) distinction equations, but in addition nonlinear dynamical structures that have been generally utilized to financial research in recent times. It reviews most crucial suggestions and theorems in distinction equations thought in a manner that may be understood by way of an individual who has uncomplicated wisdom of calculus and linear algebra. It includes recognized functions and lots of contemporary advancements in numerous fields of economics. The e-book additionally simulates many versions to demonstrate paths of financial dynamics.

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Extra resources for Discrete Dynamical Systems, Bifurcations and Chaos in Economics

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Let us consider the one-dimensional difference equation x{t + \)=f{x{t)\ ? = 0,l,--. 1. 1) is locally stable if for every s > 0 there exists S > 0 such that I < 5 implies that llx, - x* < s for all f > 1. In this book, the norm ||jc|| denotes the Euclidean norm of x, defined by 2. SCALAR LINEAR DIFFERENCE EQUATIONS 28 A fixed point that is not stable is said to be unstable. An unstable equilibrium point is called a source, or a repeller. Stability means that once we have chosen how close we want to remain x* in the future, we can find how close we must start at the beginning.

Then (i) If / ' ( / ) < 1, then x* is asymptotically stable. (ii) If |/'(x*)| > 1, then x' is unstable. Proof. In the case of If'yx") < 1, consider /? such that \r(x')\ 0. For any x in this interval, applying the mean value theorem for derivatives yields f{x)-f{x)={x-x)f{x+e(x-x\ where 0 < 0 < 1. Since x + 9\x — x j belongs [x* — £, x* + ej, we have \f{x)-f(x)\

2. SCALAR LINEAR DIFFERENCE EQUATIONS 40 Newton's algorithm for finding a zero x* of g(x) is given by the difference equation where JC0 is the initial guest of the root x*. To determine whether Newton's algorithm provides a sequence \x(t)\ that converges to x* we calculate Wl. = 0, where we use g(x*) = 0. We conclude that \imx(t) = JC*, if x0 is close enough to x' and g'(x*)* 0. Example Consider x(? + l) = 1 - Ax2(t), where xe[-l,l], Ae(0,2]. There are two equilibrium points , * l2 - 1 + ^ 1 + 4/1 2l " It is straightforward to check that x* is unstable for all A e (o, 2J; asymptotically stable i f O < A < 3 / 4 and unstable if X > 3 / 4.

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