Fractals and Chaos - opt Practical Problems

Transcription

1 Fractals and Chaos - opt Practical Problems R. Willingale October 22,

2 Contents 1 Introduction 3 2 Syllabus Fractals and processes which produce them Julia sets Iterated Function Systems Dynamical Systems Discrete systems Systems with continuous variables Routes to chaos Books and References 4 4 Getting started with SPECTRE 4 5 Problems Problem 1 - Calculating and plotting Julia sets Problem 2 - Calculating and plotting the Mandelbrot set Problem 3 - Generating a fern picture using affine transformations Problem 4 - Plotting the 4 affine transformations from the fern IFS Problem 5 - Plotting the bifurcation diagram of the logistic map Problem 6 - Plotting the Lyapunov exponent of the logistic map Problem 7 - Plotting time series and phase portraits for a driven LCR circuit Problem 8 - Plotting a bifurcation diagram for a non-linear LCR circuit Problem 9 - A practical demonstration of the varactor LCR circuit Assessment 14 2

3 1 Introduction The problems described below form part of the opt.2606 course entitled Fractals and Chaos. These problems will teach you how to generate fractals and how to simulate chaotic systems using a computer. They are to be completed in 8 hours of workshop sessions distributed throughout the course. You will be required to complete and modify skeleton programs which are written in R and C. You will also be able to compare the behaviour of a real chaotic system with a computer simulation. The programs are available on the UoL SPECTRE system and you will edit and run the programs in the same way as required for the 2nd year computing workshops. The script for this workshop can be found at: Details about the Fractals and Chaos workshops, including this script, can be found at URL: 2 Syllabus 2.1 Fractals and processes which produce them Julia sets key concepts; iteration, simple formulae leading to a very high level of complexity, fractal geometry, selfsimilarity, scaling, fractal dimensions, connectivity of sets - connected (one piece) or a Cantor set (dust of infinitely many points) - the Mandelbrot set Iterated Function Systems modelling fractals in the real world; Sierpinski triangle, affine transformations, clouds, smoking chimneys and ferns, deterministic and random systems, Koch curves, landscapes, mountains, rivers and coastlines, perimeter-area relation for fractal objects 2.2 Dynamical Systems Discrete systems Iteration of 1-D and 2-D maps, regular, quasi-periodic and chaotic behaviour, logistic equation, period doubling, bifurcation sequences, sensitivity to initial conditions, Lyapunov exponents, attractors Systems with continuous variables equations of motion, solution by iteration of difference equations, phase space, phase portraits, Poincaré sections, connection with discrete systems, basins of attraction, Fourier analysis, - power spectra, dimen- 3

4 sionality of dynamical systems, delay coordinates Routes to chaos Forced motion (dissipative systems), Hamiltonian systems, universality, Feigenbaum s number, strange attractors, the Lorenz system, fractal dimensions, non-linear dynamics, feedback, coupled equations, examples of chaotic physical systems 3 Books and References The Science of Fractal Images, Eds. Heinz-Otto Peitgen and Dietmar Saupe, Springer-Verlag, 1988, ISBN Fractals, Jens Feder, Plenum NY and London, 1988, Chaos and Fractals, H.O. Peitgen, H. Jurgens, D. Saupe, Springer-Verlag NY, 1992, Fractal Images and Chaos, H. Lauwerier, Penguin, 1991, The Pattern Book: Fractals, Art, and Nature, Editor C.A. Pickover, World Scientific Publishing Co. Pte. Ltd., 1995, X Chaotic Dynamics, G.L. Baker and J.P. Gollub, CUP, 1990, Fractals and Chaos, P.S. Addison, IoP, 1997, Period doubling and chaotic behaviour in a driven anharmonic oscillator, P.S. Linsay, Phys. Rev. Lett. 47, 1349, 1981 Evidence for the universal chaotic behaviour of a driven nonlinear oscillator, J. Testa, José Pérez and C. Jeffries, Phys. Rev. Lett. 48, 714, 1982 Chaotic Dynamics and Instabilities in solids, C.D. Jeffries, Physica Scripta Vol. T9, 11, Getting started with SPECTRE You can access SPECTRE using a X-Terminal client called NX running on the UoL IT Windows system. So first you must logon to a UoL IT Windows machine. If you have not yet used NX on your Windows account then you must install it. Programs-->Program Installer-->NX Client (click) Once installed you can start NX for a SPECTRE session. Programs-->NX Client (click) 4

5 SPECTRE (click) In order to logon to SPECTRE you must supply your UoL username and password. NX will open a window for SPECTRE. In order to do the workshop you will need to start a terminal window for SPECTRE. Applications-->System Tools-->Terminal (click) The terminal window will open within the NX window and the prompt within the terminal window will look something like: [zrw@spectre02~]$ This indicates the username (zrw in my case), the machine (spectre02 for this particular login) and the directory (~ is the home directory of the user). This terminal window serves the same function as the R Console. You type commands against the prompt and the system will execute these commands. Under Linux (or more generally UNIX) the prompt is produced by what is called a shell. This is a program which runs within the terminal and acts as an interface or wrapper around the system giving the user access to all aspects of the system. 5 Problems The skeletonrscripts and C code for the problems must be copied onto yourown directory. For example, the source code file for problem 1 is called Julia_set.Rand must be copied using the cp command. Note that the final dot in the command line puts the file into the current directory. $ cp /home/z/zrw/under/julia_set.r. The other skeleton source code files should be copied in the same way, as required. You can edit the source code using a text editor in the same way as in the 1st and 2nd year C programming workshops. To edit the R scripts or source code files you must use nedit or a similar text editor. To invoke the editor: $ nedit filename & Note the&atthe end ofthe commandline is important. It tells the shell to runthe editorasabackground job so that once the editor window is opened you can switch focus between the original terminal window and the editor window without having to close down the editor. You can edit the script or program with the editor, save the current version of the file and then switch to the terminal window to run the script. You can then return to the editor window and make changes, save again and again switch back to the terminal to run the program. 5

6 Instructions for compiling and running the programs are provided below and within the source scripts and code files. When you start up a terminal window on SPECTRE you must make R available by using the following command: $ module load R You only need to do this once every time youstart a terminal. Typingmodule load R at the start ofeach SPECTRE session is tedious. You can avoid this by putting this line into your bash (shell) profile file. Use emacs or nedit to add this line to your.bash_profile file. This file exists in your home directory so make sure you are in your home directory (using the cd command without specifying any file) before trying to edit the file. # cd $ nedit.bash_profile & Add the line module load R at the end of the file and save. Next time you start a terminal screen the command R will be available without the need to load the module. There are 2 ways to run the R scripts. Either use the source() function in R: $ R (to start R) > source("script_name.r") (the > is the R prompt) or directly at the shell prompt using Rscript: $.\script_name.r The 2nd way is quicker and easier and is recommended. Some of the R scripts use functions which are written in C. If you edit the C source code you must recompile using the following shell command: $ R CMD SHLIB source_file.c This will create a file source_file.so which is the shareble object. Actually you can perform the same task by using the C compiler directly but using the R command above ensures that the shareble object created is linked with all the required libraries. Many of the scripts produce plots. By default these are displayed on your screen using the X11() function in R. If you want to produce a file which contains the plot you can use the pdf() function: > pdf("filename.pdf") 6

7 You can copy the pdf file created (and any other files) from you SPECTRE home space to your Windows home space. Using Windows Explorer enter the following in the address bar: \\spectre01.rcs.le.ac.uk\username substituting your username. You can drag files from the Windows Explorer window on to your Desktop. When you have the files on your Windows home space you can print then etc. in the normal way. As you proceed through the problems you MUST keep notes recording what you do, results you obtain and new ideas and concept which you learn. If you try things which go beyond the scope of the script record these as well. At the end of the lectures and workshops your notes should provide a complete record of the course and what you have learnt. The tasks are broken down into sections and include questions to be answered. Keep notes on your answers to the questions. We recommend that you use a dedicated directory for this course. $ cd (to get you to your home directory) $ mkdir Rchaos (to create the new directory) # cd Rchaos (to set the current working directory) 5.1 Problem 1 - Calculating and plotting Julia sets The skeleton program R script file is Julia_set.R This implements an algorithm known as the Level Set Method for producing a rendering or picture of a Julia Set (or more strictly a filled-in Julia set). The Julia set is built around a mapping of the form: z f c (z) where c is a constant. The simplest functional form which produces complicated Julia sets is: f c (z) = z 2 +c Repeated application of this mapping produces an orbit in the complex plane: z f c (z) f c (f c (z)))... f k c (z) After k iterations we end up at the point: z k c = fk c (z) If fc k (z) never reaches, even as k, then the initial point z is a member of the filled-in Julia set characterised by the constant c. On the other hand, if the orbit escapes to then z is not a member of the Julia set. 7

8 In practise we cannot iterate an infinite number of times and we cannot measure if the orbit truly escapes to infinity. We therefore choose a large number N max and define some target set which comprises all the points at a distance greater than some radius R from the origin. If the orbit enters this target set for k < N max it is deemed to have escaped to infinity and the level of point z is defined as the number of iterations required to escape. On the other hand if the orbit remains at a radius < R after N max iterations the point is deemed to be in the filled-in Julia set. We take a square grid of points spanning part of the complex plane, usually centred on the origin. For each point we iterate as described above. If the orbit escapes we label the point with the level k. If not we label the point with N max indicating the point is very close to, or within, the set. If we increase N max or R we expect the result to be a better approximation to the limiting set. i) Copy the Julia_set.Rfile to your dedicated course directory. Use the editor to look at the file. Study the R script to see if you can work out how it creates a picture of the Julia set. You will see that the central function that performs the iteration is setlevel(z,c,level). This function is recursive. What does this mean? Can you see how the function performs the iteration? ii) Run the script. You will see that the program lists the time in seconds before and after the main calculation. How long does the main calculation take? What is the size of the image created? How long does it take to calculate each point in the image? Note: to terminate the program you must click on the image using the cursor. iii) R is an interpreted language. As the program runs an interpreter program is continuously reading and translating the script file and this takes time. The calculation will run much faster if the iteration function is compiled using a language like C or Fortran. There is another version of the Julia set script which uses a iteration function written in C. Copy the Julia_set_c.R script to your directory. Use the editor to look at the file. Note how this script differs from the original. What happens if you try to run the script and why? iv) Copy the files j_m_set.c and j_m_set.so to your directory. Try running the Julia_set_c.R again. How long does it take to calculate each point in the image? v) You will see that the script now calculates 3 images. These are the level (number of iterations to escape), the rend which is the escape angle and dist an estimate of the distance from the set to the point. So as well as running faster the C function is doing a lot more calculation. Use the editor to have a look at the C source code file j_m_set.c. Try and work out how the C function julia() is doing the iteration and how it is estimating the 3 values at each point. vi) Run the Julia_set_c.R script for different values of the constant c. Try and find examples of connected Julia sets and Cantor Julia sets. A good way to do this is to move out across the complex plane in the direction set by the inital value of the constant c. You can do this by adding a line like the following into the function. sc<- 1.1; cr<- cr*sc; ci<- ci*sc vii) Julia sets can be defined for transcendental functions and not just the quadratic mapping used above. Make a copy of the Julia_set.R $ cp Julia_set.R Julia_set_cos.R 8

9 Modify the script Julia_set_cos.R to calculate the filled-in Julia set of the mapping defined by: f c (z) = λcos(z) where the constant λ is real in the range 2.9 < λ < π. Show that if λ 2.96 the resulting pattern exhibits many of the characteristics expected of a fractal. Note that the program must calculate the cosine of a complex number. This is easy in R but not so easy in C. You could modify the C function in the j_m_set.c to calculate the cos(z) mapping but you would need to work out what this corresponds to in terms of the real and imaginary parts of z. The functions exp, cos, sin, cosh and sinh are all available in C so you might like to work out how to do this! If you have some spare time try getting this to work and give the same result as the slow R only version. 5.2 Problem 2 - Calculating and plotting the Mandelbrot set The Mandelbrot set is a sort of catalogue of the Julia sets associated with the quadratic mapping. A point z on the complex plane is a member of the Mandelbrot set if the Julia set for c = z is connected (one continuous island in the complex plane). If the point is not a member of the Mandelbrot set then the corresponding Julia set is a Cantor set (a dust of infinitely many unconnected points). We can calculate the Mandelbrot set using the Level Set Method described above. The only difference in the algorithm is that the constant c is replaced by the starting value z, strange but true! i) Copy the file Mandelbrot_set.R to your directory. $ cp /home/z/zrw/under/mandelbrot_set.r. Use the editor to look at the file. Which line in the script makes the setlevel() function calculate the Mandelbrot set rather than a Julia set? Run the script. How long does it take to calculate 1 point in the image of the Mandelbrot set? ii) Now copy the version of the script which uses a C function to do the iteration, Mandelbrot_set_c.R. How long does this script take to calculate 1 point in the image? iii) Use (and modify if need be) the script to find points which correspond to Julia sets that are connected or Cantor sets. You can make use of the locator(1) function to do this. print(locator(1)) Use the cursor to pick a position in the plot. The above code will print out the position. iv) Copy the script Mandelbrot_zoom_c.R to your directory. This allows you to zoom into the edge of the Mandelbrot set by changing the area of the complex plane plotted. Take a look at the script to see how this is done. Try modifying the script so that you can use the cursor to define a new centre position and recalculate and display a new zoomed version of the set at that position. Arrange it so that the script continues to loop until you specify a position which is outside the current area. As you zoom in print out the position and zoom factor or image width so that you know how far you have tunnelled in 9

10 at any stage. How small can the area of the image be (or alternatively how large a zoom factor can be used) before the algorithm for rendering the set starts to break down? What do you think is limiting the algorithm? 5.3 Problem 3 - Generating a fern picture using affine transformations Fractals which resemble naturally occuring patterns can be produced using Iterated Function Systems (IFS). A particularly rich form of IFS uses a combination of affine transformations. An affine transformation in two dimensional space is define by: [ x w y ] [ a11 x+a = 12 y +b 1 a 21 x+a 22 y +b 2 ] The a ij s and b i s are real constants that form a matrix A and vector B. Thus the affine transformation is specified by six real numbers. We can re-write the components of the affine transformation as: A = [ rcosθ ssinφ rsinθ scosφ ] [ h B = k ] where h and k are translations, θ and φ are rotations and r and s are scalings. A fixed point of an affine transformation is a point (x, y) which remains the same on applying the transformation. The IFS consistsoftakingagroupofaffinetransformations(4say)and allottingaprobabilitytoeachone. Starting with a fixed point of one of the transformations you iterate, choosing the next transformation at random in accordance with the probabilities. The point will move about in the XY plane slowly covering a closed set of points under the transformations. The geometry of the pattern is prescribed by the transformations while the density of points in particular areas of the pattern is determined by the probabilities. If you don t start with a fixed point the iteration may wander off in a random or open orbit. i) The script /home/z/zrw/under/affine_fern.r will plot a fractal fern-like pattern. Copy the file and try it out. ii) Use the coefficients of the affine transformations in the file to calculate the translations, rotations and scalings that make up the 4 mappings which determine the shape of the fern. You can do this either by adding code to the program (the elegant way) or doing it by hand using a calculator (the clunky way). Note that you can determine the order of the coefficients by looking at the function affine_trans. iii) The function fixed_point will return a fixed point of the transformation defined by the 6 coefficients supplied by the first argument cf[i,]. How does this function calculate the fixed point? 10

11 5.4 Problem 4 - Plotting the 4 affine transformations from the fern IFS The script Affine_plot.R plots the effect of the 4 affine transformations from Problem 3 on a test triangle. Try running the script. This will give you a better idea of how the fern geometry is determined by the transformations. i) How can you alter the transformations so that the fern bends over to the left rather than to the right? Try out the new transformations by modifying Affine_plot.R and Affine_fern.R to check that your suggested solution is correct. ii) How do the probabilities associated with each of the transformations effect the final picture? Try running the Affine_fern.R script with equal probability for all the transformations. How can you use the plot produced by the Affine_plot.R script to estimate the optimum probabilities to use? 5.5 Problem 5 - Plotting the bifurcation diagram of the logistic map The logistic map has the quadratic form: x n+1 = µx n (1 x n ) It is very similar to the quadratic mapping used for generating the Mandelbrot set except that x is real rather than complex. If the constant µ lies in the range 0 < µ < 4.0 successive values of x n are bounded 0 < x n < 1.0. If the mapping is iterated a sufficiently large number of times the sequence of x n s lie on a well defined orbit. The structure and complexity of this orbit depends on the value of µ. The script Logistic_bifurcation_diagram.R is designed to plot the orbit values x n for NPARAM values of µ. For each value of µ the iteration is repeated NSPIN times before a sequence of NPOINT orbit values are plotted. Eventually a total of NPARAM*NPOINT values of x n are plotted at their associated µ values. The complete sequence of dots forms a so-called bifurcation diagram. i) Copy the script file Logistic_bifurcation_diagram.R to your directory and run it. Look carefully at the code to make sure you understand how it works. ii) Changethe rangesofµvaluesandxvaluesplotted sothat the diagramzoomsin onthe firstbifurcation sequence. Use the plot to provide an estimate of Feigenbaum s number: µ k µ k 1 δ = lim k µ k+1 µ k where µ k is the value of the constant at the kth bifurcation. Also estimate the value of µ k for the first bifurcation sequence. Hint: you can use the cursor and the locator(1) function to find the positions of bifurcations on the plot. 5.6 Problem 6 - Plotting the Lyapunov exponent of the logistic map For a 1-D map the Lyapunov exponent is given by: 11

12 n 1 1 λ = lim log n n e f (x i ) i=0 where f (x i ) is the first derivative of the map function. This exponent is the stretching rate per iteration (divergence or convergence) averaged over the orbit. Regions of periodic behaviour of the logistic map correspond to intervals in µ over which λ < 0. At bifurcations λ 0 and if the motion becomes chaotic λ > 0. The script Logistic_lyapunov_exponent.R calculates and plots this exponent for a range of µ values. i) Copy the script Logistic_lyapunov_exponent.R to your directory. The lines of code which perform the summation to calculate λ are missing. Add these lines to make the script work. Run the script to produce the plot. ii) Modifythe scriptsothatyoucancalculatefeigenbaum snumberδ andµ k fromthefirstbifurcation sequence, as accurately as possible. Exactly how you do this is up to you. You can set the plotting ranges to zoom in on the bifurcation sequence. You could select bifurcation points using the cursor or you could search for values of the Lyapunov exponent which are very close to zero. 5.7 Problem 7 - Plotting time series and phase portraits for a driven LCR circuit The LCR circuit under consideration comprises an inductance L, a varactor diode instead of a conventional capacitor and a resistance R, in series, driven by a sinusoidal voltage. The diode has a capacitance which depends on the applied voltage. If the drive voltage is given by: V(t) = V 0 sin(ωt) with the frequency ω/2π reasonably close to resonance, the equations of motion of the current and voltage for the diode are approximately: di dt = V 0sin(ωt) RI V L dv dt = I I d(v) C(V) where the capacitance has two components, C j the junction differential capacitance and C s the storage differential capacitance: C j (V) = C j0 (1 V φ c ) 1/2 12

13 C s (V) = C s0 exp( V φ ) and the junction current is: I d (V) = I 0 (exp( V φ ) 1) The parameters C j0, φ c, C s0, φ and I 0 are measurable parameters for a given junction. We can numerically integrate these equations, using a fourth order Runge-Kutta algorithm or similar, to give the diode current and voltage as a function of time. The script LCR_circuit.R performs such an integration. In the script the list p holds the system variables. Variable cd is C s0 and the variable ct is C j0. The other parameters are self explanatory. The function deriv() should represent the equations of motion. The array components of var[nvar] are the dynamical variables, var[1], the diode current and var[2], the diode voltage. The function returns the drive voltage vd at time t, and the derivatives of the current and voltage, also at time t, according to the equations above. i) Look up further details and background to driven nonlinear oscillators in the Linsay (1981), Testa et al. (1982) and Jeffries (1985). ii) Copy the script LCR_circuit.Rto your directory and use the editor to have a look at the code. Check that you understand how the function deriv() implements the equations of motion above. The script will plot out the driving time series, the diode current time series and two phase portraits of the motion. With V 0 = 1.5 the motion should be chaotic. Demonstrate that this is the case. iii) Use the script to characterise the motion for drive amplitudes V 0 = 1.0, V 0 = 2.05 and V 0 = 4.0. iv) Change the function deriv() so that the capacitance is NOT a function of applied voltage. Demonstrate that if this is the case the motion is regular and not chaotic whatever the drive amplitude. 5.8 Problem 8 - Plotting a bifurcation diagram for a non-linear LCR circuit We can plot a bifurcation diagram for the non-linear LCR circuit described in Problem 7 above. Instead of plotting successive iterations of a 1-D mapping as we did for the logistic map in the Logistic_bifurcation.R script we now plot the amplitudes of successive maxima of the oscillation. To do this we must sample the integrated time series once every cycle. We can change the phase of the sampling by changing the start time of the integration sequence. This can be used to compensate for any phase difference between the driving voltage and the diode current. The script LCR_bifurcation.R is designed to do this. It is very similar to Logistic_bifurcation.R script except that instead of iterating the logistic equation we integrate the equations of motion of the LCR circuit through 1 cycle. i) Run the script LCR_bifurcation.R. Compare the bifurcation diagram produced with the bifurcation diagram of the logistic map. ii) Modify the script so that you can estimate the convergence ratio 13

14 V 0k V 0k 1 δ = lim k V 0k+1 V 0k for the first bifurcation sequence. How does your answer compare with Feigenbaum s number? iii) Find the drive voltage amplitudes that yield period 5 and period 3 oscillations. Are there any other periodic windows you can identify within the chaotic region? How do these windows compare with the bifurcation sequence of the logistic map? 5.9 Problem 9 - A practical demonstration of the varactor LCR circuit The LCR circuit is supplied in a black box. If you take the top off you will see the pertinent components. An op-amp is used to monitor the diode current. The input must be provided using a signal generator. The frequency should be set in the range khz. The X-output is the drive voltage and the Y-output is the diode current as seen by the op-amp. The jump leads on the top of the box allow you to select different L and R values. i) Set the circuit up and display the X and Y outputs as separate traces on a scope triggering off the Y output. Try out various combinations of L, R and drive frequency until you find a reasonable looking bifurcation sequence. Don t forget to turn on the op-amp using the switch on the box! ii) Change the oscilloscope setting to plot the X and Y outputs as a Lissajous figure. Using this display it is very easy to view the bifurcation sequence. Try following the sequence as carefully as possible through to chaos noting down the drive voltage values at each bifurcation. Also record the voltage when the sequence falls to chaos. You can use a digital meter to measure the input voltages reasonably accurately. iii) Continue increasing the drive voltage and look for period 5 and period 3 windows. Draw a sketch of the bifurcation sequence and compare it with the simulated plot from the chaos4 program. iv) Make a quantitative comparison of the simulated LCR bifurcation diagram and the observed LCR bifurcation sequence with the results from program chaos1 for the logistic equation. 6 Assessment To complete the course you must produce and submit a portfolio of work. This portfolio comprises the following: Notes on Fractals and Chaos derived from the series of lectures. Notes about the topics introduced in the lectures and workshops taken from the books and references given in the course or found during background reading. A diary of your activities in the workshops. This should include details on the way you modified the computer programs provided and how you ran the programs to answer the questions posed within each section. A report of any results obtained from using the programs in your own investigation of the properties of Fractals and Chaotic systems. 14

15 A list of references including any web link URLs, books or journal articles cited in the portfolio. The portfolio should cover all the topics in the course and all the tasks undertaken in the workshops. A typical length is 15 sides of A4 but that rather depends on the use of diagrams etc. The portfolio should not be just a copy of the lecture notes, a copy of the source code of the programs or a copy of the script from the workshops. Rather, it should demonstrate your understanding of the subject and provide an account of what you have learnt from the course. The portfolio is best produced as a word-document. When complete you should save the document as a PDF file. Submit a printout of the PDF file to the Teaching Office. 15