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Maps, Chaos, and Fractals

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Maps, Chaos, and Fractals MATH305 Summer Research Project 2006-2007 Maps, Chaos, and Fractals Phillips Williams Department of Mathematics and Statistics University of Canterbury Maps, Chaos, and Fractals Phillipa Williams* MATH305 Mathematics Project University of Canterbury 9 February 2007 Abstract The behaviour and properties of one-dimensional discrete mappings are explored by writing Matlab code to iterate mappings and draw graphs. Fixed points, periodic orbits, and bifurcations are described and chaos is introduced using the logistic map. Symbolic dynamics are used to show that the doubling map and the logistic map have the properties of chaos. The significance of a period-3 orbit is examined and the concept of universality is introduced. Finally the Cantor Set provides a brief example of the use of iterative processes to generate fractals. *supervised by Dr. Alex James, University of Canterbury. 1 Introduction Devaney [1992] describes dynamical systems as "the branch of mathematics that attempts to describe processes in motion)) . Dynamical systems are mathematical models of systems that change with time and can be used to model either discrete or continuous processes. Contin­ uous dynamical systems e.g. mechanical systems, chemical kinetics, or electric circuits can be modeled by differential equations. Discrete dynamical systems are physical systems that involve discrete time intervals, e.g. certain types of population growth, daily fluctuations in the stock market, the spread of cases of infectious diseases, and loans (or deposits) where interest is compounded at fixed intervals. Discrete dynamical systems can be modeled by iterative maps. This project considers one-dimensional discrete dynamical systems. In the first section, the behaviour and properties of one-dimensional maps are examined using both analytical and graphical methods. Fixed points, periodic orbits, and bifurcations are introduced using ex­ amples from linear and non-linear mappings. The logistic equation is used to introduce chaotic behaviour in the next section, and the concept of symbolic dynamics is introduced using the doubling map and the logistic map. The landmark theorem "period three implies chaos, is described. An example of the extension of iterative processes into the domain of fractals is given in final section. Maps Mappings or discrete dynamical systems are described by a difference equation or recur­ rence relation Xn+l = f(xn) and an initial value x0. The resulting sequence of iterates {xo, x1, x2, .... , Xn, .... } is called an orbit. The function f(xn) may involve one or more parameters, and the initial value x 0 may be varied, so many different orbits are possible for a given mapping function. Graphical and analytical methods can be used to examine the general behaviour of a mapping as the initial value or parameter is varied without having to examine every possible orbit. The mapping of the sequence of iterates may be plotted as the iterate number n vs the corresponding iterate Xn. This graph of the sequence of iterates should not be confused with the graph of the mapping function f(x) itself. Figure 1 compares the plot of the sequence of iterates of Xn+l = 2xn(1- Xn) with the graph of the mapping function f(x) = 2x(1- x). 2 (a) (b) en Q) ~ Eo.5 Q) 0.5 * * * * * * .t:: - X * * * 00 5 10 00 0.5 n X Figure 1: (a) The sequence of iterates for Xn+ 1 = 2xn (1 - Xn) and (b) The mapping function f(x) = 2x(1- x) Fixed Points and Periodic Orbits A fixed point of a mapping is a point at which the mapping or orbit remains unchanged for all subsequent iterations. Fixed points may be found analytically by solving the equation x* = f(x*). Consider the linear mapping Xn+l = Axn where A is a parameter. Fixed points of this mapping are given by: x* = f(x*) =.Ax* ----+ x* = 0 for A =I 0 The mapping converges to x* = 0 for I-AI < 1. For these values of A the fixed point is often called an "attracting" fixed point or attractor. In dynamical systems, an attractor is a set e.g. a point, a set of points, or a line, toward which the system evolves or is attracted. A simple graphical way to find the fixed point of a mapping is to plot the mapping function y = f(xn) and the line y = x on the same graph. Then the fixed points of the mapping are given by the points of intersection of the graphs of y = f(xn) andy= x. For a more formal mathematical definition of a fixed point see Martelli [1999]. A useful tool for analysing maps is a "cobweb diagram", which plots successive iterations of the mapping and illustrates the general behaviour of the function. Figure 2 demonstrates the method of preparing cobweb diagrams: • plot the graphs of y = f(xn) andy= x on the same set of axes • begin at x = x0 and find f(x0 ) (point P 1 in Figure 2) by traveling up from the x-axis to the curve y = f(xn) • from f (x 0 ) move horizontally to the line y = x to point P 2 • choose the x-coordinate of this point as the next x-value, x1 3 Y=X Figure 2: Starting a cobweb diagram: the first iteration This method works because they-coordinate of P1 is f(x0 ) as P 1 is on the curve y = f(xn)· Therefore P2 also has y-coordinate f(x0 ) since it is "level" horizontally with P1 . P2 also has f(x0 ) as its x-coordinate since it is on the line y = x, so f(x0 ) = x 1 as required. If this graph­ ical iteration scheme is repeatedly applied then the sequence of points {x0 , x1, x2, .... , Xn, .... }, i.e. the orbit of the mapping for the given x0 , is obtained. For 0 < ,.\ < 1, the mapping converges monotonically to the fixed point x* = 0. Figure 3 shows the iterative mapping and cobweb diagram for ,.\ = 0.5. In the cobweb diagram, convergence appears as a "ladder" with f(x0 ) at the top of the ladder and "steps" going down until the fixed point x* = 0 is reached. (a) (b) en Q) ~ 0.5 * .1:: X * * *'I'" 5 10 15 20 0.2 0.4 0.6 0.8 n X Figure 3: (a) The sequence of iterates for linear mapping with ,.\ = 0.5 and (b) Cobweb of the linear map with,.\= 0.5 showing convergence to the fixed point x* = 0. 4 For -1 < A < 0, the mapping shows oscillating convergence i.e. the iterations approach the fixed point from alternate sides of the fixed point. Figure 4 shows the sequence of iterates and cobweb diagram for A= -0.5. The cobweb diagram takes on the spiral shape characteristic of damped oscillations as it approaches the fixed point. (a) (b) 0.5 y='A*x (/) 0.5 Q) 1il.... Q) * .t ~ >< * 0 * * *· * * * * * * * * * * * * * -0.5 -1 0 5 10 15 20 -1 -0.5 0 0.5 n X Figure 4: (a) The sequence of iterates for linear mapping with A= -0.5 and x 0 = 1 and (b) Cobweb of the same linear map showing oscillating convergence to the fixed point X* = 0. For)..> 1, the mapping diverges monotonically, as shown by the iterative mapping and also by the cobweb diagram of the function in Figure 5. Again the cobweb shows the divergence as a ladder, but in this case f(x0 ) is at the foot of the ladder and the steps go upwards and outwards. (a) (b) 4000.----~--~--~--, 3000 2 j 2000 * >< * 1000 * 5 10 15 20 0 1 2 3 4 5 n X Figure 5: (a) The sequence of iterates for linear mapping with A= 1.5 and x 0 = 1 and (b) Cobweb of the same linear map showing divergence. 5 For A < -1, the mapping shows oscillating divergence i.e. it diverges outwards in an oscil­ lating or alternating fashion, as shown in Figure 6. This time the spiral is moving outward rather than inward toward the fixed point. (a) (b) 4000.-----~--~-~-----, 6 ='A*x y 2000 3 (/) * ~ * g .. ~ - >< * -3 -2000 * -6 L-~~~~--~~~~~~ 5 10 15 20 -5 -4 -3 -2 -1 0 1 2 3 4 5 n X Figure 6: (a) The sequence of iterates for linear mapping with A= -1.5 and (b) Cobweb of the linear map with A = -1.5 showing oscillating divergence. From the cobweb diagrams it is clear that if the slope of the line specified by mapping func­ tion AX is greater in absolute value than the slope of y = x, then the fixed point x* = 0 is repelling. If the slope of the linear map is less in absolute value than the slope of y = x, then the fixed point is attracting. This is the basis of the method of linearisation used to determine the stability of fixed points of non-linear maps. For both these cases IAI =/:. 1, and the fixed point is described as a hyperbolic fixed point. For IAI = 1 no conclusion can be drawn. In this case the fixed point is described as non-hyperbolic. The linear map Xn+l = AXn illustrates four different types of behaviour of maps: monotonic convergence to a fixed point, oscillating convergence to a fixed point, monotonic divergence to infinity, and oscillating divergence to infinity. Another type of behaviour that can occur in dynamical systems is a periodic orbit. The linear map Xn+l = axn + b is called an affine dynamical system, with fixed point x* = b/ ( 1 - a). The stability of this mapping is also determined by the slope of the linear function, hence for lal < 1 the fixed point is stable, and for lal > 1 the fixed point is unstable.
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