Figure 2.1: Driven pendulum of mass M with a torque applied at the pivot and subject to gravity. Figure 2.1 shows a pendulum of mass M subject to a torque (the rotational equivalent of a force) and to a gravitational force G. You may think, for example, of a clock pendulum or a driven swing. The angle with the vertical in a positive sense is denoted by 6 = @(t), where t € R holds for the time of the system, and we choose —7 < 6 < ft. Figure 2.2: Plots of four different types of motion of the driven nonlinear pendulum (2.1) in the (9,w)-space under variation of control parameters. look at the dynamics of a point particle with constant speed that collides elastically with the disk and the walls of the box. That is, we have specular reflection at the walls where in- and outgoing angles are the same, 6 = 6’.2, This system strongly reminds one of the idealistic case of a billiard table (without friction but with a circular obstacle) and is indeed referred to as a particle billiard in the literature. The particular example shown in Fig. 2.4 is known as the Stnaz billiard. Figure 2.3: A ball of mass M subject to gravity, that elastically bounces off a vibrating plate. Figure 2.6: Schematic representation of the stretch and fold mechanism of an ensemble of particles in a chaotic dynamical system. The dynamical system is defined as follows: we have a series of semicircles periodically continued onto the line, which may overlap with each other. A point particle of mass M now scatters elastically with these semicircles under the influence of a gravitational force G. In the simulation we study the spreading in time of an ensemble of particles starting from the same point, but with varied velocity angles. The result is schematically depictec in Fig. 2.6: We see that there are two important mechanisms determining the dynamics o: the particles, namely a stretching, which initially is due to the choice of initial velocities but later on also reflects the dispersing collisions at the scatterers, and a folding at the collisions. where the front of propagating particles experiences cusps. This sequence of “stretch” anc “fold” generates very complicated structures in the position space of the system, which look like mixing paint. Figure 2.5: A billiard where a point particle bounces off semicircles on a plate. to two nearby particles in this billiard (which do not interact with each other) with slightly different directions of their initial velocities? Since at the moment no computer simulation of this case is available to us, we switch to a slightly more complicated dynamical system as displayed in Fig. 2.5, where we can numerically explore this situation [Miy03}. Figure 3.1: Example of a trajectory in 3-dimensional phase space (note that the trajectory is not supposed to cross itself). See Fig. 3.1 for an example. Figure 3.2: A trajectory that crosses itself. Proof: Assume a single trajectory crosses itself, see Fig. 3.2. Starting with the initial condition at the point of intersection there are two choices of direction to proceed in, which contradicts uniqueness. The same argument applies to two distinct trajectories. q.e.d. Figure 3.3: stable and unstable fixed points The same reasoning applies to the simple example of a periodic orbit shown in Fig. 3.4: The left hand side depicts a circular periodic orbit. However, if we choose initial conditions that are not on this circle we may observe, for example, the behavior illustrated on the right hand side of this figure: trajectories ‘spiral in’ onto the periodic orbit both from the interior of the circle and from the exterior. An isolated closed trajectory such as this periodic orbit is called a limit cycle, which in this case is stable. Of course, as for the fixed point the opposite case is also possible, that is, the limit cycle is unstable if all nearby trajectories spiral out (you may wish to draw a figure of this). The detailed determination and classification of fixed points and periodic orbits is in the focus Figure 3.4: left: circular periodic orbit; right: approach to this orbit as a limit cycle One may now ask the further question of how large the dimensionality N of the phase space has to be for complicated behaviour to be possible. The answer is given in terms of the following important theorem:[Str94, Rob95]4 Figure 3.5: Trajectories in 2-d (left) cannot cross whereas in 3-d (right), they can. Figure 3.7: A continuous trajectory in space pierces the plane «* = K at several points : discrete time n. Figure 3.8: A sequence of functions approaching the 6-function. as shown in Fig. 3.8. It is not hard to see that in the limit of y — 0 this sequence has the desired properties, that is, 6,(x) — d(x) (y > 0). We remark that many other representations of the d-function exist. Strictly speaking the ‘d-function’ is not a function but rather a functional, respectively a distribution defined on a specific (Schwartz) function space.® The equation of motion for this dynamical system is straightforwardly derived from physical arguments [Ott93], however, we just state it here in form of where the dynamical variable 0 describes the turning angle and k := K/I is a contro parameter. As before we can rewrite this differential equation as a vector field, Two important properties of the 6-function that we will use in the following are its normal- ization, Figure 3.10: The climbing sine map. Figure 4.1: The tent map including a cobweb plot. As usual, its equations of motion are given by 2,41 = M(a,). One can easily see that the dynamics is bounded for x € [0, 1]. By definition the tent map is piecewise linear. One may thus wonder in which respect such a map can exhibit a possibly chaotic dynamics that is typically associated with nonlinearity. The reason is that there exists a point of nondifferentiability, that is, the map is continous but not differentiable at « = 1/2. If we wanted to approximate the tent map by a sequence of differentiable maps, we could do so by unimodal functions as sketched in Fig. 4.2 below. We would need to define the maxima of the function sequence and the curvatures around them such that they asymptotically approach the tent map. So in a way, the tent map may be understood as the limiting case of a sequence of nonlinear maps. Figure 4.2: Approximation of the piecewise linear nondifferentiable tent map by a sequence of nonlinear differentiable unimodal maps. Figure 4.3: Stretch-and-fold mechanism in the tent map. The tent map thus yields a simple example for an essentially nonlinear stretch-and-folc mechanism, as it typically generates chaos. This mechanism is encountered not only ir the bouncing ball billiard but also in many other realistic dynamical systems. We maj remark that ‘stretch and cut’ or ‘stretch, twist and fold’ provide alternative mechanisms fo1 generating complicated dynamics. You may wish to play around with these ideas in thought experiments, where you replace the sets of points by kneading dough. Figure 4.4: The function tan x is a homeomorphism. Both results could also have been inferred directly from Fig. 4.6. Remark 2 In typical dynamical systems the fixed points and periodic orbits are zsolated with ‘more complicated’ orbits in between, as will be discussed in detail later on. There exists also a nice fized point theorem: A continuous function F' mapping a compact interval onto itself has at least one fixed point; see [Dev89] for a proof and for related theorems. The detailed discussion of such theorems is one of the topics of the module ‘Chaos and Fractals’. Figure 4.5: Set of fixed points for F(x) = x. Figure 4.7: Cobweb plot for the map defined by the function F(x) = ses The roots of the map are F(x) =0 = Sua” => 2(3—27)=0S-2€ {0,4vV3 ~ 1.73}. We can now draw the graph of the map, see Fig. 4.7. The stability of the fixed points can be assessed by cobweb plots of nearby orbits as we have discussed before. Figure 4.9: Construction of eventually periodic orbits for an example via backward iteration. Figure 4.8: An eventually periodic orbit for F(x) = 2? Figure 4.10: Illustration of an € neighbourhood N,(p). Figure 4.11: Illustration of a set A being dense in B. Definition 15 Let A,B C R and A C B. A 1s called dense in B if arbitrarily close to each point in B there is a point in A, i.e. Vx E BVe>O N(x) N AFD, see Fig. 4.11. An application of this definition is illustrated in the following proposition: Figure 4.12: The Bernoulli shift. Proof: Let us prove this proposition by using a more convenient representation of the Bernoulli shift dynamics, which is defined on the circle.” Let Figure 4.13: Representation of a complex number z = cos¢@+isin@ on the unit circle. denote the unit circle in the complex plane [Rob95, Has03], see Fig. 4.12 There are at least two different ways of how to prove this. The idea of the first version is to use the above theorem. For convenience (that is, to avoid the point of discontinuity) we may consider again the map on the circle. Then for whatever subset U we choose AN € N such that BY(U) D S' as sketched in Fig. 4.14, which is due to the linear expansion of the map. This just needs to be formalized [Has03]. It is no coincidence that this mechanism reminds us of what we have already encountered as “mixing” . Figure 4.14: Bernoulli shift B operating on a subset of the unit circle. Figure 4.15: An example of a map exhibiting a complicated basin structure, which here is constructed by generating preimages of a given subset of the basin. Proof: This trivially follows from Definition 18. Define G(x) := F*(x) and look at G(p) = p. Then p is an attracting/repelling/marginal fixed point of G <= > p is an attracting /repelling/marginal fixed point of F* <= p is an attracting /repelling/marginal periodic orbit of F’. q.e.d. Remark 7 Pictorially speaking, this definition means that all neighbours of x (as close as desired) eventually move away at least a distance 6 from F(x) for n sufficiently large, see Fig. 5.1. ‘igure 5.1: Illustration of the idea of sensitive dependence on initial conditions. Example 19 This definition is illustrated by the following proposition: Proof: The last expression defines a time average (in the mathematics literature this is sometimes called a Birkhoff average), where n terms along the trajectory with initial condition x9 are summed up by averaging over n. These considerations motivate the following important definition: Definition 25 /All97] Let F € C' be a map of the real line. The local Ljapunov number L(xo) of the orbit {x9,%1,...,%n—1} is defined as Figure 5.2: Example of a map that has a positive local Ljapunov exponent, whereas for typical initial conditions the Ljapunov exponent is negative. Definition 27 Ljapunov exponent for periodic point. Let p € R be a periodic point of period n. Then Figure 5.3: Tentative summary of relations between different topological chaos properties. You may wish to explore yourself in which respect this property relates to what we have previously encountered as topological transitivity, see Theorem 2. Definition 28 F' 1s topologically mixing if for any two open sets U,V C JAN EN sue that Vn > N,n EN, F™U)NV £90. Figure 6.1: Left: The logistic map for control parameter r = 4 with M = 6 bins; right: Histogram for an ensemble of points iterated n times in the logistic map, computed by using these bins. Figure 6.2: Idea of deriving the Frobenius-Perron equation, demonstrated for the example of the Bernoulli shift. 2In probability theory this is called transformation of variables. Figure 6.3: Idea of deriving the Frobenius-Perron equation, demonstrated for the example of the Bernoulli shift. Figure 6.4: The “double Bernoulli shift” as an example of a map that has no unique invariant density. An interesting question is now what happens to partition parts if we let a map act onto them. Let us study this by means of the example sketched in Fig. 6.6, where we apply the map D(z) defined in the figure on the previous sample partition. The map then yields Figure 6.6: The map D(a) applied onto a partition consisting of the two parts {i, Io}. Note that here partition parts or unions of them are recovered exactly under the action of the map. This is the essence of what is called a Markov property of a map: Figure 6.5: Simple example of a partition of the unit interval. Example 27 Let us look at the map F defined in Fig. 6.7. Figure 6.7: An example of a map E(x) with a partition that is not Markov. It is elucidating to discuss an example of a map with a partition that is not Markov: Figure 6.8: Left: A map that enables a matrix representation for the dynamics of a statistical ensemble of points; right: Illustration of where the transition matrix elements come from for this map. Example 28 The map F shown in Fig. 6.8 is defined on a Markov partition (please convince yourself that this is the case). Let N’ be the number of points in partition part 7 at the nth iteration of this map. Let us start with a uniform distribution of N = >>, Ni points on (0, 1] at time step n = 0. As one can infer from the left part of Fig. 6.8, the dynamics of N’ is then given as follows: Example 29 The Markov graph for the map F' of Fig. 6.8 acting on the given Markov partition is sketched in Fig. 6.9. Figure 6.9: Markov graph for the map F' on the given Markov partition. Remark 15 Figure 6.11: Example of a map G yielding a topological transition matrix that is not directly related to a stochastic one. construct the Frobenius-Perron operator in terms of a transition matrix that is generally not a topological but a stochastic one, which acts on probabilities defined on Markov partition parts, see Section 17.5 of Ref. [Bec93] for details. For uniformly constant slope this stochastic matrix can be simplified to a topological transition matrix divided by the absolute value of the slope, which acts onto probability density vectors as given in the proposition. Remark 16 For a single vector component Eq. (6.30) reads It is not a bad idea to cross-check whether our solution indeed fulfills the Frobenius-Perron matrix equation Eq. (6.38), Figure 7.3: A delta function density yields a singular measure via integration. Example 37 For B(2) let p*(x) := $[5(a# — 4) + 6(a — 3)] be the sale Ht density — on the period 2 orbit {, 2} only, see the left part of Fig. 7.3. Then ji*( = fF dz p*( see the right hand side of this figure, is singular. Figure 7.2: A step function density and the corresponding absolutely continuous probability measure obtained by integrating the density. Figure 7.4: The two-dimensional area-preserving baker map and its iterative action onto a set of points initially concentrated on the left half of the unit square. For higher-dimensional dynamical systems we commonly have a combination of absolutely continuous and singular components of the invariant measure. This leads to another defini- tion, which we first motivate by two examples: Figure 7.5: The dissipative baker map eventually generates an invariant measure exhibiting a fractal structure in the y direction. Figure 7.6: Uniform partition of the interval J C R and a local average g; defined on the ith partition part of an observable g(x). In the following we will show that this result is not a coincidence. We do so in two steps: Figure 7.7: An example of a map that is decomposable and hence not ergodic. Remark 22 You may have noticed that the properties of a map being topologically tran- sitive and being ergodic are of a very similar nature. However, note that ergodicity requires us to use a measure, whereas topological transitivity works without a measure. In fact, for a continuous map it can be shown under fairly general conditions that ergodicity implies topological transitivity, whereas the reverse does not hold [Kat95, Rob95}.!° Figure 7.8: A simple map that we have seen before, here used in order to demonstrate the calculation of Ljapunov exponents via ensemble averages.
![Figure 3.4: left: circular periodic orbit; right: approach to this orbit as a limit cycle One may now ask the further question of how large the dimensionality N of the phase space has to be for complicated behaviour to be possible. The answer is given in terms of the following important theorem:[Str94, Rob95]4](https://figures.academia-assets.com/30606596/figure_011.jpg)
![The equation of motion for this dynamical system is straightforwardly derived from physical arguments [Ott93], however, we just state it here in form of where the dynamical variable 0 describes the turning angle and k := K/I is a contro parameter. As before we can rewrite this differential equation as a vector field, Two important properties of the 6-function that we will use in the following are its normal- ization,](https://figures.academia-assets.com/30606596/figure_016.jpg)
![As usual, its equations of motion are given by 2,41 = M(a,). One can easily see that the dynamics is bounded for x € [0, 1]. By definition the tent map is piecewise linear. One may thus wonder in which respect such a map can exhibit a possibly chaotic dynamics that is typically associated with nonlinearity. The reason is that there exists a point of nondifferentiability, that is, the map is continous but not differentiable at « = 1/2. If we wanted to approximate the tent map by a sequence of differentiable maps, we could do so by unimodal functions as sketched in Fig. 4.2 below. We would need to define the maxima of the function sequence and the curvatures around them such that they asymptotically approach the tent map. So in a way, the tent map may be understood as the limiting case of a sequence of nonlinear maps. Figure 4.2: Approximation of the piecewise linear nondifferentiable tent map by a sequence of nonlinear differentiable unimodal maps.](https://figures.academia-assets.com/30606596/figure_019.jpg)
![Both results could also have been inferred directly from Fig. 4.6. Remark 2 In typical dynamical systems the fixed points and periodic orbits are zsolated with ‘more complicated’ orbits in between, as will be discussed in detail later on. There exists also a nice fized point theorem: A continuous function F' mapping a compact interval onto itself has at least one fixed point; see [Dev89] for a proof and for related theorems. The detailed discussion of such theorems is one of the topics of the module ‘Chaos and Fractals’.](https://figures.academia-assets.com/30606596/figure_022.jpg)
![Figure 4.13: Representation of a complex number z = cos¢@+isin@ on the unit circle. denote the unit circle in the complex plane [Rob95, Has03], see Fig. 4.12](https://figures.academia-assets.com/30606596/figure_030.jpg)
![There are at least two different ways of how to prove this. The idea of the first version is to use the above theorem. For convenience (that is, to avoid the point of discontinuity) we may consider again the map on the circle. Then for whatever subset U we choose AN € N such that BY(U) D S' as sketched in Fig. 4.14, which is due to the linear expansion of the map. This just needs to be formalized [Has03]. It is no coincidence that this mechanism reminds us of what we have already encountered as “mixing” . Figure 4.14: Bernoulli shift B operating on a subset of the unit circle.](https://figures.academia-assets.com/30606596/figure_031.jpg)
![The last expression defines a time average (in the mathematics literature this is sometimes called a Birkhoff average), where n terms along the trajectory with initial condition x9 are summed up by averaging over n. These considerations motivate the following important definition: Definition 25 /All97] Let F € C' be a map of the real line. The local Ljapunov number L(xo) of the orbit {x9,%1,...,%n—1} is defined as](https://figures.academia-assets.com/30606596/figure_034.jpg)
![Figure 6.8: Left: A map that enables a matrix representation for the dynamics of a statistical ensemble of points; right: Illustration of where the transition matrix elements come from for this map. Example 28 The map F shown in Fig. 6.8 is defined on a Markov partition (please convince yourself that this is the case). Let N’ be the number of points in partition part 7 at the nth iteration of this map. Let us start with a uniform distribution of N = >>, Ni points on (0, 1] at time step n = 0. As one can infer from the left part of Fig. 6.8, the dynamics of N’ is then given as follows:](https://figures.academia-assets.com/30606596/figure_043.jpg)
![Figure 6.11: Example of a map G yielding a topological transition matrix that is not directly related to a stochastic one. construct the Frobenius-Perron operator in terms of a transition matrix that is generally not a topological but a stochastic one, which acts on probabilities defined on Markov partition parts, see Section 17.5 of Ref. [Bec93] for details. For uniformly constant slope this stochastic matrix can be simplified to a topological transition matrix divided by the absolute value of the slope, which acts onto probability density vectors as given in the proposition. Remark 16 For a single vector component Eq. (6.30) reads](https://figures.academia-assets.com/30606596/figure_046.jpg)
![Figure 7.3: A delta function density yields a singular measure via integration. Example 37 For B(2) let p*(x) := $[5(a# — 4) + 6(a — 3)] be the sale Ht density — on the period 2 orbit {, 2} only, see the left part of Fig. 7.3. Then ji*( = fF dz p*( see the right hand side of this figure, is singular.](https://figures.academia-assets.com/30606596/figure_049.jpg)
