dynamical system动力学系统

IX: Introduction to the theory of dynamical systems, stability and bifurcationsParts:1. Introduction.2. Diskrete dynamical systems.3. The Lyapunov exponent.4. Julia and Mandelbrot sets.5. Continuous dynamical systems.6. Some introductory examples.7. Classification of critical points.8. The general solution of a linear system.9. Classification of equilibrium points in specific systems.10. Exercises.1. Introduction.A dynamical system is a phenomenon that changes with tim, for instance the position of a pendulum, the weather, the amount of predators and prey in a lake, et cetera. The traditional way of describing a dynamical system is to use a linear system of differential equations. In this case we have a pretty simple theory to solve the problem (see for instace part 8).A more realistic model however often leads to nonlinear systems of differential equations. In this case it is much more complicated to describe the behavios in the long run, but with help of computers and existing theories we can sometimes obtain the solution as an attractor to the system. In many other cases we will instead get bifurcations or chaos. Chaos means that it is hard (or impossible) to determine the long term behavior; small changes in indata gives dramatic changes in the long term behavior. Some attractors can be described as fractals, some particular self similar sets (a small part of the set has the same structure as the whole set). Such attractors are sometimes called strange attractors.Remark: This fascinating and important part of mathematics is still under rapid development and we can expect many more fundamental discoveries in the future from this research.Remark: Two important concepts that we will study are stability and bifurcations.Example: A simple example of attractor that can be illustrated with a pocket calculator is Kaprekar's constant. Pick a four-digit number p1 where not all four digits are the same. Order the digits in descending order, call this number p1'. In the same way we construct the number p1'', but now with ascending digits. Now construct the number p2=p1'-p1''. Repeat the procedure. Whatever number we are starting with, we will allways end up with 6174and then stay there. The number 6174is an attractor. We illustrate the procedure below:p1=2873gives p1'=8732and p1''=2378. Then p2=p1'-p1''=6354. We proceed according top2'=6543, p2'=3456, p3=p2'-p2''=3087.p3'=8730, p3'=0378, p4=p3'-p3''=8352.p4'=8532, p4'=2358, p5=p4'-p4''=6174.p5'=7641, p5'=1467, p6=p5'-p5''=6174.We realize that we are stuck with 6174. It can be proved that at most seven iterations are needed irrespective of starting number. On this page, the routine is described for numbers with more or fewer digits than four.2. Discrete dynamicalsystems.Letwhere V is some set, be a continuous function. If x0 is a given starting point, we define the orbit: x0, x1, x2,... of the system bythat is, x n is the state of the dynamical system at time step n.Main problem: What can we say about x n for large values of n?We have chaos if small changes in x0 implies large (or unpredictable) changes in x n.An attractor is a stable state for the dynamical system for large values of n.Example 1:Verhulst's population model. Letbe a parameter and defineThis is a useful description of the growth of a population. The orbit x0, x1, x2,...is obtained in the following way: Assume that we have a starting value x0 year 0. Then computewhere x n is the population year n.As we see above, different values of r give very different behavior of the behavior of the orbit (try yourself with a calculator).We see that the orbit converges to an attractor in the cases where we do not have chaos. Describe the attractor in the different cases!It can be shown that the following holds: There is a sequence of parameters r1,r2,r3,... such that* if 0<r<r1=1, the sequence {x n} converges to the attractor 0.* if 1=r1<r<r2=3, the sequence {x n} converges to a (constant) attractor 1-1/r.* if 3=r2<r<r3 the sequence {x n} converges to a periodic state with two different values.* if r3<r<r4the sequence {x n} converges to a periodic state with four different values.* if r k<r<r k+1 the sequence {x n} converges to a periodic state with 2k-1 different values.* if r>r* we have chaos. The long term behavior is not periodic and the behavior of the orbit extremely depends on the starting value x0. This is not the case for r<r*. The value r*=3.569945672...For more information about this behavior, see this page. The behavior above r2=3is called period doubling and can be illustrated in a so called bifurcation diagram.We can now define Feigenbaum's constant, as the limit of the quotientThe interesting about this constant is that it also appears for several other functions f r(x) and thus also other values of r k!3. The Lyapunov exponent.We define f n(x) for a given function f(x) and a given starting value x0 byBy the chain rule we then have thatBy taking logarithms and dividing by n on both sides we getThis convergence can be proved by using a particular mathematical technique. The limit is called the Lyapunov exponent of the dynamical system.Now suppose that we have two (close) starting values x0 and y0. By using the mean value theorem we see thatsinceandThe conclusion is thata) if we have thatthat is, the system is not sensitive to the starting value.b) if thendiverges to infinity with exponential growth.4. Julia and Mandelbrotsets.Let z be a complex number and consider the functionfor a given parameter c. We choose a starting value z0and investigate the orbiti) The case c=0. We have three choices:This is no fractal.ii)The case . We now define the set J c={z0: z n is inside a fix boundary in the plane}. This boundary is called a Julia set and is an example of a fractal.For different starting points the orbits will either stay inside a bounded region or diverge to infinity. Those starting values that have bounded orbits are usually colored black while those diverging to infinity are colored in other colors depending on the speed with which they diverge. The results will be figures as those shown below. Note that the sets consists of the differentstarting values z0 for a given value of c. This implies that there are infinitely many different Julia sets.It can be shown that there is only two main types of Julia sets. Either the area inside boundary is connected or it is broken into infinitely many parts forming a cloud of points, with a fascinating fractal structure. The latter type is usually called a (general) Cantor set.The Mandelbrot set is the set of all c such that the Julia sets are connected. This means that we can see the Mandelbrot set as a map over all Julia sets. The set is illustrated below with the values c for the above Julia sets.Note that the Mandelbrot set is a set of values of c, in contrast to the Julia sets. The Mandelbrot set has a detailed structure on all scales. If we zoom in to look closer we see small copies of the Mandelbrot set everywhere. We also see that the set is everywhere connected. This self similar structure is typical for fractals. See the figure below.5. Continuous dynamicalsystems.In the most general case we will study systems of the typewhereis an element in a vector space for every fix choice of t.In this course we will mainly study systems on the formwhere x=x(t)and y=y(t)are regular real valued functions. In this caseThe solutions are defined on some intervaland can be drawn in the state space.Alternatively they can be represented as a parametric curve in the xy-plane, also known as the phase plane.The arrows in the figure show how the system is developed in time. HereThe equilibrium points are obtained by solving the systemSincewe obtainA phase portrait of the system is all orbits and equilibrium points in the phase plane. A quick way to get a phase portrait is to:1) Find all equilibrium points by solving the system2) Let a standard software (e.g. MATLAB) plot the solutions ofIn part 9 we describe a more careful way to create phase portraits of linear dynamical systems, which we can get as approximations of general nonlinear systems by a first order taylor expansion. We give several concrete examples of continuous dynamical systems in the next part.6. Some introductoryexamples.Example 2: Consider the systemIf we differentiate the first equation on time we getthat inserted into the second implies thatThis differential equation has the solutionfor some constants a and b. Since y is the derivative of x we haveFinally we note thatwhere c is a constant. The only equilibrium point obviously is (0,0). The phase portrait looks like this:Remark: The relation x2+y2=c2 may also be obtained in the following way:whereExample 3: Consider the systemThis decoupled system has the solutionwhere c1 and c2 are two arbitrary constants. To draw the phase portrait we solve the equationThe phase portrait thus becomes:Example 4: Every second order differential equationcan be written as a system of first order differentialequationsExample 5:The equation of the pendulum (see e.g. chapter 7)can be written as the systemWe get the equilibrium points whenthat is, they areHereare stable equilibrium points (colored green in the phase portrait) andunstable equilibrium points (colored red in the phase portrait).Example 6:Van der Pols equationcan be written aswhere is a (small) parameter. Here we have an (unstable) equilibrium point (0,0)(red color) and if we look at the phase portrait we see that we also have a limit cycle.This limit cycle (green color) is stable (all solutions in the neighborhood will be attracted to the limit cycle). We have plotted the phase portrait below for .Example 7: Predators and prey, Volterra's model. There are two different species of fish in a lake: A(prey), that feeds off sea weeds (which is abundant) and B (predator), that feeds off the prey A.The more prey A we have, the higher growth of A since the sea weed can be supposed to be enough to feed all. On the other hand, the growth of A is limited by the number ofencounters between A and B, since they lead to that A being eaten. The more predators B, the lower growth of B since then we will have bigger concurrence about the prey A. On the other hand, the growth increases with the number of encounters between A and B, since B gets food this way. If conclude the above reasoning, we get Volterra's modelwhere a, b, c, d are positive constants, x=x(t) the population of A and y=y(t) the population of B at time t.We get the equilibrium points by solving the systemthat isWe have two possibilities for both equations to become zero. Either we have (x,y)=(0,0) or we have(x,y)=(b/d,a/c). If we draw the phase portrait we see thatthe first point is unstable while the second is stable.This model is useful to explain many well known biological phenomena when we have some kind of competition between different animals, for instance foxes and rabbits.7. Classification ofcritical points.An equilibrium point to a systemi said to be isolated if there is a neighborhood to the critical point that does not any other critical points. There are four different types of isolated critical points that usually occur. They are center, node, saddle point and spiral.An equilibrium point can be stable, asymptotical stabl e or unstable. A point is stable if the orbit of the system is inside a bounded neighborhood to the point for all times t after some t0. A point is aymptotical stable if it is stable and the orbit approaches the critical point as . If a critical point is not stable then it is unstable. In the figure above we see that a center is stable but not asymptotically stable, that a saddle point is unstable, that a node is either asymptotically stable (sink) or unstable (source) and that a spiral either is asymptotically stable or unstable.In certain nonlinear systems we might also have "mixtures" of the above types for higher order critical points. See the example above.8. The general solution of alinear system.Consider the linear dynamical systemthat isWe look for solutions on the formIf we instert these to the system we getthat isThis equation system has nontrivial solutions if and only ifthat is, if and only ifThis is the characteristic equation of the system and the solutions its eigenvalues. If we now for each eigenvalues solve the equation systemwe obtain the eigenvectorsThen we have the general solution of our original dynamical system asthat iswhere C1 and C2 are arbitrary constants.Example 8: Assume that the matrix A isThis matrix has characteristic equationwith the eigenvaluesand corresponding eigenvectorsThis means that the general solution of the corresponding dynamical system isthat isExample 9: Solve the systemSolution: We consider the matrixand its characteristic equationthat isThe eigenvalues thus arewith corresponding eigenvectorsThis means that the dynamical system has the general solutionthat isThese are all complex solutions. We are actually only interested in the real solutions. With help of Euler's formula we getIf we now pick arbitrary real constants D1 and D2 and putwe get the general real solution9. Classification ofequilibrium points inspecific systems.Again consider the systemThe equilibrium point obviously is (0,0). Let and be the eigenvalues to the matrix A. We have the following different possibilities:1) IfLetbe the eigenvectors belonging to the corresponding eigenvalues and . Then the general solution of the system above isWe again have a number of possibilities:i)c2=0, c1>0: x(t)is a curve along l1+(away from origo for increasing t).ii)c2=0, c1<0: x(t)is a curve along l1-(away from origo for increasing t).iii)c1=0, c2>0: x(t)is a curve along l2+(away from origo for increasing t).iv)c1=0, c2<0: x(t)is a curve along l2-(away from origo for increasing t).v) neither c1 nor c2 are zero. Then forwe havehencefor large (in absolute value) negative t. In particular we have thatWhenwe havefor large positive t. Hence x(t)diverges to infinity with a slope asympotically v2as t goes to (positive) infinity. This means that (0,0) is an unstable node. The lines defined by the eigenvectors v1 and v2 are calledseparatrices. The behavior is shown below in the phase portrait.2) IfIn the same way as above we realize that (0,0)is a stable node. The phase portrait looks the same but with reversed arrows.3) IfWe have the following possibilities:i)c2=0, c1>0: x(t) is a curve along l1+ (towards origo for increasing t).ii)c2=0, c1<0: x(t) is a curve along l1- (towards origo for increasing t).iii)c1=0, c2>0: x(t)is a curve along l2+(away from origo for increasing t).iv)c1=0, c2<0: x(t)is a curve along l2-(away from origo for increasing t).We have a saddle point. See the phase portrait below.4) IfWe have two cases:a) We have two linearly independent eigenvectorsThen we can write the solutionwhere a1and a2are arbitrary. This means that every curve is a halfline towards origo. See the phase portrait below.b) We only have one eigenvectorThen the solution on the formwhere the vectorsatisfiesFor large t this means that the solution isIn this case the phase portrait will look like this.Both these cases are examples of a stable node.5) IfThis case is analogous to case 4)above, but with reversed arrows; (0,0) is an unstable node.6) IfBy using similar arguments as in Example 9we realize that the real solutions areThree cases:i) : Then x(t) and y(t) are periodic with periodThe point (0,0) is a center.ii) : The amplitude of x decreases and we thus have a stable spiral.iii) : The amplitude of x increases and we thus have an unstable spiral.Example 10: Consider the systemThe coefficient matrixhas the characteristic equationand thus the eigenvaluesThis corresponds to case 3) above and we conclude that (0,0) is an unstable saddle point. The eigenvectors are obtained by solving the linear equation system:Eigenvalue 1:implies thatthat is, the corresponding eigenvector isEigenvalue 2:implies thatthat is, the corresponding eigenvector isThe general solution of the system thus isthat isThe eigenvectors define the directions of the separatrices.Example 11: Consider the systemHere we have the critical point (2,1). We make the change of variables x1=x-2 and y1=y-1 and rewrite the system asBy using the result from example 10 above we see that the point (2,1) is an unstable saddle point and that the solution to the original system isWe get the phase portrait by taking that from example 10 and move it two steps in the x-direction and one step in the y-direction.In the same way as in Example 11 we may instead study the more general systemThe equlibrium point is here (a0,b0). By making the change of variables x1=x-a0and y1=y-b0we can transfer the system to the ones studied above with equilibrium point (0,0).10. Exercises.9.1) Draw the phase portrait for the system9.2) Draw the phase portrait for the system9.3)Find the general solution and draw the phase portrait for the system9.4) Draw the phase portrait for the system9.5) The equation for a damped harmonic oscillator isRewrite the equation as a system by introducing the variableShow that (0,0) is a critical point. Describe the properties and stability of the critical point in the cases:a)a=0,b) a2-4km=0,c) a2-4km<0,d) a2-4km>0.9.6)Describe Verhulst's population model. In particular describe how this model can be used to illustrate the notions of attractor and chaos. What is Feigenbaum's constant?9.7) Describe how to illustrate Julia sets. What is the famous Mandelbrot set?。