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Do numerical trajectories represent true trajectories?
Nuclear Physics B (Proc. Suppl.) 2 (1987) 601 North-Holland, Amsterdam
DO NUMERICAL
601
T R A J E C T O R I E S R E P R E S E N T T R U E TRAJECTORIES?
Stephen M. Harnmel, I James A. Yorke, 2 and Ceiso Orebogl I University of Maryland, College Park, Maryland 20742 Chaotic processes have the property t h a t relatively small numerical errors tend to grow exponentially fast.
In an i t e r a t e d process, if errors double each iterate and
numerical calculations have 50-bit (or 15 digit) accuracy, a true t r a j e c t o r y through a poInt can be expected to have no correlation with a numerical t r a j e c t o r y a f t e r 50 iterates.
On the other hand, numerical studies o f t e n Involve hundreds of thousands of
iterates. It is therefore of crucial importance to uvAerstand how much of what we see In a picture of a chaotic a t t r a c t o r is an a r t i f a c t due to chaos-amplified rouadoff error. While a numerical t r a j e c t o r y will diverge rapidly from the true t r a j e c t o r y with the same initial poInt, there o f t e n exists a different true t r a j e c t o r y with a slightly different initial point which stays near the noisy t r a j e c t o r y for a long time.
Anosov and Bowen showed
t h a t systems which are uniformly hyperbolic will have the shadowing property:
a
numerical (or noisy) t r a j e c t o r y will stay close (shadow) a true t r a j e c t o r y for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report
rigorous results
for non-hyperbolic systems:
numerical trajectories
typically can be shadowed by true trajectories for long time periods. We have rigorous numerical procedures to test whether there exists a true t r a j e c t o r y near the noisy t r a j e c t o r y for a long time.
Calculations were done on a Cray X-MP which thereby
defines the roundoff procedure (= 14 digit precision). With the help of the computer we were able to prove the following results for the logistic map x n = aXn(1 - Xn). Let Pn denote the nth point of the C r a y - g e n e r a t e d t r a j e c t o r y of the logistic map using a = 3.8 and initial point P0 = 0.4. THEOREM. There is a true t r a j e c t o r y { Xn} of the logistic map for which Pn is within a distance of 10- 7 of x n for 107 iterates (i.e., for each n = 0,1,2 ..... 107). We
aiso
have
a
similar
theorem
for
the
H6non map.
More
generally,
if
low-dimensional maps are i t e r a t e d with 2M-digit accuracy, we expect there to exist true trajectories staying within 10-M of the numerical t r a j e c t o r y for an average of ~ 10M iterates before separation. 1. Laboratory for Plasma and Fusion Energy Studies. 2. Institute for Physical Science and Technology and Department of Mathematics. 0920-5632/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
DO NUMERICAL
601
T R A J E C T O R I E S R E P R E S E N T T R U E TRAJECTORIES?
Stephen M. Harnmel, I James A. Yorke, 2 and Ceiso Orebogl I University of Maryland, College Park, Maryland 20742 Chaotic processes have the property t h a t relatively small numerical errors tend to grow exponentially fast.
In an i t e r a t e d process, if errors double each iterate and
numerical calculations have 50-bit (or 15 digit) accuracy, a true t r a j e c t o r y through a poInt can be expected to have no correlation with a numerical t r a j e c t o r y a f t e r 50 iterates.
On the other hand, numerical studies o f t e n Involve hundreds of thousands of
iterates. It is therefore of crucial importance to uvAerstand how much of what we see In a picture of a chaotic a t t r a c t o r is an a r t i f a c t due to chaos-amplified rouadoff error. While a numerical t r a j e c t o r y will diverge rapidly from the true t r a j e c t o r y with the same initial poInt, there o f t e n exists a different true t r a j e c t o r y with a slightly different initial point which stays near the noisy t r a j e c t o r y for a long time.
Anosov and Bowen showed
t h a t systems which are uniformly hyperbolic will have the shadowing property:
a
numerical (or noisy) t r a j e c t o r y will stay close (shadow) a true t r a j e c t o r y for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report
rigorous results
for non-hyperbolic systems:
numerical trajectories
typically can be shadowed by true trajectories for long time periods. We have rigorous numerical procedures to test whether there exists a true t r a j e c t o r y near the noisy t r a j e c t o r y for a long time.
Calculations were done on a Cray X-MP which thereby
defines the roundoff procedure (= 14 digit precision). With the help of the computer we were able to prove the following results for the logistic map x n = aXn(1 - Xn). Let Pn denote the nth point of the C r a y - g e n e r a t e d t r a j e c t o r y of the logistic map using a = 3.8 and initial point P0 = 0.4. THEOREM. There is a true t r a j e c t o r y { Xn} of the logistic map for which Pn is within a distance of 10- 7 of x n for 107 iterates (i.e., for each n = 0,1,2 ..... 107). We
aiso
have
a
similar
theorem
for
the
H6non map.
More
generally,
if
low-dimensional maps are i t e r a t e d with 2M-digit accuracy, we expect there to exist true trajectories staying within 10-M of the numerical t r a j e c t o r y for an average of ~ 10M iterates before separation. 1. Laboratory for Plasma and Fusion Energy Studies. 2. Institute for Physical Science and Technology and Department of Mathematics. 0920-5632/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)