4452 Mathematical Modeling Lecture 13: Chaos and Fractals

Transcription

1 Math Modeling Lecture 13: Chaos and Fractals Page Mathematical Modeling Lecture 13: Chaos and Fractals Introduction In our tetbook, the discussion on chaos and fractals covers less than 2 pages. Contrast that with the book Chaos and Fractals: New Frontiers of Science [1] which clocks in at over 9 pages. You can imagine that this field is a large one, and we will onl be able to give it a cursor glance in this course. I want to give ou a flavour of what chaos is all about, but I certainl am not going to be anwhere near complete! The importance of chaos in modeling is that chaotic sstems can have limit sets (the limit set being the long time behaviour of the solution, which until now we have generall thought of as either an equilibrium fied point or unbounded divergence) which are fractal in nature. Linear sstems do not have limit ccles (closed loops), but it is surprising that simple nonlinear dnamical sstems will have limit ccles, and can also ehibit chaos. It is important to note that the presence of a limit ccle does not mean the sstem is chaotic the presence of a fractal limit ccle, however, does mean that the sstem is chaotic. Since chaos deals with the long term behaviour of solutions (tpicall, chaos is thought of as sensitivit to initial conditions), sstems can ehibit degrees of chaos. What I mean is, if we have two sstems, the both ma be chaotic, but one ma be more chaotic than the other. There are was to measure the degree of chaos in the sstem (Lapunov eponent), which I will discuss, but not epect ou to calculate. Continuous Models: The Hénon Heiles Model There is a continuous dnamical sstem which was constructed to model the motion of a star moving in the field of its gala, known as the Hénon Heiles model [3]. It is a two dimensional dnamical sstem based on Hamilton s equations of motion, which incorporates the position and momentum of the particle, making it a four dimensional model in the variables,, p, p. p p p 2 p 2 + (1) This is sstem of two coupled nonlinear oscillators, and is unbounded. That means that for some initial conditions, the sstem will run off to infinit. This can be solved using a numerical technique such as Euler s method, although we must be ver careful that we choose the step size to be small enough to ensure our results are accurate. Although this is a four dimensional problem, we will be plotting onl the portion of the phase space and (t) to get an idea about how the sstem is behaving. I have plotted the solution to the Hénon-Heiles sstem (found using Euler s method, although Runge Kutta would be far more efficient) in Fig. 1. Looks like a spirograph, doesn t it? These seem to be solutions which are ver regular, and not ver chaotic. How can we estimate how chaotic a trajector is? B doing this, we will be performing a sensitivit analsis of this trajector to its initial conditions.

2 Math Modeling Lecture 13: Chaos and Fractals Page Figure 1: The phase space solution to the continuous sstem in Eq. (1). On the left the inital point is ().1, ()., p ()., p ().11, and on the right the initial point is ().2, ().1, p ().4, p ().. Lapunov Eponents The measure of the degree of chaos in a continuous sstem is epressed in the form of the Lapunov eponent, which represents the long term behaviour of the nonlinear sstem. What follows is easil understood, but ma be a bit much for us to implement in this class. One wa of deriving the Lapunov eponent is in terms of the long time behaviour of the monodrom matri (sometimes this matri is called the Jacobi field matri). For eample, if ou have a 2 dimensional sstem 1(t) f 1 ( 1 (t), 2 (t)) 2(t) f 2 ( 1 (t), 2 (t)). Then the sensitivit to initial conditions (which is one wa of thinking of chaos) means we should take the derivative of the sstem with respect to the initial conditions. This would give us the rate of change of the variables along the solution curve as a function of time. ( ) 1 (t) 1 () ( ) 1 (t) 2 () ( ) 2 (t) 1 () ( ) 2 (t) 2 () 1 () f 1( 1, 2 ) 2 () f 1( 1, 2 ) 1 () f 2( 1, 2 ) 2 () f 2( 1, 2 ) f 1 ( 1, 2 ) 1(t) 1 1 () + f 1 ( 1, 2 ) 1(t) 1 2 () + f 2 ( 1, 2 ) 1(t) 1 1 () + f 2 ( 1, 2 ) 1(t) 1 2 () + f 1 ( 1, 2 ) 2(t) 2 1 () f 1 ( 1, 2 ) 2(t) 2 2 () f 2 ( 1, 2 ) 2(t) 2 1 () f 2 ( 1, 2 ) 2(t) 2 2 () (2) We ve had to use the chain rule to epand the partial derivative. The monodrom matri has elements which

3 Math Modeling Lecture 13: Chaos and Fractals Page 3 are called Jacobi fields, J ij (t) i (t)/ j (), and it is given b M 1 (t) 1 () 2 (t) 1 () 1 (t) 2 () 2 (t) 2 (). The Lapunov eponents are defined as 1 λ ij lim t t ln J ij, which is the same as assuming the Jacobi fields grow as J ij ep(λ ij t). Tpicall, the largest of the Lapunov eponents is used to determine the degree of chaos present in the trajector (different trajectories have different Lapunov eponents). Sometimes, a coarse run of a great man solutions of the dnamical sstem and monodrom matri is performed and the Lapunov eponents averaged to estimate the choaticit of the entire sstem. There are other was of determining the Lapunov eponents as well, in terms of integrals. All well and good, but to get λ ij we need to solve the 4 dimensional sstem in Eq. (2)! The sstem is initialized using the fact that at time zero the monodrom matri is the identit matri. For the Henon Heiles sstem, we would solve for the phase space trajector and monodrom matri together, which would entail solving for 2 (4 phase space variables, 16 Jacobi fields) time evolved quantities at once. In m eperience, with molecular collisoins, I am forced to calculate the monodrom matri anwa, so it is eas to get Lapunov eponents. However, ou do not want to calculate the monodrom simpl as a means to get the Lapunov eponents. Simpl rerunning the analsis with a small change in the initial conditions and eeballing the two solutions to see if there is a big difference ma be good enough. The Hénon Heiles models actuall ehibit a ver low degree of chaos. The Rössler Attractor The Rössler attractor (found in 1976 b Otto Rössler) is a beautiful, elementar, eample of chaos in continuous sstems. Rössler s dnamical sstem is: ( + ) +.2 z.2 + z.7z (3) I solved this sstem in Mathematica using an interpolating function using the following commands. The result is output in Fig. 2. tf 21; sol NDSolve[{ [t] -[t] - z[t], [] -.4,

4 Math Modeling Lecture 13: Chaos and Fractals Page 4 [t] [t] +.2 [t], [] -.3, z [t].2 + [t]z[t] -.7 z[t], z[].2}, {,, z}, {t,, tf}, Method -> RungeKutta, MaSteps -> 2] plot1 ParametricPlot3D[Evaluate[{[t], [t], z[t]} /. sol], {t,, tf}, PlotPoints -> 6, PlotRange -> All] Equation (3) is called an attractor since for different initial condition which are in the region of the orbit shown in Fig. 1, the long term behaviour of the solution will look ver similar. The trajector is not approaching an orbit in the sense of approaching a circle or some other stable, fied, curve. The trajector in phase space is bound to follow a pattern like the one in Fig 2, but the specific trajector is chaotic over long time periods. The trajector is bound, et chaotic since where it is within that boundar is sensitive to initial conditions. Figure 3 shows this, in the sense that two trajectories which begin close together remain in the same general region, but move in different parts of the region after a period of time. Hence, an attractor pulls in neighbouring points, and for a chaotic attractor orbits of nearb points must diverge from each other due to the sensitive dependence on initial conditions. To be an attractor, one of the Lapunov eponents must be negative; to be chaotic, one of the Lapunov eponents must be positive. The Lorenz attractor is another beautiful chaotic sstem which is a model of thermal convection. tetbook discusses it. Our Figure 2: The phase space solution to the continuous sstem in Eq. (3), the Rössler attractor. The initial conditions are ().4, ().3, z().2.

5 Math Modeling Lecture 13: Chaos and Fractals Page Figure 3: The (t) vs t solution to the continuous sstem in Eq. (3) for two trajectories: on the left we have ().4, ().3, z().2; in the middle we have ().4, ().3, z().2; on the right we have the difference in (t) between the two trajectories. Although the value of (t) remains bounded b the attractor, two trajectories which begin close together become far apart for t > 1. Henon Map: Discrete Attractor The Henon map is an eample of a discrete sstem which ehibits an attraction to a limit ccle. The Henon map is given b n n n 1 n.3 n 1, n 1, 2, 3,... (4) Figure 4 shows the Henon map phase space for two different initial conditions. We can see that in both cases, the same limit ccle is approached. So we have an attractor, but it is chaotic (since where we are on the limit ccle is dependent on initial conditions) Figure 4: The phase space solution to the discrete sstem in Eq. (4), the Henon Map. The initial conditions for the graph on the left are () 1, () ; for the graph on the right () 1, ().3. The number of points is 12. In economics, models of Brownian motion (the motion of a small particle suspended in water) are used to predict the stock market. This is a chaotic sstem, since the behaviour is random but the particle stas in some localized volume. The banking industr snatched up man phsicists and engineers about 1 ears ago since the had eperience with the models the banking industr wanted to use. Now there are schools who train economists in the theor directl, so the crossover doesn t happen as much toda. You ma have heard the term strange attractor. Strange attractors are the point where chaos and fractals meet. As dnamical sstems, strange attractors are chaotic. As geometric shapes, strange attractors are fractals. I guess we should talk about fractals, eh?

6 Math Modeling Lecture 13: Chaos and Fractals Page 6 Fractals There are certain smmetries in plane mathematical geometr. You can reflect an object, ou could rotate an object, or ou could translate an object. Objects ma possess no smmetr, some of these smmetries, or all of these smmetries. A circle looks same under all these smmetries; the letter A does not. There is another smmetr, the smmetr of magnification. An object which possesses this tpe of smmetr is called a fractal. Fractals are objects which look the same on an scale. Thus, fractals are self similar. Their are man famous eamples of fractal geometr, and with the eplosion of the internet fractal art has also eploded. It is ver eas to search the internet and find a fractal landscape that someone has created to stir our souls. People also create fractal music, although I haven t heard an mself. The reason fractals pops up when people talk about discrete dnamical sstems is because fractal concepts like the Julia set, the Mandelbrot set, continued fractions, are all computed using an iterative technique. For eample, here is how the Julia set is constructed b an iteration technique: z n ± z n 1 c () where z n is a comple number, and c is a comple constant. At each step we randoml decide if we want the positive or negative square root. If we plot z in the comple plane, we get a fractal image. In Fig. I have plotted the Julia set on two different scales. The self-similarit of the Julia set is evident Figure : The comple plane representation of the iteration of the Julia set in Eq. (). The initial point was z() 1/2, and c i. The number of points used was. Notice that changing the scale b an order of magnitude the geometr looks the same. This is an indication of self similarit, and fractal behaviour. Notice that the initial conditions do not affect the final shape of the Julia set (Figs. and 6). The initial conditions affect the transient solution, which is the first few iterations the sstem goes through before it fall into the limit ccle of the Julia set; once this transient solution is passed, the points will be constrained

7 Math Modeling Lecture 13: Chaos and Fractals Page 7 to fill out the Julia set. Once the transient solution has passed, the solution is chaotic (it fills out the Julia set, but how it fills it out is sensitive to the initial condition). Fractals can be used to model objects in realit that possess magnification smmetr. Sa, the bottom of the ocean, or a coastline. The are also used for data compression Figure 6: The comple plane representation of the iteration of the Julia set in Eq. (). The initial point was z() 1 i, and c i. The number of points used was. This looks the same as Fig., ecept for the initial transient solution. References [1] H. Peitgen, H. Jürgens & D. Saupe, Chaos and Fractals: New Frontiers of Science, Springer Verlag (New York) [2] The Nonlinear Lab: bfraser/version1/nonlinearlab.html. Blair Fraser, [3] M. Hénon & C. Heiles, Astron. J. 69, 73 (1964). [4] M. Meerschaert, Mathematical Modelling, 2nd ed., Academic Press (San Diego) 1999 (p2).