Chaos and Liapunov exponents

Transcription

1 PHYS347 INTRODUCTION TO NONLINEAR PHYSICS - 2/22 Chaos and Liapunov exponents Definition of chaos In the lectures we followed Strogatz and defined chaos as aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions. This means: Aperiodic behaviour: A chaotic orbit never settles down, it does not converge to a fixed point or any periodic cycle. Deterministic: Chaotic trajectories look pretty random, yet they are generated by a deterministic system. There is no randomness in the iteration of the logistic map for example, once have you picked r and an initial x the trajectory x, x 2,... is fully determined. Sensitity to initial conditions: nearby trajectories separate exponentially fast, they are characterized by a positive Liapunov exponent. Now this is one definition of chaos, but not the only one. Although it should be said that all definitions more or less agree that the three properties above are necessary for a system to be chaotic. Other characteristic features that are often discussed are the following: Periodic orbits are dense: What this means is that for any initial condition x resulting in a chaotic orbit you can find a second initial condition x + δ with an arbitrarily small δ such that starting the iteration from x + δ results in a periodic orbit. For any initial condition of a chaotic orbit there is therefore an nearby initial condition which gives a periodic orbit. One says periodic orbits are dense. The system is topologically mixing: Take for example the logistic map. The iteration maps the interval [, ] onto itself. The statement that a map is topologically mixing means the following: if you take any arbitrarily small interval (a, b) [, ] and all the orbits generated by starting points in (a, b) then you will eventually come arbitrarily close to any point in [, ]. I.e. the interval (a, b), no matter how small it is, is stirred around (mixed) in the larger interval [, ] by the system to the degree that it comes infinitesimally close to any point in [, ] even those far away from (a, b). This is illustrated in Fig.. 2 Liapunov exponent The Liapunov exponent λ is a measure for the sensitivity to initial conditions. Before we define it, one crucial remark: the Liapunov exponent is not so much a property of the a given system itself, but instead a property of a particular orbit of that system. For example a dynamical system

2 .8 x n n Figure : Tpological mixing of the logistic map at r = 4. We plot different trajectories, all started in the interval.3 x.35. For the first few time steps (n =, 2, 3, 4) the interval remains relatively compact, it jumps around a bit, but it remains an interval. After six or so iterations the trajectories are essentially spread all over the place and cover the entire interval from to. There are gaps, but this is mostly because we have only iterated trajectories out of the initial interval. If we take more of them, and run the map for a little longer these gaps will close. could have a stable periodic cycle for one initial condition, but a chaotic trajectory for another initial condition. The cycle would have a negative Liapunov exponent, the chaotic orbit a positive exponent. 2. Growth factor To understand the notion of the Liapunov exponent consider the map x n+ = f(x n ), and two different initial conditions x and x + δ. The quantity δ is therefore the initial separation of the two orbits. Carrying out one iteration one finds x = f(x ), x + δ = f(x + δ ), where we have denoted the separation of the two orbits at time step by δ. Carrying out a Taylor expansion to first order one finds δ = f (x ) δ. 2

3 This statement of course holds more generally, if the deviation at iteration n is δ n then in linear approximation we have δ n+ = f (x n ) δ n. This statement should be very obvious to you by now. If you do not understand it, or can t derive it then you should do a serious revision of what we have done up to now in the course. We will write g n = f (x n ) for the growth factor at iteration n. To put the last equation in other words: The growth factor of the perturbation at iteration n is given by f (x n ), the magnitude of the perturbation grows (or decays) by precisely this factor in iteration n. 2.2 Average growth factor Now let us consider the following cases: The system approaches a fixed point eventually, i.e. x n x as n. At long times the only point the trajectory visits is x, i.e. the relevant asymptotic growth factor is f (x ). This is exactly the stability criterion for fixed points: stable if f (x ) <, and unstable if f (x ) >. The system approaches a 2-cycle define by p and q. So asymptotically the trajectory reads..., p, q, p, q, p, q, p, q,.... Now what is the net growth factor here? If the system is at p then the growth factor is f (p), if the system is at q then the growth factor is f (q). The total growth factor in two iterations is therefore f (p)f (q) and the 2-cycle is stable if this quantity is smaller than, and unstable if it is greater than. For a general k-cycle (cycle of period k), defined by say p, p 2,..., p k, the stability is governed by f (p )f (p 2 ) f (p k ) in a similar fashion, this quantity is the total growth factor over k iterations. Now, the Liapunov exponent is a measure of the average growth factor per iteration. What does this mean? Well it means that we have to consider all growth factors along the trajectory (for example a k-cycle) and average them somehow. Naively one might think one could do something like k k i= f (p k ). But this is no good, in the case of the two cycle we would get 2 [ f (p) + f (q) ], which is not really consistent with the above stability criterion. The mistake we have made is to take the arithmetic mean of a multiplicative quantity (at each step the magnitude of the perturbation is multiplied by f (x n ) ). What we should do is take the geometric mean. For the case of the k-cycle this would be µ = f (p )f (p 2 ) f (p k ) k, i.e. multiply all the growth factors along the cycle with each other and then take the k-th root of their product. This is then something like the average multiplication factor µ per iteration. The attractor is then stable if µ < (average growth factor is less than, so perturbations decay on average) and unstable if µ > (average growth factor is greater than, so perturbations grow on average). Let us check whether this is consistent with what we said above: Fixed points: for fixed points there is nothing to check, µ = f (x ) = f (x ). 3

4 2-cycles: µ = f (p)f (q) 2. Now one sees that µ < f (p)f (q) < so the stability criterion µ < is equivalent to f (p)f (q) <. A similar statement holds for k-cycles (you may want to convince yourself of this). 2.3 Liapunov exponent Now instead of the average growth factor µ one frequently looks at its logarithm, λ = ln µ. Note that µ by definition. Recalling µ = f (p )f (p 2 ) f (p k ) k one finds λ = ln µ = [ ] ln f (p ) + ln f (p 2 ) + + ln f (p k ). k We found that the attractor is stable if µ <, this translates into λ <. If however λ > then µ > so the attractor is unstable. We also note that the expression on the RHS of the last equation is the arithmetic mean of ln f (x n ) along the trajectory. So stability is determined by the geometric mean of the multipliers f (x n ) or equivalently by the arithmetic mean of their logarithms. The Liapunov exponent of a k-cycle p, p 2,..., p k is now precisely defined as λ = [ ] ln f (p ) + ln f (p 2 ) + + ln f (p k ), k it is the (arithmetic) average of the logarithm of the growth rate along the k-cycle, or equivalently the logarithm of the geometric-mean growth rate per iteration λ = ln f (p ) f (p k ) k. Now this is fine for periodic orbits of period k. What about chaotic trajectories, by definition these are not periodic? Well, one applies the same principle and computes the average (arithmetic) loggrowth rate per iteration. The only difference is that the average is not not over a finite number of points the trajectory visits (p,..., p k in the case of the cycle), but instead over an infinite number of points, namely all points along the chaotic orbit: λ = lim m m [ ln f (x ) + ln f (x 2 ) + + ln f (x m ) Incidentally, this last equation also captures the cases of a fixed point and of a k-cycle. For the fixed point all x i are equal to x, so you get λ = ln f (x ). For a cycle the sequence of the {x i } is periodic, and this last equation reproduces earlier results for the cycle, λ = k k i= ln f (p i ). You should make sure you understand why this is so. In order to measure a Liapunov exponent of a chaotic orbit in practice we obviously can t average over an infinite number of points, instead one picks a sufficiently large value of m, and averages over the first m points of the attractor. Typically m will do. One can make things a little more reliable by dropping the first few hundred iterations (to allow for equilibration ), the average is then performed over the remainder of the trajectory. But the basic message stays the same: if λ < the orbit is stable, on average perturbations about the orbit decay, and the orbit is an attractor. If λ > then perturbations grow on average, and the orbit is repelling. If µ = (which may happen) one has λ =. Such attractors are called super-stable. ]. 4

5 x n λ cum g n Figure 2: Logistic map at r = 4. Upper panel: Two orbits started from nearby initial conditions. The separation of the two trajectories is clearly visible. Middle: Growth factor g n = f (x n ) = 4 2x n at each iteration for one of the two orbits (the one corresponding to the black line in the upper panel). As seen in the figure the growth factors are mostly >, but occasionally they are also smaller than one. So most of the time the separation will grow. Lower panel: This panel shows the cumulative best estimate for the Liapunov exponent up to iteration n as a function of n. I.e. at iteration n we plot λcum = n n i= ln f (x i ). Formally one should take the limit n in this expression to evaluate the Liapunov exponent. As seen in the figure the cumulative average settles down to λ.7 > pretty quickly, the trajectory we are analyzing is definitely chaotic. It should now be clear to you why a positive Liapunov exponent, λ >, indicates sensitivity to initial conditions. Positive Liapunov exponents are indeed a necessary ingredient of chaos according to our definitions in Section of this document. An illustration for the logistic map can be found in Fig Higher dimensional maps Let us now briefly consider higher dimensional maps. In 2 dimensions we have x n+ = f(x n, y n ), y n+ = g(x n, y n ) for example. A trajectory is now a sequence of pairs (x n, y n ). If we now write the separation between two trajectories as (δ n, η n ) (i.e. the second trajectory is (x n + δ n, y n + η n )), then one finds 5

6 (you should check this): ( δn+ η n+ ) = J (xn,yn) where J (xn,y n) is the Jacobian of the map evaluated at (x n, y n ). Now the perturbation is a vector, and as such it may change its magnitude at each iteration, but also its direction. This is illustrated in Fig 3. In fact the vector (δ n, η n ) lives in a 2-dimensional space that moves along with the orbit of the map (a so-called co-moving frame or Frenet frame), see Fig 3.This space is often called the tangent space of the map, the perturbation vector evolves in this space 2. Given that this space has two dimensions, there are two linearly independent contributions to the perturbation vector. These can either be both stable, or both unstable, or one stable the other one unstable, just as we had stable and unstable directions in the case of 2D stability analysis for continuous-time systems. For this reason we will have two Liapunov exponents, λ and λ 2, one for each of the two linearly independent directions in tangent space. A d-dimensional map will therefore have a d-dimensional tangent space, and accordingly there are d Liapunov exponents. This is referred to as the Liapunov spectrum. A trajectory of such a system chaotic if there is at least one unstable direction in tangent space, i.e. if at least one Liapunov exponent is positive. Frequently it is therefore sufficient to compute the largest Liapunov exponent numerically. This is much easier than computing all Liapunov exponents 3. If this largest exponent is greater than zero, then the system is sensitive to initial conditions, trajectories separate exponentially and the orbit is likely to be chaotic. If the largest exponent is negative, then all others are negative as well, then the orbit can not be chaotic as there is no sensitivity to initial conditions. ( δn η n ), 2 If you want to know more about tangent spaces etc then you should take a course on differential geometry. There is a whole theory around this, and dynamical systems can be described on an entirely different level using the language of differential geometry. 3 Measuring all Liapunov exponents requires the tracking of the growth of all d base vectors in tangent space. This can be tricky as they all tend to align along the fastest growing direction. The components perpendicular to that direction become very small very quickly, making it hard to tell the different vectors apart. Reliable numerical procedures then often involve frequent re-setting using a Gram-Schmidt orthonormalization procedure. 6

7 y (x 4,y 4 ) y (x 3,y 3 ) (x 2,y 2 ) (x,y ) x x Figure 3: Left: Four points (x n, y n ) of a trajectory of a 2D map, (x n+, y n+ ) = (f(x n, y n ), g(x n, y n )). Right: The co-moving tangent space spanned by (δ n, η n ). The arrow in the coordinate systems illustrates the possible behaviour of a perturbation vector: (i) its magnitude grows in each iteration, (ii) its direction in 2D tangent space changes over time and (iii) the δ n -component shrinks, but the η n component grows. This system has one positive Liapunov exponent (corresponding to the growth of perturbations in δ-direction, and one negative Liapunov exponent, corresponding to the decay of perturbations in the η-direction. 7