Nonlinear Dynamics and Chaos - a set of class notes

A ''dynamical system'' consists of ''states'' and a ''rule'' giving the present state in terms of past states. For a deterministic dynamical system this rule prescribes a unique determination of the present from the past and there is no randomness in the rule. \n\nThere are two types of systems: discrete and continuous. ''Discrete systems'' are given as functions of discrete variables. For ''continuous'' systems the functions depend on continuous variables. A typical variable is time. Time can flow continuously, or we could mark it off in discrete values, such as a number of days, months, years, etc. \n\nWe will first look at one dimensional discrete systems. Then states are {//x~~n~~//} for //n=0,1,2,...//, and a rule is given by //x~~n~~=f(x~~n-1~~)//. One needs an ''initial state'', //x~~0~~//, to generate future states, //x~~1~~//, //x~~2~~//, etc. The set {//x~~0~~, x~~1~~,x~~2~~, ...//} is called an ''orbit'', or a ''trajectory''. One often refers to these rules as ''maps''. A map is a function whose domain is the same as its range. \n\nThere are special points that help in describing the behavior of a given system. One typically looks at fixed points and period points, or orbits, and the orbits in the neightborhood of these special points.\n\nA ''fixed point'', //p// for the map described by //f(x)// satisfies //f(p)=p//.\n\n<<<\n''Example'' //f(x)=x^^2^^//. \n\nTo find the fixed point, we solve //p^^2^^=p//. The solutions are //p=0,1//.\n<<<\n\n<<<\n''Example'' //f(x)=cos(x)//. \n\nTo find the fixed point, we have to solve //cos(p)=p//. This is a transcendental equation. We can solve it graphically, or use Newton's method, //x~~n+1~~ = x~~n~~ - g(x~~n~~)/g'(x~~n~~)//.\n<<<\n\n<<<\n''Example'' //f(x)=rx(1-x)//, //x// in [0,1]. \n\nThis is the famous ''logistic map''. It is a very simple quadratic map which exhibits some of the key features of one dimensional maps as the parameter //r// is varied. We will spend some time looking at the behavior of this map. \n\nIt's graph is a parabola with vertex at (1/2, //r///4). Thus, for it to be a map of [0,1] to [0,1], the values of //r// are restricted to be positive and no larger then 4. \n\nThe fixed points satisify //rp(1-p)=p//. Solutions of this equation are //p=0// and //p=1-1/r//. Note that for //1-1/r// to be in [0,1], one needs $r\sge 1$. Thus, there is one fixed point for $ r\sle 1$ and two otherwise.\n<<<\n\n

We now look at two dimensional discrete systems (or, 2D maps). The states are points in the plane {//(x~~n~~,y~~n~~)//} for //n=0,1,2,...//, and the rule is given by a pair of functions\n//x~~n~~ = f(x~~n-1~~,y~~n-1~~)//, \n//y~~n~~ = g(x~~n-1~~,y~~n-1~~)//. \nOne also needs an ''initial state'', (//x~~0~~,y~~0~~//), to generate future states.\n\nSometimes the system is defined with a vector valued function ''F''//(x, y)// = //(f(x, y), g(x, y))//.\n\n<<<\n''Examples'' \n\nThe Henon Map: //''f''(x, y)=(a - x^^2^^ + by, x)//. \n\nLinear Maps: //''f''(x, y)=(ax + by, cx + dy)//. \n\nThe [[Standard Map]]: \n //y~~n+1~~ = y~~n~~ + k //sin //x~~n~~// ,\n //x~~n+1~~ = x~~n~~ + y~~n+1~~// (mod //2 $\spi$// ) \n<<<\n\nChecking the above Standard Map link shows that the dynamics for nonlinear two dimensional maps is more interesting than we have seen for one dimensional maps. \n\nAs with maps in one dimension, we will first look at fixed points. The general map takes the form //''f'' = (f(x,y), g(x,y)//. We consider the resulting 2D discrete map \n//x~~n~~ = f(x~~n-1~~,y~~n-1~~)//, \n//y~~n~~ = g(x~~n-1~~,y~~n-1~~)//. \nA fixed point of this map, //(x*,y*)//, satisfies the equations\n//x* = f(x*,y*)//, \n//y* = g(x*,y*)//. \nSo, one solves this system. In general there are several fixed points. \n\nNext, one considers the stability of the fixed points. For each fixed point, one considers the behavior of all points in the neighborhood of the fixed point. Denote these points as //(x~~n~~,y~~n~~) = (x*,y*)+($\snu_n$,$\stau_n$)//. Inserting into the system, one has\n//x~~n~~ = f(x~~n-1~~,y~~n-1~~) = f(x*+$\snu_n$,y*+$\stau_n$)//, \n//y~~n~~ = g(x~~n-1~~,y~~n-1~~) = g(x*+$\snu_n$,y*+$\stau_n$) //. \n\nDenoting //''x''~~n~~ = (x~~n~~,y~~n~~), // // ''x''* = (x*,y*) // and $\sxi_n$ = ($\snu_n$,$\stau_n$), we can write this equation in vector form as\n//''x''~~n~~ = ''f''(''x''*+$\sxi_{n-1}$ )//.\nUsing the multivariate Taylor seriesexpansion, we have \n//''x''*+ $\sxi_n$ $\sapprox$ ''f''(''x''*)+D''f''(''x''*)$\sxi_{n-1}$ //.\nHere //D''f''(''x''*)// is the Jacobian matrix evaluated at the fixed point. Since //''f''(''x''*) = ''x''*//, \n//'$\sxi_n$ = D''f''(''x''*)$\sxi_{n-1}$ //.\n\nThus, the behavior of points in the neighborhood of a given fixed point depends on the Jacobian matrix.This last equation is a linear map for the system of //'$\sxi_n$ = ''x''~~n~~-''x''*//. Recall that the Jacobian matrix is a matrix of first derivatives. For our two dimensional map, it is given by\n$$D{\smathbf f} = \sleft(\sbegin{array}{cc}\n \sfrac{\spartial f}{\spartial x} & \sfrac{\spartial f}{\spartial y} \s\s\n \sfrac{\spartial g}{\spartial x} & \sfrac{\spartial g}{\spartial y} \s\s\n\send{array}\sright)$$\nIn order to understand nonlinear dynamics, we will first need to investigate linear maps. \n\n

This is an experimental use of [[TiddyWiki|http://www.tiddlywiki.com/]] for a course on //Nonlinear Dynamics and Chaos//. for the [[Mathematics and Statistics Department|http://www.uncw.edu/math]] at [[UNC Wilmington|http://www.uncw.edu]]. The course webpage is at [[my web site|http://people.uncw.edu/hermanr/chaos/]]. \n\nThis page is built using Tiddlers, which are chunks of information that can be brought up in a nonlinear order. You can select the Tiddlers you want displayed by clicking on the Menu items. (Click on any highlighted topic.) You can close tiddlers using close, close all, or close others. Links to other pages will appear in new browser windows. \n\nYou can print a collection of Tiddlers or create a link to this view them using permaview. The one caveat is that Miscrosoft Internet Explorer sometimes renders these pages differently than Firefox, the preferred browser for viewing this content.

In the [[Introduction to Chaos]] we noted that the [[Exploratorium|http://www.exploratorium.edu/]] has a nice short [[description of chaos|http://www.exploratorium.edu/complexity/lexicon/chaos.html]]: \n<<<\nThe qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems - by Dr. Stephen H. Kellert, //In the Wake of Chaos//\n<<<\n\nThe definition that we will use of a chaotic orbit is one which is ''not asymptotically periodic'' and whose ''Lyapunov exponent is greater than zero''. Of course, one needs to define these terms. \n\nAn orbit is asymptocially periodic if it converges to a periodic orbit. Namely, let be $\s{x_1, x_2, \sdots \s}$ be the orbit of interest and let $\s{y_1, y_2, \sdots, y_k, y_1, \sdots \s}$ be a period $k$ orbit. Then, the first of these is asymptotically periodic if and only if $$\slim_{n\srightarrow\sinfty} |x_n-y_n|=0.$$\n\nNot only should such orbits be forbidden from approaching periodic orbits, but neighboring orbits should diverge rapidly. This is needed for sensitive dependence on initial conditions. One measures this (exponential) divergence using Lyapuonv exponents. We first define the Lyapunov number $L$ of the orbit as $$L(x_1) = \slim_{n\srightarrow\sinfty} \sleft(|f'(x_1)|\sdots |f'(x_n)| \sright)^{1/n}.$$ Then the Lyapunov exponent is given as $$h(x_1) = \slim_{n\srightarrow\sinfty} \sfrac 1n \sleft[\sln|f'(x_1)|+\sdots +|\sln f'(x_n)| \sright].$$ Of course, these definitions only apply when the limits exist and when they do one has, $\sln L=h.$\n\nThe tent map is an example of a map that has infinitely many chaotic orbits. First of all, the Lyapunov exponent of an orbit is $\sln 2$ since the absolute value of the slope of $T(x)$ is 2 for $x\sne \sfrac 12.$ Also, the itineraries of asymptotically periodic orbits of the tent map must have repeating tails. But, there are an infinite number of nonrepeating itineraries! So, there are an inifinite number of orbits satisfying the above conditions for a chaotic orbit. \n\n\n

Due to the complexity that even simple systems can exhibit, visualizations are common in the study of nonlinear dynamics and chaos. Here are a few common images that show typical features encountered in the course.\n\n A magnetc toy with two rotating arms.\n[img[Magnetic Toy|http://people.uncw.edu/hermanr/Chaos/toy.gif]]\n\n|!Standard Map|!Nonlinear Pendulum|\n|[<img[Standard Map|http://people.uncw.edu/hermanr/Chaos/chaos.1.gif]]|[>img[Nonlinear Pendulum|http://people.uncw.edu/hermanr/Chaos/chaos.2.gif]]|\n|!Damped Pendulum|!Forced Pendulum|\n|[<img[Damped Pendulum|http://people.uncw.edu/hermanr/Chaos/chaos.3.gif]]|[>img[Forced Pendulum|http://people.uncw.edu/hermanr/Chaos/chaos.4.gif]]|\n|!Homoclinic Orbit|!Perturbed Homoclinic Orbit|\n|[<img[Homoclinic Orbit|http://people.uncw.edu/hermanr/Chaos/chaos.5.gif]]|[>img[Perturbed Homoclinic Orbit|http://people.uncw.edu/hermanr/Chaos/chaos.6.gif]]|\n\n@@clear(left):clear(right):display(block): ''More images of chaos.''@@\n\n*The Chaos Game\n[img[Chaos Game|http://mathworld.wolfram.com/images/eps-gif/ChaosGameHalf_1000.gif]] \n\n*The Lorenz Attractor\n[img[LorenzAttractor|http://www.advancedforecasting.com/weathereducation/images/chaos.jpg]] \n\n*The Mandelbrot Set\n[img[Mandelbrot Set|http://picturethis.pnl.gov/im2/madelbrot_set0/madelbrot_set.jpg]]\n\n*Zooming into the Mandelbrot Set\n[img[Mandelbrot Zoom|http://www.cems.uvm.edu/math/classes/math266/images/mandel_b.jpg]]\n\n*A Julia Set\n[img[Julia Set|http://www-cc.gakushuin.ac.jp/~xyazawa/dynamics/externalrays/externalraysfilledjulia.gif]]

<<tabs tabsClass\nLayout "Layout Templates" SystemLayout\nSystem "System Tiddlers" SystemTiddlers\nStyle "StyleSheets" StyleSheets\n"Side Bar" "Side Bar Elements" SystemSideBars\nOptions "Personal Preferences" SystemOptions\nShadows "Hidden System Pages" ShadowPages\n>>\n\nThis page is a slightly modied version of [[Jack Parke's configuration|http://jackparke.googlepages.com/jtw.html#Config]]. Another one is at [[Julian Knight's site|http://knighjm.googlepages.com/knightnet-default-tw.html#Configuration]]. You can view the code. You will need to create several new tiddlers and add the corresponding code: SystemLayout, SystemTiddlers, StyleSheets, SystemSideBars,

Required Text: //Chaos: An Introduction to Dynamical Systems//, by Alligood, Sauer and Yorke, 2000.\n\n''Nonlinear Dynamics and Chaos (3 Hours)'' This course is an introduction to discrete and continuous dynamical systems leading to the study of stability and chaos in dynamical systems. Topics to be covered are one dimensional and two dimensional maps, fixed points, periodic points, bifurcation, fractals, continuous dynamical systems, nonlinear oscillations, and chaotic attractors. Applications in a variety of fields will be presented.\n\n[[Course Website|http://people.uncw.edu/hermanr/Chaos/index.htm]]

[[Course Description]]

''Example of an Eigenvalue Problem'' Determine the eigenvalues and eigenvectors for\n$$A =\sleft(\sbegin{array}{cc}\n 1 & -2 \s\s\n -3 & 2 \s\s\n\send{array}\sright)$$\n\nIn order to find the eigenvalues and eigenvectors of this equation, we need to solve\n\sbegin{equation}\nA{\sbf v} = \slambda {\sbf v}.\n\send{equation}\n\nLet ${\sbf v}=\sleft(\sbegin{array}{c}\n v_1 \s\s\n v_2 \s\s\n\send{array}\sright).$ Then the eigenvalue problem can be written out. We have that\n\sbegin{eqnarray}\nA{\sbf v} &=& \slambda {\sbf v} \s\s\n\sleft(\sbegin{array}{cc}\n 1 & -2 \s\s\n -3 & 2 \s\s\n\send{array}\sright)\sleft(\sbegin{array}{c}\n v_1 \s\s\n v_2 \s\s\n\send{array}\sright) &=& \slambda \sleft(\sbegin{array}{c}\n v_1 \s\s\n v_2 \s\s\n\send{array}\sright) \s\s\n\sleft(\sbegin{array}{c}\n v_1-2v_2 \s\s\n -3v_1+2v_2 \s\s\n\send{array}\sright) &=& \sleft(\sbegin{array}{c}\n \slambda v_1 \s\s\n \slambda v_2 \s\s\n\send{array}\sright).\n\send{eqnarray}\n\nSo, we see that our system becomes\n\sbegin{eqnarray}\nv_1-2v_2=\slambda v_1, \s\s\n-3v_1+2v_2 = \slambda v_2.\n\send{eqnarray}\nThis can be rewritten as\n\sbegin{eqnarray}\n(1-\slambda) v_1-2v_2=0, \s\s\n-3v_1+(2-\slambda)v_2 = 0.\n\send{eqnarray}\n\nThis is a homogeneous system. We can try to solve it using elimination, as we had done earlier when deriving Cramer's Rule. We find that multiplying the first equation by $2-\slambda$, the second by 2 and adding, we get $$[(1-\slambda)(2-\slambda)-6]v_1=0.$$ If the factor in the brackets is not zero, we obtain $v_1=0$.Inserting this into the system gives $v_2=0$ as well. Thus, we find ${\sbf v}$ is the zero vector. However, this does not get us anywhere. We could have guessed this solution. This simple solution is the solution of all eigenvalue problems and is called the trivial solution. When solving eigenvalue problems, we only look for nontrivial solutions!\n\nSo, we have to stipulate the the factor in the brackets is zero. This means that $v_1$ is still unknown. This situation will always occur for eigenvalue problems. The general eigenvalue problem can be written as $$A{\sbf v}-\slambda {\sbf v} = 0,$$ or by inserting the identity matrix, $$A{\sbf v}-\slambda I{\sbf v} = 0.$$ Finally, we see that we always get a homogeneous system, $$(A-\slambda I){\sbf v}=0.$$ The factor that has to be zero can be seen now as the determinant of this system. Thus, we require \n\sbegin{equation}\n\smbox{det}(A-\slambda I) =0.\n\send{equation}\n\nWe write out this condition for the example at hand. We have that\n$$\sleft|\sbegin{array}{cc}\n 1-\slambda & -2 \s\s\n -3 & 2-\slambda \s\s\n\send{array}\sright|=0.$$\nThis will always be the starting point in solving eigenvalue problems. Note that the matrix is $A$ with $lambda$'s subtracted from the diagonal elements.\n\nComputing the determinant, we have $$(1-\slambda)(2-\slambda)-6=0,$$ or $$\slambda^2-3\slambda-4=0.$$ We therefore have obtained a condition on the eigenvalues! It is a quadratic and we can factor it: $$(\slambda-4)(\slambda+1)=0.$$ So, our eigenvalues are\n$\slambda=4,-1$.\n\nThe second step is to find the eigenvectors. We have to do this for each eigenvalue. We first insert $\slambda=4$ into our system:\n\sbegin{eqnarray}\n-3 v_1-2v_2=0, \s\s\n-3v_1-2v_2 = 0.\n\send{eqnarray}\nNote that these equations are the same. So, we have one equation in two unknowns. We will not get a unique solution. This is typical of eigenvalue problems. We can pick anything we want for $v_2$ and then determine $v_1$. For example, $v_2-1$ gives $v_1=-2/3.$ A nicer solution would be $v_2=3$ and $v_1=-2.$ These vectors are different, but they point in the same direction in the $v_1v_2$ plane.\n\nFor $\slambda=-1$, the system becomes\n\sbegin{eqnarray}\n2 v_1-2v_2=0, \s\s\n-3v_1+3v_2 = 0.\n\send{eqnarray}\nWhile these equations do not at first look the same, we can divide out the constants and see that once again we get the same equation, $$v_1=v_2.$$ Picking $v_2=1$, we get $v_1=1$.\n\nIn summary, the solution to our eigenvalue problem is\n$$\slambda=4, \squad {\sbf v}=\sleft(\sbegin{array}{c}\n -2 \s\s\n 3 \s\s\n\send{array}\sright)$$\n$$\slambda=-1, \squad {\sbf v}=\sleft(\sbegin{array}{c}\n 1 \s\s\n 1 \s\s\n\send{array}\sright)$$

Here are some examples of dynamical systems that may be of interest to you. Some of these could later be explored as projects.\n\n[[The Perfect Shuffle]]\nHyperion\nKirkwood Gaps\nFractals and Neurons\nFractal Music\nMandelbrot and Julia Sets\nFractal Image Compression\nTurbulence\nControlling Chaos\nStrange Attractors\n\n

Fractals are defined in terms of self-similar structures. There are many sites devoted to displaying fractal images generated in numerous ways. Exploring some of these sites will lead to a better understanding of what a fractal is and why they are so fascinating.\n!!General Fractal Pages\n*[[Google Images|http://images.google.com/images?q=fractals&hl=en&lr=&sa=X&oi=images&ct=title]]\n*[[The Spanky Fractal Database|http://spanky.triumf.ca/]]\n*[[Self Similarity|http://local.wasp.uwa.edu.au/~pbourke/fractals/selfsimilar/]] from [[An Introduction to Fractals|http://local.wasp.uwa.edu.au/~pbourke/fractals/fracintro/]]\n*[[James Sprott's Fractal Gallery|http://sprott.physics.wisc.edu/fractals.htm]]\n*[[What Are Fractals?|http://www.math.umass.edu/%7Emconnors/fractal/intro.html]]\n*[[Fractal Geometry|http://classes.yale.edu/Fractals/]]\n*More [[Fractal Geometry|http://www.math.vt.edu/people/hoggard/FracGeomReport/FracGeomReport.html]]\n!! Cantor Sets and Other 1D Fractals\n*[[The Cantor Set|http://personal.bgsu.edu/~carother/cantor/Cantor1.html]]\n*Applets from [[Shodor|http://www.shodor.org/]] \n**[[Cantor's Comb |http://www.shodor.org/interactivate/activities/CantorComb/?version=1.5.0_02&browser=MSIE&vendor=Sun_Microsystems_Inc.]]\n**[[Koch's Snowflake|http://www.shodor.org/interactivate/activities/KochSnowflake/]]\n**[[Fractal Dimensions|http://www.shodor.org/interactivate/activities/FractalDimensions/]]\n*[[L System Based Fractals|http://ejad.best.vwh.net/java/fractals/lsystems.shtml]]\n!!Mandelbrot and Julia Sets\n*[[The Mandelbrot and Julia Sets Anatomy |http://www.people.nnov.ru/fractal/MSet/Contents.htm]]\n*[[Benoit Mandelbrot|http://www-history.mcs.st-andrews.ac.uk/~history/Biographies/Mandelbrot.html]]\n*[[Gaston Julia|http://www-history.mcs.st-and.ac.uk/Biographies/Julia.html]]\n*[[Julia Sets|http://en.wikipedia.org/wiki/Julia_set]]\n*[[Mandelbrot Explorer|http://www.softlab.ece.ntua.gr/miscellaneous/mandel/mandel.html]]\n*[[Fractint Homepage|http://spanky.triumf.ca/www/fractint/fractint.html]]\n*[[Julia Set Generator|http://facstaff.unca.edu/mcmcclur/java/Julia/]]\n*[[Xaos Fractal Zoomer|http://xaos.sourceforge.net/index.php]]\n!!Iterated Function Systems\n*[[David Arnold's IFS page| http://online.redwoods.cc.ca.us/instruct/darnold/ifs/]] [[Chaos game|http://online.redwoods.cc.ca.us/instruct/darnold/ifs/chaos.m]]\n*[[Matlab Code for the Fern|http://www-rohan.sdsu.edu/~rcarrete/teaching/M-538/lectures/codes/fractals/fern.m]]\n*[[Classic Iterated Function Systems|http://ecademy.agnesscott.edu/~lriddle/ifs/ifs.html]] and [[IFS construction Kit|http://ecademy.agnesscott.edu/~lriddle/ifskit/]]\n!!Fractal Dimension\n*[[Fractal Dimensions|http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html]]\n*[[Fractal Dimension Calculator|http://local.wasp.uwa.edu.au/~pbourke/fractals/fracdim/]]\n*[[FracTop|http://www.csu.edu.au/faculty/sciagr/eis/fractop/]] - Fractal Dimensions of Images

!!Introductions\n[[Course Web Site|http://people.uncw.edu/hermanr/chaos/]]\n[[A Brief History of Chaos Theory|http://www.schuelers.com/ChaosPsyche/part_1_3.htm]]\n[[Chaos Hypertextbook|http://hypertextbook.com/chaos/]]\n[[Chaos on the Web|http://haides.caltech.edu/~mcc/Chaos_Course/Outline.html]]\n[[Nonlinear Dynamics and Chaos|http://www.deas.harvard.edu/climate/eli/Courses/2004fall/]]\n[[Chaos and Time Series|http://sprott.physics.wisc.edu/phys505/index.htm]]\n[[Chaos and Fractals: Predicting the Unpredictable|http://www.xscite.com/MichaelThompson/_MichaelThompson_CFnew.html]] Video\n[[Iterated Function Systems|http://ecademy.agnesscott.edu/~lriddle/ifs/ifs.html]]\n[[Introduction to Nonlinear Dynamics and Chaos|http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/]] with applets\n[[Chaos - Classical and Quantum|http://chaosbook.org/]]\n[[Nonlinear Dynamics Links|http://archives.math.utk.edu/topics/nonlinearDynamics.html]] at Mathematics Archives\n[[Videos at Texas A&M|http://www.math.tamu.edu/%7Empilant/math614/videos.html]] - A set of videos based on our text.\n!!Nonlinear Programs\n*[[TISEAN Nonlinear Time Series analysis|http://www.mpipks-dresden.mpg.de/~tisean/TISEAN_2.1/index.html]]\n*[[Math Archive|http://archives.math.utk.edu/software/msdos/dynamics/.html]] - not all links work\n*[[Phaser|http://www.phaser.com/]] Update of well known program, not free but 21 day trial period\n*[[Files for Lynch's Dynamical Systems ... with MATLAB|http://www.mathworks.com/matlabcentral/fileexchange/loadAuthor.do?objectId=757442&objectType=author]]\n*[[Visualization of Complex Dynamical Systems|http://www.cg.tuwien.ac.at/research/vis/dynsys/]]\n!!Articles and Research\n*[[Dynamical Systems Portal|http://www.dynamicalsystems.org/]]\n*[[arXiv - Chaotic Dynamics|http://arxiv.org/list/nlin.CD/recent]]\n*[[arXiv - Dynamical Systems|http://arxiv.org/list/math.DS/recent]]\n!!Other\n*[[Nonlinear dynamics and chaos: Lab demonstrations Video|http://dspace.library.cornell.edu/handle/1813/97]]\n*[[Ninety + thirty years of nonlinear dynamics: Less is more and more is different|http://tutorials.siam.org/dsweb/enoc/]]\n*[[Recurrence Plots|http://www.recurrence-plot.tk/]]

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The [[Exploratorium|http://www.exploratorium.edu/]] has a nice short [[description of chaos|http://www.exploratorium.edu/complexity/lexicon/chaos.html]]: \n\n<<<\nThe qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems - by Dr. Stephen H. Kellert, //In the Wake of Chaos//\n<<<\n\nBy the end of this course we will understand more specifically what this definition really means. We will study both discrete and continuous nonlinear systems. Nonlinear systems are generally not solvable and exhibit a vareity of behaviors. Historically, they were either ignored ar approximated by linear systems. But with the advent of computers (and later desktop computers), nonlinear systems have been explored numerically. This uncovered a strange new world and has spurred on more analysis. \n\nOne of the key elements of chaotic behavior in deterministic systems is that of the sensitivity of initial conditions, often exemplified by the butterfly effect.The [[butterfly effect|http://www.pha.jhu.edu/~ldb/seminar/butterfly.html]] is attributed to [[Edward N. Lorenz|http://en.wikipedia.org/wiki/Edward_Lorenz]] using computers to study a [[simple model|http://haides.caltech.edu/~mcc/chaos_new/Lorenz.html]] of the weather. \n\nHowever, the concept of [[sensitive dependence on initial conditions|http://www-chaos.umd.edu/misc/poincare.html]] goes back to the work of [[Poincaré|http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Poincare.html]] and his studies of the stability of the solar system. Later in the course we will discuss the stability of the solar system when we talk about the properties of continuous systems. \n\nWe will begin with discsrete systems. A discrete system can be thought to evolve in discrete time. For example, you can look at the system once a month, year, decade, etc. Thus, the system is described in terms of sequences. One typically studies population dynamics as a fist example. \n\nAn interesting example is that of card shuffling. We will open up the class by showing how the perfect shuffle can be viewed as a dynamical system. You will explore this example later in a homework problem. The introduction will be provided via a [[powerpoint presentation|http://people.uncw.edu/hermanr/Chaos/ShuffleTalk/CardShuffling.ppt]].

Itineraries were presented in one slide of the shuffling example. The slide as shown below summarizes the discussion of itineraries in the course text. One looks at a point //x~~n~~// of a given orbit and determines which half of the unit interval it lives. For example, //1/4// lives in [0 0.5]. This is on the left, so it is denoted by an //L//. On the other hand, //5/6// lives in the other half, denoted by an //R//. The study of labeling of orbits with simple symbols is called ''symbolic dynamics''. We will investigate this further later in the course.\n\nThe itinerary for an orbit is denoted by the string of //L//'s and //R//'s obtained by locating each member of the orbit. For example, we start with //x~~0~~// = 1/3. This number lies in the left half, so the orbit starts with an //L//. The next number //x~~1~~// = //f(x~~0~~)// = 8/9 is in the right half. So, we add an//R// to get //LR//. The next value is //x~~2~~//=32/81. This is in the left half. So, for the orbit {1/3, 8/9, 32/81, ...} we have the itinerary //LRL...//.\n\n[<img[Itineraries|itineraries.jpg]]\n\nWhat is interesting about itineraries is that when one gets repeating symbols, like //LRLRLR...//, then one can identify periodic orbits. In this case the string //LR// repeats. Furthermore, we see that an orbit approaches a periodic orbit if the tail of the itinerary begins to repeat. So, //LLRRLRLRRLRRLRR...// tends to repeat the string //LRR// after many iterations. Thus, it approaches a period three orbit.\n\nWe can determine iterations using Maple of MATLAB. In the software section is a Maple Worksheet which we will use later in the course. The MATLAB code is simple for producing itineraries and isa good example for showing how to form and output strings. The function for doing this is \n{{{\nfunction itstring=logit(x,N)\n%\n% Determine itinerary based on logistic map G = 4x(1-x)\n% Start with x and locate points of orbit in Left or Right half of [0,1]\n%\n\nG=@(z) 4*z.*(1-z);\n\nif x<0.5\n itstring='L';\nelse\n itstring='R';\nend\nfor i=1:N\n x=G(x);\n if x<0.5 \n itstring=[itstring,'L'];\n else\n itstring=[itstring,'R'];\n end\nend\n}}}\n\nCopy the above function, paste it into MATLAB's editor and save as {{{logit.m}}}. Now, try \n\n{{{>>logit(.25,8)}}}\n\nThis produces the itinerary for 0.25 of length 8.\n\nYou can also do this for the tent map. Typing the tent map function is slightly more involved. In the code below a sub function is added to the function. Save it as {{{tentit.m}}} and try it out. \n\n{{{\nfunction itstring=tentit(x,N)\n%\n% Determine itinerary based on the tent map\n% 2x, if x<= 0.5 \n% T(x) = \n% 2(1-x), if x>= 0.5\n%\n% Start with x and locate points of orbit in Left or Right half of [0,1]\n%\n\nif x<0.5\n itstring='L';\nelse\n itstring='R';\nend\nfor i=1:N\n x=tent(x);\n if x<0.5 \n itstring=[itstring,'L'];\n else\n itstring=[itstring,'R'];\n end\nend\n\nfunction y=tent(x)\nif x<=0.5\n y=2*x;\nelse\n y=2*(1-x);\nend\n}}}

A linear map is given by //''f''(x, y)=(ax + by+e, cx + dy+f)//. The discrete system takes the form\n//x~~n+1~~ = a x~~n~~ + b y~~n~~ + e//\n//y~~n+1~~ = c x~~n~~ + d y~~n~~ + f//,\nwhere //a, b, c, d, e, f// are constants. By a simple translation, this system can be transformed into\n//x~~n+1~~ = a x~~n~~ + b y~~n~~//\n//y~~n+1~~ = c x~~n~~ + d y~~n~~//,\nwhich has a fixed point at the origin, (0,0).\n\nThis system can be written in matrix form:\n$$\sleft(\sbegin{array}{c}\n x_{n+1} \s\s\n y_{n+1} \s\s\n\send{array}\sright) = \sleft(\sbegin{array}{cc}\n a & b \s\s\n c & d \s\s\n\send{array}\sright)\sleft(\sbegin{array}{c}\n x_{n} \s\s\n y_{n} \s\s\n\send{array}\sright) ,$$ \nor, ${\sbf x}_{n+1} = A {\sbf x}_n$. Iterating from the initial condition, one can show that ${\sbf x}_{n} = A^n {\sbf x}_0.$ Thus, we need to understand how $A^n$ behaves for large //n//.\n\nFor 2x2 matices, //A// is similar to one of the following matrices: \n$$A_1=\sleft(\sbegin{array}{cc}\n a & 0 \s\s\n 0 & b \s\s\n\send{array}\sright), \squad a\sne b, \squad A_2=\sleft(\sbegin{array}{cc}\n a & 1 \s\s\n 0 & a \s\s\n\send{array}\sright), \squad A_3=\sleft(\sbegin{array}{cc}\n a & -b \s\s\n b & a \s\s\n\send{array}\sright)= r\sleft(\sbegin{array}{cc}\n \scos\stheta & -\ssin\stheta \s\s\n \ssin\stheta & \scos\stheta \s\s\n\send{array}\sright), $$\nwhere $r=\ssqrt{a^2+b^2}$ and $\stan\stheta = \sfrac ba. $\n\nRecall, a matrix //A// is similar to a matrix //B// if there exists an invertible matrix //S// such that //A = S^^-1^^BS//. Then, \n//A^^n^^ = (S^^-1^^BS)^^n^^ = (S^^-1^^BS)(S^^-1^^BS)...(S^^-1^^BS) = S^^-1^^B^^n^^S.// Thus, the large //n// behavior or //A^^n^^// is related to that of //B^^n^^//. \n\nAlso, //A// and //B// have the same eigenvalues. (See the section on [[Eigenvalue Problems]].) This is shown using the eigenvalue equation: $$0=|A-\slambda I| = |S^{-1}BS-\slambda S^{-1}S| = |S^{-1}(B-\slambda I)S| = |S^{-1}||S||B-\slambda I| =|B-\slambda I| $$\n \nReturning to the matrices //A~~1~~//, //A~~2~~//, and //A~~3~~//, we can write out the large //n// behavior. Namely,\n$$A_1^n=\sleft(\sbegin{array}{cc}\n a^n & 0 \s\s\n 0 & b^n \s\s\n\send{array}\sright), \squad A_2^n=a^{n-1}\sleft(\sbegin{array}{cc}\n a & n \s\s\n 0 & a \s\s\n\send{array}\sright), \squad A_3= r^n\sleft(\sbegin{array}{cc}\n \scos n\stheta & -\ssin n\stheta \s\s\n \ssin n\stheta & \scos n\stheta \s\s\n\send{array}\sright).$$\n\nFor //A~~1~~// we have three cases:\n* |//a//|<1, |//b//|<1, a sink or attracting point.\n* |//a//|>1, |//b//|>1, a source or repelling point\n* |//a//|<1, |//b//|>1, or |//a//|>1, |//b//|<1, a saddle point\n\nThe //A~~2~~// case is slightly more complicated. The origin appears to be a source (|//a//|>1) or a sink (|//a//|<1).\n\nFor //A~~3~~// we have three cases:\n* |//r//|=1, we have a center, points rotate in an ellipse about the origin.\n* |//r//|<1, the points spiral in towards the origin, called a stable focus.\n* |//r//|<1, the points spiral in away from the origin, called an unstable focus.\n

[[Course Description]]\n[[Notes]]\n[[General Links]]\n[[Chaos Images]]\n----------------------------------\n[[Examples]]\n[[Software]]\n[[MATLAB Wiki|http://people.uncw.edu/hermanr/wiki/matlabwiki.html]]\n[[Papers]]\n----------------------------------\n[[What are Tiddlers?]]\n[[My Homepage|http://people.uncw.edu/hermanr]]\n-----------------------\n{{small{\n<<search>>\n<<closeAll>>\n<<permaview>>\n<<newTiddler>>\n<<saveChanges>>}}}\n-----------------------\n{{smaller{[[Site created and maintained by\n Dr. R. Herman.|http://people.uncw.edu/herman]]}}}\n

[[Introduction to Chaos]]\n[[1D Maps]]\n[[The Logistic Map]]\n[[Itineraries]]\n[[2D Maps]]\n[[Linear Maps]]\n[[Eigenvalue Problems]]\n[[Chaos]]\n[[Fractals]]\nDifferential Equations\n[[System of ODEs]] - Solution via Eigenvalues\nPopulation Models\nLorenz Attractor\nForced Oscillations\nSynchronization\nControlling Chaos

Over the semester links to papers will be posted here. Students may find interesting topics for projects in this list. \n\n[[Introduction to Chaos in Deterministic Systems|http://arxiv.org/ftp/nlin/papers/0308/0308023.pdf]], Carlos Gershenson, PDF\n[[ArXiv.org|arXiv.org]] search on [[chaos|http://arxiv.org/find/grp_q-bio,grp_cs,grp_physics,grp_nlin,grp_math/1/ti:+chaos/0/1/0/all/0/1]], [[nonlinear dynamics|http://arxiv.org/find/grp_nlin/1/ti:+AND+nonlinear+dynamics/0/1/0/all/0/1]]

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Dr. Russell L. Herman, a mathematical physicist, is a Professor in the [[Department of Mathematics and Statistics|http://www.uncw.edu/math/]] at [[UNC Wilmington|http://www.uncw.edu/]], teaches for the [[Department of Physics and Physical Oceanography|http://www.uncw.edu/phy/]] and has been a Faculty Associate for the [[Center for Teaching Excellence|http://www.uncw.edu/cte/]]. He is ~Editor-in-Chief of [[The Journal of Effective Teachinghttp://people.uncw.edu/hermanr/ET/]] and has recently been honored with [[teaching awards|http://people.uncw.edu/hermanr/teaching.htm]].\n\nHis interests include topics in nonlinear evolution equations, soliton perturbation theory, fluid dynamics, relativity, quantum mechanics, chaos and dynamical systems, signal analysis and investigations into instructional uses of technology in mathematics and science.\n\n''Office:'' Bear Hall 124\n''Phone:'' (910)962-3722\n''Web Page:'' [[http://people.uncw.edu/hermanr/|http://people.uncw.edu/hermanr/teaching.htm]]\n''Email:'' [[hermanr@uncw.edu|hermanr@uncw.edu]]

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a set of class notes

Nonlinear Dynamics and Chaos

Links to useful information on software programs and code for nonlinear systems will be posted here.\n\n[[An exhaustive set of links to fractal software|http://home.att.net/~Paul.N.Lee/Fractal_Software.html]]\n[[Xaos|http://wmi.math.u-szeged.hu/~kovzol/xaos/doku.php]]\n[[Fractint|http://spanky.triumf.ca/www/fractint/fractint.html]]\n[[Exploring Fractals|http://classes.yale.edu/fractals/Software/Software.html]] A suite of Java applets\n[[Iterated Function System Applet|http://www.cut-the-knot.org/Curriculum/Geometry/ifs.shtml]]\n\n[[The Quadratic Map|http://haides.caltech.edu/~mcc/chaos_new/Scalemap.html]]\n[[Cobweb Plot Applet|http://www.emporia.edu/math-cs/yanikjoe/Chaos/CobwebPlot.htm]]\n[[Nonlinear Web|http://math.bu.edu/DYSYS/applets/nonlinear-web.html]]\n[[Period Doubling|http://www.lboro.ac.uk/departments/ma/gallery/doubling/]]\n[[Bifurcation Diagram for Logistic Map|http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/BifArea/]]\n\nA ''MATLAB'' wiki is under construction as [[matlabwiki|http://people.uncw.edu/hermanr/wiki/matlabwiki.html]]\n\n''Maple Code:''\n*Iteration Worksheet [[IterMap.mws|IterMap.mws]]\n*Testing Round Off [[IterDigits.mws|IterDigits.mws]]\n*[[Iteration|http://people.uncw.edu/hermanr/Chaos/QuickIter/]], \n*[[Binary and Itinerary Code|http://people.uncw.edu/hermanr/Chaos/Itinerary/]] and the [[Maple File|http://people.uncw.edu/hermanr/Chaos/Itinerary/Itinerary.mws]]

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The simplest example is a system of linear differential equations of the form\n\sbegin{eqnarray}\n\sfrac{dx}{dt} &=& ax+by \s\s\n\sfrac{dy}{dt} &=& cx+dy.\n\send{eqnarray}\nWe note that this system is coupled. We cannot solve either equation without knowing either $x(t)$ or $y(t).$ A much easier\nproblem would be to solve an uncoupled system like\n\sbegin{eqnarray}\n\sfrac{dx}{dt} &=& \slambda_1x \s\s\n\sfrac{dy}{dt} &=& \slambda_2y.\n\send{eqnarray} The solutions are quickly found to be\n\sbegin{eqnarray}\nx(t) &=& c_1 e^{\slambda_1t}, \s\s\ny(t) &=& c_2 e^{\slambda_2t}.\n\send{eqnarray}\nHere $c_1$ and $c_2$ are two arbitrary constants.\n\nWe can determine particular solutions of the system by specifying $x(t_0)=x_0$ and $y(t_0)=y_0$ at some time $t_0.$ Thus,\n\sbegin{eqnarray}\nx(t) &=& x_0 e^{\slambda_1t}, \s\s\ny(t) &=& y_0 e^{\slambda_2t}.\n\send{eqnarray}\n\nWouldn't it be nice if we could transform the more general system into one that is not coupled? Let's write our systems in more general form. We write the coupled system as $$\sfrac{d}{dt}{\sbf x} = A{\sbf x}$$ and the uncoupled system as $$\sfrac{d}{dt}{\sbf y} = \sLambda{\sbf y}$$, where $$\sLambda = \sleft(\sbegin{array}{cc}\n \slambda_1 & 0 \s\s\n 0 & \slambda_2 \s\s\n\send{array}\sright).$$ We note that $\sLambda$ is a diagonal matrix.\n\nNow, we seek a transformation between ${\sbf x}$ and ${\sbf y}$ that will transform the coupled system into the uncoupled system. Thus,\nwe define the transformation\n\sbegin{equation}\n{\sbf x} = S{\sbf y}.\n\send{equation} Inserting this transformation into the coupled\nsystem we have\n\sbegin{eqnarray}\n\sfrac{d}{dt}{\sbf x} &=& A{\sbf x} \s\s\n\sfrac{d}{dt}S{\sbf y} &=& AS{\sbf y} \s\s\nS\sfrac{d}{dt}{\sbf y} &=& AS{\sbf y}.\n\send{eqnarray}\n\nMultiply both sides by $S^{-1}$. [We can do this if we are dealing with an invertible transformation; i.e., a transformation in which\nwe can get ${\sbf y}$ from ${\sbf x}$ as ${\sbf y} = S^{-1}{\sbf x}$.] We obtain $$\sfrac{d}{dt}{\sbf y} = S^{-1}AS{\sbf y}.$$ Noting that\n$$\sfrac{d}{dt}{\sbf y} = \sLambda{\sbf y},$$ we have\n\sbegin{equation}\n\sLambda =S^{-1}AS.\n\send{equation}\n\nThe expression $S^{-1}AS$ is called a similarity transformation of matrix $A$. So, in order to uncouple the system, we seek a\nsimilarity transformation that results in a diagonal matrix. This process is called the diagonalization of matrix $A$. We do not\nknow $S$, nor do we know $\sLambda$. We can rewrite this equation as $$AS =S\sLambda. $$ We can solve this equation if $S$ is real symmetric, i.e, $S^T=S$. [In the case of complex matrices, we need the matrix to be Hermitian, $\sbar{S}^T=S$ where the bar denotes complex conjugation.] So, we know that $$AS = S^T \sLambda = (\sLambda S)^T. $$\n\nOnce can thus show that the columns of $S$ (denoted ${\sbf v}$) satisfy an equation of the form\n\sbegin{equation}\nA{\sbf v} = \slambda {\sbf v}.\n\send{equation} This is an equation for vectors ${\sbf v}$ and numbers $\slambda$ given matrix $A$. It is called an eigenvalue problem. The vectors are called eigenvectors and the numbers are called eigenvalues. In principle, we can solve the eigenvalue problem and this will lead us to solutions of the uncoupled system of differential equations. See [[Eigenvalue Problems]]

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The logistic map is a standard example for introducing periodic orbits, bifurcations and displaying the onset of chaos in one dimensional maps. It was introduced at the end of [[1D Maps]]. Some of the behavior is shown in some MATLAB examples. See [[Discrete Dynamics I|http://http://people.uncw.edu/hermanr/wiki/matlabwiki.html#%5B%5BDiscrete%20Dynamics%20I%5D%5D]] in the [[MATLAB Wiki|http://people.uncw.edu/hermanr/wiki/matlabwiki.html]]. On that page you will see how to use MATLAB to determine fixed points and periodic orbits.\n\nYou can also use Maple to look at orbits. However, one needs to be careful because Maple prefers to do things symbollically and can easily be brought to a halt when doing too many iterations. Some code is provided [[here|http://people.uncw.edu/hermanr/Chaos/QuickIter/]] and under the [[Software]] section of this site.\n\nThe logistic map is a simple quadratic map given by //f(x)=rx(1-x)// for //x// in [0,1] and //r// in (0,4]. As //r// is varied one sees a variety of behaviors in the orbits. As we saw in [[1D Maps]], there is one fixed point, the origin, for //r<1// and two fixed points, the origin and //r = 1 -1/r//, for $1\sle r \sle 3$. \n\nBelow are some plots for different valuesof //r//.\n\n|! //r = 0.5// |! r = 2.0 |\n|[<img[Orbit for r=0.5|logisticr05.jpg]]|[>img[Orbit for r=2.0|logisticr2.jpg]]|\n|! //r = 3.1// |! r = 3.5 |\n|[<img[Orbit for r=3.1|logisticr31.jpg]]|[>img[Orbit for r=3.5|logisticr35.jpg]]|\n|! //r = 3.56// |! r =4.0 |\n|[<img[Orbit for r=3.56|logisticr356.jpg]]|[>img[Orbit for r=4.0|logisticr4.jpg]]|\n\nWe see that as //r// is increased the points asymptotically tend to either the fixed points or periodic orbits. However, for //r = 4// the points seem scattered all over. What happens is that the origin becomes unstable at //r = 1.0// and a new fixed point is born. As //r// is increased, this fixed point becomes unstable and a period 2 orbit is born. This becomes unstable and we get a period 4. Then we get a period 8. This is called ''period doubling''. There is a rich set of behavior as //r// is increased. \n\nThis is captured by a bifurcation diagram. We plot the asymptotic values of the orbit vs //r//. The result is the bifurcation diagram below. Drawing a vertical line at a specific //r// value helps to determine the orbit behavior. For //r = 4// all periodic orbits are unstable and thus we get the above set of scattered points in the orbits.\n\n[<img[The Bifurcation Diagram|bifur.jpg]]\n\n

The riffle, or faro, shuffle is a traditional method of shuffling. The deck is divided in half and one riffles the cards: the two halves of the deck are interleaved together. With practice one can sometimes obtain a perfect shuffle with every other card coming from alternating hands.\n\nLinks:\n\n[[The Magic of Perfect Shuffles|http://www.maa.org/mathland/mathtrek_8_3_98.html]]\n[[Card Shuffling|http://www.flatrock.org.nz/topics/art_of_playing_cards/how_to_win_at_poker.htm]]\n[[A Perfect Shuffle Applet|http://www.math.fau.edu/richman/Liberal/shuffle.htm]]\n[[How to do a perfect shuffle|http://www.virtualmagicshow.com/stuff/]] - Video

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