Chaos Lecture Notes 4j4929

Nonlinear Dynamics and Chaos III: Introducing Chaos David Brantley 12 December, 2012

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Lecture Introduction

In the past couple lectures we have introduced the concepts and analysis techniques of nonlinear dynamics. Through examples we have seen how such systems can exhibit extremely rich state space structure and how sensitive this structure can be to even minute changes in parameters. In this lecture we focus on a specific type of nonlinear system that epitomizes the most unique and exciting aspects of nonlinear dynamics: chaos. Chaos, not to be confused with randomness, means that the long term state of a system cannot be predicted given any previous state. The surprising thing is that chaos arises from completely deterministic equations, the mapping or ODE governs exactly where the trajectory is going. This may seem contradictory, however the two statements are really looking at two levels of description. It’s these multifold levels of description that makes chaos so exciting and challenging for current science.

1.1

Lesson Plan

1. Mappings and Chaos (a) What is a Map? (b) Introducing Chaos 2. Lyapunov Exponents (a) Lyapunov Exponents of Maps 1

(b) Lyapunov Exponents of Multidimensional Systems 3. Stretching and Folding in State Space (a) Qualitative Description of Strange Attractors (b) Fractal Dimensions 4. General Features of Chaotic Systems (a) Driven Damped Pendulum

2 2.1

Mappings and Chaos What is a Map?

Consider for a moment a simple model of an ODE integrator algorithm. The algorithm takes your current position in XYZ space, computes the velocity field at that point, then finds the next position in space using the familiar d = vt. Assuming the time step t is small enough this should give you a good approximation to the trajectory. This type of algorithm generates a point sequence that tells you where the next position is given a previous position. This is what is referred to as an Iterative Mapping. More precisely: Definition 1. A map is a function f :S→S which defines an evolution rule with discrete time and a continuous state space S. Evolution is defined by iteration of the map. xn+1 = f (xn )

n∈Z

The orbit, or trajectory, is thus the point sequence {x0 , x1 , x2 . . .}. So how do you find the fixed points of a map? If we go back to our analogy of the ODE integrator, a fixed point will occur where the velocity field is zero. Thus after every iterative time step d = 0 and the next iterate is simply the same point. Thus we define

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Definition 2. A fixed point of a map is a point x∗ such that f n (x∗ ) = x∗

∀n∈Z

This defines a diagonal line in the plot of f (xn ) versus xn plot. Thus any point of the map which intersects with this line will have a point at which f (xn ) = xn , a fixed point. So what do maps have to do with chaos?

2.2

Introducing Chaos

Now that we know what a mapping is, let’s do a little analysis on them. Given a one dimensional discrete mapping xn+1 = f (xn ) we can ask the question: how does the point sequence {x0 , x1 , x2 . . . } differ from the point sequence {x¯0 , x¯1 , x¯2 . . . } produced from slightly perturbed initial conditions x0 + δx0 ? We know from the definition of mappings that f n (x¯� ) = f n (x� + δx� ) = {x¯0 , x¯1 , x¯2 . . . } Which implies that in general, x¯n = xn + δxn = f (xn−� + δxn−� ) Assuming we have chosen n such that δxn is still small, but far enough away from δx0 so that the change is significant, we can Taylor expand the right side of the above equation. Taking only the first order expansion we find. x¯n ≈ f (xn−1 ) + f � (xn−1 )δxn−1 Comparing δxn = f � (xn−1 )δxn−1 We can recycle this same argument recursively for all of the δxn ’s giving us δxn = f � (xn−1 )f � (xn−2 ) . . . f � (x0 )δx0 �n−1 � � ≡ f � (xl ) δx0 l=0

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Looking back on our derivation we see that the ratio of δxn versus δxn−1 is the derivative f � (xn−1 ). Thus the f � (xl ) are a measure of how xn and x¯n move toward or away from eachother. In order to get a feel for what this equation actually implies, let’s consider the special case of periodic motion. Let’s define the point sequence xl as the Poincar`e map (with the stroboscopic sample time equal to one period). In this case all xl and their derivatives are equal, giving |δxn | = |f � (x0 )|n |δx0 | Defining σ = ln |f � (x0 )| our equation becomes |δxn | = enσ |δx0 |

(1)

Note that this equation is only valid for periodic Poincar`e maps, however we really only care about how the trajectories change on average since this will tell us about the trajectorys asymptotic behavior. So we may define σ for non-periodic trajectories by averaging over all of the xl giving us n−1

1� σ = lim ln |f � (xl )| n→∞ n l=0

(2)

We see that (2) is a logarithmic measure for the mean expansion rate per iteration(unit time) of two infinitesimally close trajectories. We call σ the Lyapunov Exponent of the trajectory. This very important measure can tell us about the existence of chaos and more in a system.

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Lyapunov Exponents

3.1

Lyapunov Exponents of Maps

Looking at (2) once more we see that depending on the average magnitude of f � (xl ) we can get three flavors of exponents. 1.

σ<0

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(a) In light of (1) a negative Lyapunov exponent means that nearby trajectories will converge exponentially upon each other. (b) This is characteristic of orbits that are attracted to stable fixed or periodic points as well as trajectories that exhibit asymptotic stability. (c) Dissipative or non-conservative systems also have negative Lyapunov exponents. 2.

σ=0 (a) Characteristic of nearby trajectories that neither converge nor diverge but stay a fixed distance from eachother. (b) Systems in steady state and conservative have this type of exponent. (c) For example Simple Harmonic Oscillators with different amplitudes but same frequency will exhibit this type of stability.

3.

σ>0

The most important case for this lecture!

(a) Exponential divergence of nearby trajectories. (b) Extreme sensitivity to initial conditions, since any δx0 explodes to arbitrary size rapidly. (c) Chaotic systems have positive Lyapunov exponents! A positive Lyapunov exponent means that practically, chaotic systems are unpredictable. Since any physical parameter or computer simulation is determined with limited precision, there is always a nonzero δx0 that becomes large exponentially. Furthermore, attempting to overcome this by more precisely fixing initial conditions is doomed to failure, since doing so would require an exponential increase in precision. Notice that our system is entirely deterministic, the mapping precisely fixes the system. There is no stochastic term that would add noise and uncertainty.

The Tent Map Consider the following map. � µxn 0 ≤ xn < 12 xn+1 = µ(1 − xn ) 12 ≤ xn ≤ 1 5

Our space here is bounded on the unit interval xn ∈ [0, 1], which fixes our range of µ values such that µ ∈ [0, 2].

Graphically our map looks like a tent! The derivative at every point on this map f � (xl ) = ±µ! Therefore looking back at (2) we see that n−1

n−1

1� 1� σ = lim ln |f � (xl )| = lim ln µ = ln µ n→∞ n n→∞ n l=0 l=0 Giving σ > 0 when µ > 1. Thus the tent map exhibits chaos whenever this parameter is greater than one. Graphically this can be seen in the cobweb map for µ > 1. A typical trajectory in this chaotic system will explore the space more and more finely, never falling into a repeating cycle.

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In case the students are unfamiliar with cobweb plots here is a simple explanation. You start with a point on the xn axis. You find the equivalent xn+1 by drawing a vertical line to the curve. Then you use this value as your new xn by reflecting it over the f (xn ) = xn line and then find your new xn+1 and repeat. This will trace out the trajectory.

3.2

Lyapunov Exponents of Multidimensional Systems

The techiniques for deriving the Lyapunov exponents for the one dimensional mappings can be generalized to higher dimensions and continuous time. Consider: � A flow field (ie trajectory field) in state space given by Φ(t) with the velocity field given by � dΦ � = F� (Φ) dt Just like in the mapping case we consider a slightly perturbed initial condition and see how the two evolve. ie we consider: � � Φ(0) → Φ(0) + �(0) 7

and how �(t) evolves in time. Let’s look at the Taylor expansion � d � dΦ ∂Fi � + Mij (t)�� (Φ + �) ≈ + |� �(t) = F� (Φ) dt dt ∂Φj Φ(t) Where Mij (t) is the Jacobian of the velocity field evalueated at the position � and time Φ(t). Comparing we see that d�� = Mij (t)�� dt

(3)

This has the formal solution �(t) = L(t)��(0) Note that in general (3) has no analytical solution. This is because of it’s dependence upon the initial perturbation �(0) and in general the time dependence of Mij (t) is not known. If we make a few simplifying assumptions we can pull something sensible out of (3).

Time Varying Eigenvalues Assume: 1. M(t)’s eigenvalues but not eigenvectors vary with time. 2. State space coordinates correspond to M(t)’s eigenvectors. 3. System is ergodic: Time averaged quantities along the trajectories are equivalent to averaged quantities over state space.

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Let



 X(t) � Φ(t) =  Y (t)  Z(t)

We consider small displacements in the reference frame of the eigenvectors along the directions of the eigenvectors. δX(t), δY (t), δZ(t) Plugging into (3) we get 

  �   ˙ A[Φ(t)] 0 0 δ X(t) δX(t)  δ Y˙ (t)  =  0 � B[Φ(t)] 0   δY (t)  ˙ � δZ(t) δ Z(t) 0 0 C[Φ(t)]

Where A,B, and C are the time-dependent eigenvalues. Looking at just ˙ δ X. � δ X˙ = A[Φ(t)]δX(t) Has the solution δX(t) = δX(0) exp

��

t 0

� � )] dt A[Φ(t �

�

The integral in the exponential is due to the chain rule. Rearranging and taking the logarithm we find � � � t � δX(t) � � � )] �= ln �� dt� A[Φ(t δX(0) � 0 Like in the mapping case we are only interested in what happens on average to the nearby trajectories. Here we define the time average of A[Φ(t)] by dividing the integral by the total time. This gives us � � � 1 �� δX(t) �� 1 t � � � ln = dt A[Φ(t )] t � δX(0) � t 0

This expression by itself is still dependent upon the initial starting position δX(0), however recall our last assumption: time averages along trajectories are the equivalent to the averages over state space. Thus if we average over long enough times, we will in effect average over all of state space. 9

� � �A� = time average of A[Φ(t)] = state space average of A for all possible flows Φ(t) This allows us to say that the average above is independent of starting position, since any initial position will eventually explore all of state space. Now we may define: � � � 1 t � � � 1 �� δX(t) �� σ1 = lim dt A[Φ(t )] = lim ln � t→∞ t 0 t→∞ t δX(0) �

� As one of the three Lyapunov exponents for Φ(t). Comparing the two sides we see that, just like the map case, σ is a logarithmic measure of the mean expansion rate in time of the perturbation(measured by the eigenvalue in this case). Due to the ergodic system assumption, this is valid, however it is important to note that Lyapunov exponents measure divergence in state space. In general we cannot assume that an average in time over a trajectory will average over state space since the trajectory may not explore all of state space. In general for an N-dimensional system there are N linearly independent perturbations with N associated Lyapunov exponents. For an autonomous(not explicitly time dependent) system with three degrees of freedom we have 3 Lyapunov exponents that may be defined for each attractor region of the system. The sign conventions from the one-d map case still hold for each σ, therefore . The combination of the three signs characterizes the attractor.  (−, −, −)    (0, −, −) (σ1 , σ2 , σ3 ) =  (0, 0, −)    (+, 0, −)

fixed point limit cycle (1 - periodic direction) torus (2 periodic directions) strange (chaotic) attractor

While the other three cases are interesting in their own right, for the remainder of this lecture we are going to focus on the properties of chaotic attractors.

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Stretching and Folding in State Space

4.1

Qualitative Description of Strange Attractors

Consider: A system with a positive Lyapunov exponent and a restricted region of phase space(such as an attractor). Let’s take a look at a small state space volume element ∆V just as a parameter is switched and the region becomes a chaotic attractor.

Going from left to right. 1. The direction corresponding to the negative Lyapunov exponent will experience an exponential convergence of points, flattening that dimension. The positive direction will experience a stretching of all points in that dimension. 2. Eventually the stretched volume will reach the boundary of the attractor. By definition, all trajectories within the attractor must remain in the attractor, therefore the volume has no choice but to fold back on itself over and over again. 3. After an infinite number of foldings, the original ∆V is crushed so thin that it no longer has any volume, yet it is not quite a surface since it has a type of ”depth” due to the folding. This ”broken” dimensional object is characteristic of strange attractors and is called a fractal. Note that this also explains chaotic attractors have a high sensitivity to initial conditions. No matter how close two trajectories are, in this infinitely folding space they will end up on different folds and lose all correlation between their trajectories. So how do we describe these chaotic attractors and what are their properties?

4.2

Fractal Dimensions

Chaotic attractors are fractals and are objects with noninteger dimension, neither volumes nor surfaces or lines. How do we therefore find their 11

dimension? Here we discuss one technique called the capacity or Hausdorff dimension. Consider: A metric space(where distance has meaning) that is covered by a grid of uniform linear dimension �. The ”volume” of an object in this space can be estimated by adding all of the individual ”volumes” of the boxes that contain pieces of the object.

V (�) = N (�)�Dh As we take smaller and smaller boxes V (�) gets closer and closer to the ”true” volume of the object. I.e. lim V (�) → constant �→0

The exponent Dh acts as the dimension of the space. �� � � D N (�) 1 h Dh = lim ln = lim ln �→0 V (�) �→0 � ln N (�) − ln V (�) ln N (�) = lim 1 �→0 �→0 ln � ln 1�

⇒ Dh = lim

V (�) remains finite and does not contribute to the limit. Therefore the Hausdorff or capacity dimensionDh of an object may be defined as ln N (�) �→0 ln 1�

lim

(4)

The Middle Third Cantor Set Consider: The unit interval [0, 1]. Now remove the middle third of the set giving two new dist intervals [0, 1/3] ∪ [2/3, 1] and again cutting out the middle third of these two new intervals gives four intervals [0, 1/9] ∪ [2/9, 1/3] ∪ [2/3, 7/9] ∪ [7/9, 1] and so on for a total of 2n intervals. 12

The subdivisions always have a length of one third of the previous interval size giving �n = 31n . Using these two values we can compute the fractal dimension of the middle third cantor set as ln N (�n ) ln 2n ln 2 = lim = ≈ 0.63 1 n n→∞ n→∞ ln 3 ln 3 ln �n

Dh = lim

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General Features of Chaotic Systems

So we have looked at various aspects of chaotic systems in mappings and in state flows. What can we say generally about such systems? There is no universally acceptd mathematical definition, however there are a few key features which all chaotic dynamical systems share. 1. Sensitivity to Initial Conditions (a) Characterized by a positive Lyapunov exponent. (b) Note that while this is a necessary condition for chaos it is not sufficient to define chaos. 2. Topological Transitivity Definition 3. For every pair of non empty open sets U and V in state space, there exists some integer k, such that f k (U ) ∩ V �= Ø where f is the dynamical function. This means that any two regions in state space, no matter how far apart, will eventually overlap eachother. From the perspective of the folding state space this makes sense since by repeated folding, every region of state space gets streteched and deformed until eventually they overlap every region at least once. Note: As mentioned above σ > 0 is not a sufficient definition of chaos. For example: Consider the map xn+1 = rxn

for r > 1

Has f � (xn ) > 1 everywhere and therefore σ > 0. However this map has no attracting regions. Therefore trajectories do not fold in on themselves and there is no transitivity. This map is not chaotic. 13

no attractors ⇒ no topological transitivity ⇒ no chaos 3. Density of Periodic Points This one is a bit abstract so hang on. Definition 4. For any point X on a chaotic attractor, within any neighborhood � there exists another point Y which is periodic. Consider the period doubling bifurcation described in the previous lecture. Now imagine this cycle repeating over and over again each time doubling the period until you get an infinite periodic solution. Now you have an infinite number of periodic points in your state space. However, these periodic points are all rational multiples of the fundamental period and this makes them countably infinite. The state space operates on a set of Rn and therefore is uncountably infinite i.e you could go out infinite decimal places while a rational has a finite amount of decimal places. This means that even though there are an infinite number of periodic points, the probability of selecting one is zero, since there are infinitely more real numbers. This is what is meant by density of periodic orbits.

Driven Damped Pendulum In our last example we look at a pendulum with a periodic driving force and linear drag. The equation of motion can be found using sum of torques to yield: mL2 θ¨ = −bL2 θ˙ − mgL sin θ + LA cos Ωt

This can be put in the simplified form by setting ω0 = 1 giving θ¨ + β θ˙ + sin θ = f cos Ωt The equations can be further simplified by putting them in autonomous form dω = −βω − sin θ + f cos φ dt dθ =ω dt 14

dφ =Ω dt Notice that we have three parameters to choose from β, Ω, f . We are only going to explore f the driving strength as the bifurcation parameter. We set Ω = 2/3 and β = 0.5.

In the figure above the vertical axis is ω and the horizontal is θ. The respective f value is shown in the upper right corner of the plots. When f is near in value to β we expect that the dissipated energy nearly balances the energy input by the driving, thus we should see the state plot to be approximately circular as in the top left most plot. As we increase the driving strength the stable periodic attractor bifurcates into two attractors, each with a period of half the original period as shown in the middle plot. Further increases result in more frequency doublings, starting to fill in the phase space as shown in the bottom left corner. At a critical value of f = 1.09 the pendulum ceases to have any finite period attractors and instead begins exploring arbitrary regions of the now chaotic attractor. This is shown in the Poincar`e map in the bottom middle plot. In the case of nonchaotic motion this map would be comprised of a finite series of points and would reach some steady state after a period of time. In this case however the map continues to fill in finer and finer detail 15

as the pendulum explores more regions of phase space. This is exemplified by the self similar fractal structure of the last plot on the far right.

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Conclusion

In summary we have explored chaos in mappings and differential equations. We have shown that chaotic systems exhibit distinct character from ordinary nonlinear systems and explored some of these features in detail. The final message to take away is that while chaotic systems may not be fully predictable, it does not mean that there is no information to be taken out of it. With the continual improvement of computational power comes the ability to analyze these systems in greater detail and improve our understanding of how they operate. I hope this has been informative.

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