Chow, S.-N.

Palmer, K. J.

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7Shui-Nee Chow; K. J. PalmerThe accuracy of numerically computed orbits of dynamical systemsIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 74--76.Persistent URL: http://dml.cz/dmlcz/702394Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czTHE ACCURACYX)F NUMERICALLY COMPUTED ORBITS OF DYNAMICAL SYSTEMS CHOW S.-N., ATLANTA, GA, U.S.A. PALMER K.J., CORAL GABLES, FL, U.S.A. In their papers [2,3] Hammel, Yorke and Grebogi have given a procedure which determines the accuracy of numerically computed orbits of dynamical systems. They apply their procedure to maps which exhibit a large amount of hyperbolicity. However their procedure does not use the hyperbolicity explicitly. In this paper we give a procedure for one-dimensional maps which does use the hyperbolicity explicitly. Unlike the procedure of Hammel et al., our procedure works forward. After N iterates we can decide whether our theorem applies and, if it does, we can estimate how far the computed orbit is from a true orbit. Now we state the main theorem. Let / : [0,1] —> [0,1] be a C2 function and let {^n/n-J)1 be a pseudo-orbit of this map, i.e., | yn+i — f(yn) \ is small for n = 0 ,1 , ...,jV. We define the quantities a = s ip £ | Df(yn)-1Df(yn+1)-1...Df(ym)-1 |, n=0 m=n which measures the expansiveness of the map, and H I "=o m=n т = sup | £ Df(yn)-1Df(yn+1)-1...Df(ym)-1\ym+1 - f(ym)} | It turns out that r gives a good measure of how close the pseudo-orbit (of course, our numerically computed orbits will be pseudo-orbits) is to a true orbit. THEOREM. Let f : [0,1] -+ [0,1] be a C2 function with M = sup{| D2f(x) |: 0 < x < 1}. Let {y^n^o1 be a pseudo-orbit of f such that 2MOT < 1. Then there is an exact orbit {xn}n=0 with (1 + 1/2(1 + v / 1 - 2 M a r ) ) - 1 r < sup | xn - yn |< 2(1 + y/l-2MCT)~1T. n=0 Outline of proof. Denote by S the set of sequences x = {xn}%=0 with | xn - yn |< e for n = 0,1,..., N, where s = 2r/(l + ^/I~-2MOT). S is a compact convex subset of HN+1. We define a mapping T on S. If x £ S we define (Tx)n = yn - E Df(yn)-1Df(yn+1)-1....Df(ym)-1hm (n = 0 , . . . ,JV) , m=n where hn = f(xn) - yn+i - Df(yn)(xn - yn). 74 It turns out that T is a continuous mapping of $ into itself and so, by Brouwer's fixed point theorem, has a fixed point x = {xn}N=0. This is the exact orbit that we wanted. Note that the idea of this proof was suggested by the proofs of the shadowing lemma given in Palmer [4] and Chow, Lin and Palmer [l]. Tht Mtthod of Computation Let / : [0,1] —> [0,1] be a C2 mapping. Suppose our computer starts with a number y0 in [0,1] and computes an orbit {yn}N=o of / in single precision. {yn} will be, in fact, a pseudo-orbit. To use the theorem we have to find the quantities a and r. For large N it would not be practical to compute the sums Y^m=n- Instead we calculate the quantities min(n+p,N) a, = sffp £ \Df(yn)-K..Df(ym)-i\, "=0 m=n min(n+p,N) r, = sffp | £ #/(y„)-1....D/(2/,.)-1[ym+i - /(y»)l | , "=° m=n where p is an integer, 0 < p < N, such that It, = tm; | Df(yn)-\..Df(yn+t)-1 |< 1. n=0 It turns out that » < ( l - d , ) - ' « - t < ( l - ( S ) " V (-) The computation of fipiapiTp is done in double precision. We have fully analyzed the effect of round-off error on these computations. Unless the hyperbolicity if very weak (i.e. a is large and np < 1 only for large p), it turns out that the effect of round-off error is very slight. Examplt. We consider the quadratic map f(x) = ax(\ — x) with a = 3.8. Then M = 2a = 7.6. The computations were done on an IBM compatible computer using Microsoft Quickbasic. For N = 426,000,p = 30 and yQ = .3, we find that \ip = 2.297433184600331 * 10"3, ap = 375.6005726956602, TP = 9.60282364278178 * 10"6. Using the inequalities (l) and taking into account the round-off error, we find that a < 376.4658, r < 9.624939 * 10~6. Then 2Mar < .05507661 and . 2r/(l + yj\ - 2MOT) < 9.76125 * 10"6. Our theorem enables us to conclude that during 426,000 iterates our computed orbit differs by at most 1/105 from a true orbit. Note that the orbit was computed only in single precision, that is to an approximate accuracy of 7 decimal digits. So over 426,000 iterates we have only lost two digits of accuracy. 75 References 1. S.N, CHOW, X.B. LIN and K.J. PALMER, A shadowing lemma with applicàtions to semilinear parabolic equations, SIAM J. Math. Anal.. 20(1989). 2. S.M. HAMMEL, J.A. YORKE and C GREBOGI, Do numerical orbits of chaotic dy-namical processes represent true orbits, J. Complexity 3(1987), 136-145. 3. S.M. HAMMEL, J.A. YORKE and C GREBOGI, Numerical orbits of chaotic processes represent true orbitsy Bull. Ameг. Math. Soc. 19(1988), 465-470. 4. K.J. PALMER, Exponential dichotomies, the shadowing lemma and transversal homo-clinic points, Dynamics Repoгted 1(1988), 265-306. 76 

The accuracy of numerically computed orbits of dynamical systems

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The accuracy of numerically computed orbits of dynamical systems

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Institute of Mathematics AS CR, v. v. i.