TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 26

CHAPTER 51 Distinguishing Random Variables from Variables Created by a Deterministic Chaotic Process. Dynamical systems are either described as recursive functions (discrete time) or as differential equations. With discrete time, recursive functions | CHAPTER 51 Distinguishing Random Variables from Variables Created by a Deterministic Chaotic Process Dynamical systems are either described as recursive functions discrete time or as differential equations. With discrete time recursive functions recursive functions are difference equations discrete analog of differential equations one can easily get chaotic behavior. . the tent map or logistic function. The problem is how to distinguish the output of such a process from a randomly generated output. The same problem can also happen in the continuous case. First-order differential equations can be visualized as vector fields. 1083 1084 51. DETERMINISTIC CHAOS AND RANDOMNESS An attractor A is a compact set which has a neighborhood U such that A is the limit set of all trajectories starting in U. That means every trajectory starting in U comes arbitrarily close to each point of the attractor. In R2 there are three different types of attractors fixed points limit cycles and saddle loops. But in R3 and higher chaos can occur . the trajectory can have a strange attractor. Example Lorenz attractor. There is no commonly accepted definition of a strange attractor it is an attractor that is neither a point nor a closed curve and trajectories attracted by it take vastly different courses after a short time. .I 11 m it c mii e I I I i c i. .r. oe lim og V g j .1 No w lexcLdi imeiisioiis rst t e xj-diu-s xl iniensi n cxs inig__0 l 1 i i cating the exponent with which the number of covering pieces N e increases as the diameter of the pieces diminishes. Examples with integer dimensions for points we have N e 1 always therefore dimension is 0. For straight lines of length L N e L e therefore we get log L g log S g Q liiiig _o iog 1 s 1 an or an area wiL sunace it is iuig q og 1 g 2. Famous example of set with fractal dimension is the Cantor set start with unit interval take middle third out then take middle third of the two remaining segments out etc. For e 1 3 one gets N e 2

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