CHAOS -SOME BASIC CONCEPTS

CHAOS -SOME BASIC CONCEPTS Anders Ekberg INTRODUCTION This report is my exam of the "Chaos-part" of the course STOCHASTIC VIBRATIONS. I m by no means any expert in the area and may well have misunderstood some concepts. So read the following with a supsiciuous mind. This report was finished in June 1997. 1. WHEN IS CHAOS POSSIBLE? Chaos is characterized by the way a dynamic system doesn t repeat itself, even though the system is governed by deterministic equations. This is due to the fact that an initial disturbance grows exponentially (as compared to the case of non-chaotic systems, where an initial disturbance grows linearly). Two conditions that must be fulfilled, if the system is to behave chaotic, are The system has to have at least three indepent dynamic variables The equations of motion must include a non-linear term that couples several of the variables. These conditions are necessary but not sufficient if the dynamic system is to behave chaotic. Since the governing differential equations are non-linear, it is often necessary to solve them numerically. 1

2 (10) 2. PHASE SPACE The phase space is a mathematical space. The orthogonal coordinates of this space each represents one indepent variable of the dynamical system. The phase space is just a tool to visualize the behavior of a dynamical system and does not provide any additional information about the system. As an example, one can define a phase space as a plane with angular velocity on one axis and angle on the other axis, and then plot a curve by expressing the angular velocity and the angle as a function of time (which is then the parameter of the curve). The resulting curve (in the 2D-case) is called a trajectory and is valid for a given set of indata. Since a chaotic system has to have at least three indepent variables, it should be mapped in, at least, a three dimensional space. Some interesting features of a phase space diagram is that, for a deterministic system, two trajectories will not cross. Also, for a conservative system (where the energy of the system is constant in time), the points found in an area of the phase space move in such a way that, at a later time, the area occupied by these points remains the same. For a dissipative system, this area will decrease. To examine whether the system is conservative or dissipative, one can evaluate the expression F, where F is contains the time derivative of the dynamic variables. If this expression equals zero, the system is conservative. If it is less than zero, the system is dissipative. As an example, a driven pulum is governed by the equation 2 d θ 1dθ + -- + sinθ = gcos( ω d q d t D t) t 2 (1) which can be rewritten as dω dt 1ω = ------ sinθ + gcosφ q dθ = ω dt dφ = dt ω D (2) and the dissipation inequality becomes F d d d ω 1 1 = --- sinθ + g cosφ ωω dω dθ dφ q D = -- + 0 + 0 = -- q q Thus, this system is dissipative. (3)

3. THE POINCARÉ-SECTION A Poincaré-section is obtained from the phase space diagram by viewing this stroboscopically, i.e. sampling points in the phase space at regular intervals. The Poincaré section can give information about the relationship between the strobe frequency and the natural frequency of the dynamic system. For example, if these are the same, the Poincaré-section would be a single point if the Poincaré-section is projected from a 3D-phase space onto a phase plane. If the system has a direct current, in addition to oscillating movements, there may be a nonzero average displacement and the offset will generally be asymmetric. If the relation between the strobe frequency and the natural frequency is irrational, the strobe points will never repeat and the result will be a filled pattern (e.g., a curve). if the system is dissipative, the points of the Poincaré-section will move towards the attractor. The main feature of the Poincaré-section (and also of projections) is its ability to decrease the amount of information in a systematic manner, thus bringing some order in chaos. Superposing Poincaré-sections for all possible values of the strobe frequency will bring back the full phase space diagram. 3 (10) 4. MAPPING Consider a difference equation. This is an equation that analyses the conditions at discrete instants and defines the conditions at time n+1 as a function of the conditions at time n, i.e. x n + 1 = f( x n ). If x is a real number on the interval (x min, x max ), The function f, is then called a map of the interval (x min, x max ) onto itself, since it generates a value in this interval for another given value in the interval. A dynamical system whose phase space is three-dimensional can, at least in theory, be converted to a mapping through the Poincaré-section since the Poincaré-section defines the conditions at given intervals and the differential equations that govern the motion at these intervals are deterministic. Maps are much easier to analyze than differential equations. Thus, they are very useful to investigate sensitivity to initial conditions, mechanisms of bifurcation etc. 4.1. LOGISTIC MAP The logistic map is defined as + = µx n ( 1 x n ) x n [ 01, ] x n 1 (4) This map (or rather, the corresponding differential equation, was originally used to model population growth. The interesting feature of this map is that, although the governing difference equation is

4 (10) very simple, it will, for appropriate values of µ, predict a chaotic behavior. The two files logistic.m and logbif.m are MATLAB-files that visualize the behavior of the logistic map. The file logistic.m, plots the values of x n vs. x n+1. By varying µ, it is found that, for values below approximately 3, the plot will converge to one point. For values above 3, the plot will oscillate between two (or more) values. In Fig. 1 a plot is shown for µ=3.7 which results in a chaotic motion, i.e. the plot will oscillate between infinitely many values. Fig. 1 Logistic map a) Plot of x n vs. x n+1 for µ=3.7 b) Bifurcation diagram, plot of µ vs. x n. The file logbif.m plots a bifurcation diagram for the logistic map. In this diagram, µ is varied and all possible values of x n (after initial transients have faded out), for the chosen value of µ, are plotted. In the bifurcation diagram, note that chaos is reached through a sequence of periodic doublings (i.e. x n can have 1 value, then 2, then 4, 8,...). This equals an oscillation between 1, 2, 4, 8,... values in Fig. 1. The ratio of spacings between consecutive values of bifurcations approaches a constant, which is called the Feigenbaum number and is defined as µ k µ lim ------------------------ k 1 = δ = 4.6692016 k µ k + 1 µ k This is a universal property of the periodic doubling route to chaos for maps that have a quadratic maximum (which [probably] means that the function has a quadratic term and a maximum). Periodic windows can also occur in the bifurcation diagram. In these windows, the map regains its periodicity. For instance, there is a periodic window at µ=3.83. As µ increases, a periodic doubling sets in and the map will return to a chaotic behavior. At the onset of a periodic window, the map can show a nearly periodic motion which is disturbed by occasional irregular bursts. This is called a type I intermittency. (5) 4.2. CIRCLE MAP A circle map is used as an approximation of cyclic motions (such as a driven pulum). A

5 (10) specific type of the circle map, is the standard map, which is defined as + = θ n + Ω ----- K sin( 2 πθ) 2π θ n 1 (6) This is a map consisting of 2 parameters, (θ,k). Note that the map has to be used with periodic boundary conditions, for instance in the interval [0,1] (i.e. 0.25=1.25) if it is to approximate cyclic motions. The file circle.m evaluates the circle map for given values of Ω and K. In Fig. 2, a plot is made to show resulting values for two different sets of values of Ω and K. Note that for the first case, where K=0, the result is oscillating between 5 values. In the second case, the pattern repeats itself after many cycles. In the third case, Ω is an irrational number and the pattern does not repeat itself (infinitely many points). However, for some values of K, phaselocking can occur and the pattern can repeat itself even though Ω is irrational. This can be seen in the fourth plot. Fig. 2 The circle map. a) Resulting values for Ω=0.4 and K=0 b) Resulting values for Ω=0.403 and K=1.2 c) For Ω = 2 and K=1 d) r Ω = 2 and K=1.2 4.3. HORSESHOE MAP The horseshoe map is a little different map. It acts on a unit square and consists of three steps. An expansion in the y-direction by a factor m>2 A contraction in the x-direction by a factor λ ( 01, 2).

6 (10) A folding of the stretched and contracted surface Parts of the mapped square outside the original unit square are trimmed. The horseshoe map is graphically shown in Fig. 3. Fig. 3 The horseshoe map The main feature of the horseshoe map is its ability to simulate configurations in phase space, where there are regions of strong contraction and expansion. 5. ATTRACTORS An attractor is a point (or a closed curve or a surface), in the phase space diagram, towards which the graph for a given set of initial coordinates will converge. An example is the point ω=θ=0 (i.e. no motion) towards which a damped pulum, with no applied loads, will converge in due time. As an other example, one can consider a ligthly driven pulum with damping. This pulum will (after steady-state has been reached) follow an elliptic path in phase space. Thus, in phase space, the attractor is a curve (with dimension 2). If this pulum is studied by the use of a Poincaré-section, the result will be a point. Thus, the attractor in the Poincarésection is a point (with dimension 0). The same result will be obtained for a non-driven (after initial input) and undamped pulum. In analyzing the behavior of the system near these attractors, one can make a linear approximation of the governing differential equations. The solution to this linear equations will show the behavior of the system. For instance, if the solutions are complex values with negative real parts, the solution will be stable and the curve will spiral inwards towards a focus. 5.1. STRANGE ATTRACTOR For chaotic motion, an attractor is more complex. It consists of a fractal with dimension greater than two in phase space. Such attractors are called strange attractors. For instance, the strange attractor for a chaotic motion of a driven pulum, consists of closely spaced sheets. This attractor has a dimension between 2 and 3. (If it had been just one sheet, the dimension had been 2 and if it would have occupied the entire volume (of some part of the

7 (10) phase space) the dimension would have been 3). A fractal is a structure that consists of an infinite number of infinitely small points (or layers). 5.2. BASIN OF ATTRACTION A basin of attraction is the region in phase space that contains all initial conditions on the dynamic variables which will lead to a convergence towards a specific attractor. A border between two basins of attractions is called a separatrix. Such a boundary is a fractal, i.e. the basins are interwoven near the boundaries and a small disturbance in the initial conditions can result in an exchange of attractor for the point studied. 6. AUTONOMOUS SYSTEM Autonomous systems are systems that are governed by equations that are not explicitly time-depent. This means that the resulting curves in phase space don not dep on absolute time even though time can be a variable in the governing differential equations. As an example, stream lines in steady flow are not time depent even though the differential equations which governs the fluid flow are time depent. Trajectories of autonomous systems do not cross. Often, you can transform non-autonomous systems to autonomous by introducing variables which are not explicitly time depent. 7. BIFURCATION DIAGRAM A bifurcation is the change in numbers of solutions to a differential equation as one variable is varied (while all other variables are kept fixed). Thus, for some values of a variable, there may exist only one possible solution (in terms of velocity, etc.), while, for other values of this variable, there may exist two or more possible solutions (after a long time ). Which of these solutions that will actually exist, is depent on the initial conditions or the motion can be cycling between the different values. For certain values of the parameter varied, there are infinitely many possible solutions. In these cases, the motion is chaotic. The strength of a bifurcation diagram is its possibility to visualize parameter regimes for a dynamical system where chaos (or harmonic motion) can occur. An example of a bifurcation diagram is shown in Fig. 1b.

8 (10) 8. THE LYAPUNOV EXPONENT The Lyapunov exponent λ, gives a measure of the sensitive depence upon initial conditions for a map (or a dynamical system). It is defined by the following relation. Consider a system that is evolving from two initial states x and x+ε, where ε is a small disturbance After n steps of the mapping from these two initial states, the divergence of the resulting values may be expressed as ε( n) εe λn (7) Note that the divergence must be taken as an average for many x and ε:s. The Lyapunov exponent λ, will give a measure of the average rate of divergence in the sense that if λ is negative, slightly different initial conditions will t to the same resulting values, the evolution is not chaotic. If λ is positive, evolutions from slightly different initial conditions will t to separate, the motion is chaotic. It can be shown that the Lyapunov exponent can be calculated as λ = n 1 1 lim -- f' ( x n ln i ) n i = 0 (8) APPENDIX Listing of MATLAB-programs that are used to generate the plots in this report. Note that the codes are neither optimized or quality assured in a proper manner (i.e. they are quick hacks ). Please feel free to use them in any manner you wish, but bear in mind that I take no responsibility for the code. Also, a reference to the source would be appreciated. LOGISTIC.M % MATLAB-program to evaluate a logistic map defined by: % x(i+1)=my*x(i)*(1-x(i)) % Create the two vectors xn and xnn, where x(i) and x(i+1) % are stored respectively xn=zeros(1,100) xnn=zeros(1,100) % Give an initial value for xn(1) and my

9 (10) xn(1)=0.1 my=3.7 % Now, start looping for i=1:99 xnn(i)=my*xn(i)*(1-xn(i)) xn(i+1)=xnn(i) xnn(100)=my*xn(100)*(1-xn(100)) plot(xn,xnn,'o') xlabel('xn') ylabel('xnn') grid LOGBIF.M % MATLAB-program to evaluate a logistic map defined by: % x(i+1)=my*x(i)*(1-x(i)) % and create a bifurcation diagram % Create matrix where xn-numbers for the % different values of my are stored. bif=zeros(1101,470); % Loop through the interesting values of my my=2.9:0.001:4; % Create the two vectors xn and xnn, where x(i) and x(i+1) % are stored respectively xn=zeros(1,100); xnn=zeros(1,100); for j=1:1101 % Give an initial value for xn(1) and my xn(1)=0.1; % Now, start looping for i=1:499 xnn(i)=my(j)*xn(i)*(1-xn(i)); xn(i+1)=xnn(i); % Store values when transient, initial parts are excluded if i>30 bif(j,i-30)=xnn(i); xnn(500)=my(j)*xn(500)*(1-xn(500)); bif(j,470)=xnn(500); % plot(xn,xnn,'o')

10 (10) % xlabel('xn') % ylabel('xnn') % grid plot(my,bif,'.') CIRCLE.M % MATLAB-program to evaluate a circle map defined by: % x(i+1)=x(i)+o-k/2/pi*sin(2*pi*x(i)) % Create the two vectors xn and xnn, where x(i) and x(i+1) % are stored respectively xn=zeros(1,10000); xnn=zeros(1,10000); % Give an initial value for xn(1) and my xn(1)=0.3; o=sqrt(2); k=1.2; % Now, start looping for i=1:9999 xnn(i)=xn(i)+o-k/2/pi*sin(2*pi*xn(i)); % Apply periodic boundary conditions l=1; while xnn(i)>l l=l+1; xnn(i)=xnn(i)-l+1; xn(i+1)=xnn(i); xnn(10000)=xn(10000)+o-k/2/pi*sin(2*pi*xn(10000)); % Check periodic boundary conditions l=1; while xnn(10000)>l l=l+1; xnn(10000)=xnn(10000)-l+1; plot(xn,xnn,'o') xlabel('xn') ylabel('xnn') grid