12] Paul J. Schweitzer. Perturbation theory and nite. Markov chains. Journal of Applied Probability, 5(3):401{. 404, 1968. 13] Alistair I. Mees. Dynamical systems ...
Computing Physical Invariant Measures Gary Froyland Department of Mathematical Engineering The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, 113 Tokyo, JAPAN
Abstract| We consider the problem of describing the long term behaviour of a dynamical system in a statistical sense. For many chaotic systems, the asymptotic distribution of points in a trajectory fT i(x)g1 i=0 often appears to be independent of x. We present new results on the numerical approximation of this common asymptotic distribution, called the physical invariant measure of T . I. Introduction & Problem Description
Let (M; T ) be a discrete dynamical system, where T : M ?is a continuous mapping from some smooth space M to itself. The system (M; T ) will often arise as a model of some dynamical process, be it physical, chemical, or biological. The map T and the phase space M used in the model may have been obtained: (i) directly from physical (or other) principles, or (ii) indirectly from a stream of observed data from a physical (or other) process, combined with standard embedding techniques. Our concern is with producing a numerical estimate of the long term distribution of most orbits of our system (M; T ). The computational techniques we present may be used both in situations where the map T is known (case (i) above), and in situations where one must reconstruct the map from a given set of time series data (case (ii) above). In the latter situation, our method provides an automatic way to reconstruct the dynamics.
A. Physical Invariant Measures
For simplicity, we will assume that M is a compact subset of Rd . Let E (M; T ) denote the set of ergodic T -invariant Borel probability measures on M ; typically, E (M; T ) is an in nite set. Elements of E (M; T ) may be thought of as distributions of mass on M that are invariant under the action of T . Each of these invariant mass distributions describes the asymptotic behaviour of one or more orbits of T . Formally, 2 E (M; T ) satis es the property that cardfi : T i(x) 2 A; 1 i ng=n ! (A) (1)
This work was completed at The University of Western Australia
as n ! 1 for -almost all x 2 M and every measurable A M . That is, the frequency with which -almost all orbits enter a set A is given by the measure of A with respect to . In this way, the invariant measure provides a complete description of the long term behaviour of -almost all orbits of T . However, the orbits that a given invariant measure describes may not be a particularly large proportion of the totality of orbits for the system (M; T ). For example, if x0 is a xed point of T , then the delta measure x0 is a T -invariant probability measure and an element of E (M; T ), but only describes the long term behaviour of the xed point x0 . We really would like to nd a measure so that (1) holds for Lebesgue-almost all x 2 M , as this would mean that describes the long term distribution of a \randomly selected" starting point. If there is a such a measure , it must be unique, and we will call it the physical invariant measure of the system (T; M ). It is the physical measure that shows up in most computer simulations of a system.
II. Finite Approximation using Markov Chains For many stochastic systems, one does not have the problem of having to choose one of in nitely many invariant measures; often (in the case of ergodic systems, for example) the stochastic system has a unique invariant measure. We will make use of this fact to select an invariant measure from E (M; T ) by adding some random noise to T . Thiswill produce a stochastic dynamical system T : M ?, where is a measure of the magnitude of the noise. We will do this in such a way that (M; T ) is an ergodic stochastic system, so that T has a unique invariant probability measure, denoted . We then let the noise go to zero ( ! 0), and extract a limiting measure = lim!0 . The idea of selecting an invariant measure for a deterministic system as a limit of measures of noisy systems goes back to Kolmogorov, and the limiting measure is sometimes called a Kolmogorov measure. There are two main conditions that must be satis ed for such a technique to useful. Firstly, the invari-
ant measures must be easier to compute than the physical measure , otherwise there is no point to introducing the noisy systems. Secondly, there must be some associated theory to show that the limiting measure is actually the physical measure of T .
A. An easily computable
We may easily satisfy the rst condition by choosing a type of noise that will produce a stochastic system describable by a nite state Markov chain. In this case, the unique invariant measure may be simply computed as a left eigenvector of the transition matrix for our Markov chain. A straightforward way to impose a nite description of the dynamics on M is to partition M into a nite number of connected, positive Lebesgue measure sets Pn := fA1 ; : : : ; An g and to consider the movement of points between these sets. The diameter of the partition sets will determine the amount of noise added; to decrease the magnitude of the noise , we need to decrease the maximum size of our partition sets, in other words, increase n = n(). For concreteness, de ne a transition matrix ?1 Pn;ij = m(Aim\(AT ) Aj ) ; (2) i where m is the standard volume measure on M . The number Pn;ij represents the probability that a point x 2 Ai will move into Aj in one iteration; that is, Pn;ij = Prob(T (x) 2 Aj jx 2 Ai ). We may now introduce a stochastic system on M with transition function Pn : M B(M ) ! [0; 1] de ned by
Pn(x; E ) =
m(E \ Aj ) P ; n;ix j m(Aj ) i=1
n X
(3)
where Aix is the unique partition set containing x. A simulation of the noisy system T would proceed as follows (see Figure 1): (i) Suppose that x 2 Aix , (ii) Choose an image set Aj according to probability Pn;ix j , (iii) Randomly select a point y 2 Aj (using a uniform distribution on the partition set Aj ), (iv) Put y = T (x). It is relatively straightforward to check that the Markov chain Pn has an invariant measure n given by
n (E ) = m(mE(A\ A) i ) pn;i ; i i=1 n X
(4)
where pn is the unique xed vector of Pn , (pn Pn =
pn ).
T Tx x
Figure 1: Our random perturbation. Darker partition sets represent a higher probability of the set containing the random image of x. Our construction above is a generalisation of a method originally proposed by Ulam [1] to approximate absolutely continuous invariant measures of interval maps. The matrix Pn;ij may also be thought of as a nite-dimensional approximation of the PerronFrobenius operator for T [1, 2].
B. A good
While it may be shown that the limiting measure is T -invariant ([3, 4, 5]), there is no guarantee that is anything like the physical measure of T . In fact, it is shown in [6] that by simply modifying the nonzero entries of Pn;ij , one may produce an in nity of nonphysical limiting measures . Still, numerical experiments suggest [7, 5, 8] (and in restricted situations, mathematical theory proves [2, 9, 10]) that the limiting measure is the physical measure when Pn;ij is de ned as in (2). Whether or not = in the generality we are considering is still an open problem, and it is this question that we discuss here. From the above remarks, it appears that the Lebesgue measure of the fraction of Ai that moves into Aj in one step, is a good choice for Pij . Why is this the case? We restrict our attention to systems for which m; that is, has a density with respect to Lebesgue that is bounded above, and away from zero. Let us suppose that we use a di erent transition matrix to de ne another noisy system P~ n , namely ?1
P~n;ij = (Aim\(AT ) Aj ) : i
(5)
It is easy to check that the invariant measure of the Markov chain governed by P~ n is
~n (E ) =
m(E \ Ai ) (A ): i m(Ai ) i=1
n X
(6)
So, in fact ~n gives exactly the correct weight to each partition set Ai , namely (Ai ). In this sense, the matrix P~n is the best nite approximation to the dynamics of T on the partition Pn . It may be shown (Lemma 3.6 [11]) that the entries of Pn and P~n are close, in fact jP~n;ij ? Pn;ij j =
O(n?1=d ) provided that T is C 1+Lip and the elements then the sequence of invariant measures fng1 n=n0 of Pn are all roughly the same shape and size. In fact, the two matrices P~n and Pn are close in the L1 matrix norm, with kP~n ? Pn k1 = O(n?1=d ) under the same conditions. This suggests that the xed vector p~n of P~n (recall p~n;i = (Ai )) is close the xed vector pn of Pn , which in turn tells us that ~n is close to n . To make this reasoning rigorous requires a good deal of work.
III. Mixing Properties
An important inequality concerning the di erence p~n ? pn is [12] kp~n ? pn k1 kP~n ? Pn k1 kZnk1 ; (7)
k where Zn := (I ? (Pn ? Pn1 ))?1 = I + 1 k=1 (Pn ? 1 Pn ) is the fundamental matrix of the Markov chain governed by Pn . We see that the multiplier kZnk1 depends on the rate at which Pnk approaches the limiting equilibrium matrix Pn1 (which has pn as its rows). That is, the faster our Markov chain approaches equilibrium from some non-equilibrium state, the less sensitive the equilibrium vector pn is to perturbations in the entries of the transition matrix Pn . In other words, the faster our stochastic system mixes the state space, the more robust its invariant density is to perturbations. Now, we already know that kP~n ? Pn k1 = O(n?1=d ), so if kZnk1 = O(log n), we would have that kp~n ? pnk1 ! 0 as n ! 1 (as we re ne our partition sets). This will mean that the sequence of invariant measures fn g of our randomly perturbed systems converges to the physical measure of the original system, as we decrease the noise to zero. In general, it is dicult to connect the mixing rates of stochastically perturbed systems with their unperturbed counterparts. A rigorous result is the following [11]. Theorem 1: Let M Rd and T : M ? be a C 1+Lip map with a unique absolutely continuous invariant measure . Suppose that the density of is strictly positive and that T is mixing with respect to . Let fPn g1 n=n0 be a sequence of (n n) transition matrices generated by (2) from a sequence of regular partitions fPn g1 n=n0 whose maximal element diameter goes to zero as n ! 1. De ne Rn and rn to be the least values satisfying P
max 1in
n X j
=1
Pijk ? pj Rn rnk
for all n; k 0.
If (i) rn r, for all n n0 and some r < 1, and (ii) Rn = O(np ); p > 0
de ned by (4) converges strongly to the unique absolutely continuous measure of T . The rate of convergence is O(log n=n1=d). Condition (i) says that we may nd a universal bound r for the rate of mixing of each of the stochastic systems Pn . This means that the rate of approach to equilibrium is no slower than O(rk ) for every system Pn , n n0 . Such a property seems reasonable as one would not expect the mixing rate of a system to decrease signi cantly when a small amount of noise is added. Condition (ii) says that the constant Rn grows no faster than polynomially with the number of partition sets. As the n n matrices Pn grow in size with n, it is to be expected that the corresponding Markov chains Pn will take longer to reach equilibrium as the starting distributions may be concentrated in partitions sets Ai of ever decreasing size. Numerical experiments [11] suggest that conditions (i) and (ii) will often be satis ed, and it appears that such a method of proving that n ! has the potential to be applied to a very wide class of systems. It has been proven [11] that conditions (i) and (ii) are satis ed for multidimensional piecewise linear expanding Markov maps and piecewise C 1+Lip expanding interval maps.
IV. An Example: Computing Physical Measures from Time Series
As an illustration of Ulam's method of approximation, weapply it to a nonlinear torus map T : S 1 S 1 ? de ned by T (x; y) = (y + 0:1 sin(2x) + 2x + 0:2 cos(2x); x + 0:1 cos(2x)) (mod 1): (8) A plot of a 10000 point orbit of T is shown in Figure 2. The trajectory appears chaotic and the distribution of points is typical when compared with other orbits begun at di erent initial points. We would like to approximate this density. To make things a little more dicult, we are only going to be allowed to observe a time series of length 11. From these 11 points, we will reconstruct a map, and use this reconstructed map to de ne our transition matrix P . We use the tessellation reconstruction method [13] to simultaneously partition the torus into triangles, and produce a piecewise linear reconstruction of T which is linear on each triangle; see also [14]. The resulting estimate of the invariant density is shown in Figure 3, where the vertices of the triangles are 10 of the 11 data points. This partition was re ned by splitting up the triangles into smaller ones, producing a partition of 200 triangles. The new estimate of the invariant density is shown in Figure 4. This density compares
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Figure 2: Orbit of length 10 000 generated by
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Figure 3: Absolutely continuous approximation of the invariant measure of (8) using an 11 point orbit and a partition of 20 triangles. Darker shades of grey represent regions of higher density. favourably with the distribution of points in Figure 2. The reader should keep in mind that this approximation was obtained from a time series of just 11 points.
V. Acknowledgements
This work was partially supported by a grant from the Australian Research Council.
References
[1] S. M. Ulam. Problems in Modern Mathematics. Interscience, 1964. [2] Tien-Yien Li. Finite approximation for the FrobeniusPerron operator. A solution to Ulam's conjecture. Journal of Approximation Theory, 17:177{186, 1976. [3] R.Z. Khas'minskii. Principle of averaging for parabolic and elliptic di erential equations and for Markov processes with small di usion. Theory of Probability and its Applications, 8(1):1{21, 1963.
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Figure 4: Absolutely continuous approximation of the invariant measure of the nonlinear torus map (8) using an 11 point orbit. [4] Yuri Kifer. Random Perturbations of Dynamical Systems, volume 16 of Progress in Probability and Statistics. Birkhauser, Boston, 1988. [5] Gary Froyland, Kevin Judd, Alistair I. Mees, Kenji Murao, and David Watson. Constructing invariant measures from data. International Journal of Bifurcation and Chaos, 5(4):1181{1192, 1995. [6] Gary Froyland. Estimating Physical Invariant Measures and Space Averages of Dynamical Systems Indicators. PhD thesis, The University of Western Australia, Perth, 1996. Available at http://maths.uwa.edu.au/gary/. [7] Tohru Kohda and Ryuji Shinji. A simple algorithm to approximate asymptotic measure on chaotic attractor of Henon map. In Proceedings of the 1992 Symposium on Nonlinear Theory and its Applications, Kanagawa, July 1992, pages 99{102, 1992. [8] C. S. Hsu. Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems, volume 64 of Applied Mathematical Sciences. Springer-Verlag, New York, 1987. [9] Jiu Ding and Ai Hui Zhou. Piecewise linear Markov approximations of Frobenius-Perron operators associated with multi-dimensional transformations. Nonlinear Analysis, Theory, Methods & Applications, 25(4):399{408, 1995. [10] Gary Froyland. Finite approximation of Sinai-BowenRuelle measures of Anosov systems in two dimensions. Random & Computational Dynamics, 3(4):251{ 264, 1995. [11] Gary Froyland. Approximating physical invariant measures of mixing dynamical systems in higher dimensions. Nonlinear Analysis, Theory, Methods, & Applications. In press. [12] Paul J. Schweitzer. Perturbation theory and nite Markov chains. Journal of Applied Probability, 5(3):401{ 404, 1968. [13] Alistair I. Mees. Dynamical systems and tesselations: Detecting determinism in data. International Journal of Bifurcation and Chaos, 1(4):777{794, 1991. [14] Alistair I. Mees, Kenji Murao, Kevin Judd, and Gary Froyland. Triangulations on tori and density estimation. In Proceedings of the 1993 International Symposium on Nonlinear Theory and its Applications, Hawaii, December 1993, volume 1, pages 275{280, 1993.
