Computers, used in what I shall call the “heuristic mode,” can greatly enhance our understanding of the mathematics of nonlinear dynamical processes—a field that is taking on increasing importance as we explore the complex behavior that even simple systems often exhibit. As I shall try to show, computers allow us to penetrate into unexplored regions of mathematics and to discover unforeseen linkages among ideas.

1.
H. H. Goldstine, The Computer from Pascal to von Neumann, Princeton U.P., Princeton, N.J. (1972).
2.
H. H. Goldstine, J. von Neumann, in Collected Works of John von Neumann, A. Taub, ed., MacMillan, New York, (1963), volume 5, page 1.
Another formulation is in a 1949 paper. Collected Works, volume 6, page 437.
3.
S. M. Ulam, A Collection of Mathematical Problems, Wiley–Interscience, New York (1960).
4.
N. J.
Zabusky
,
J. Comput. Phys.
43
,
195
(
1981
). This comprehensive review paper contains an extensive bibliography of papers and monographs on computational synergetics.
5.
E. Fermi, J. R. Pasta, S. M. Ulam, Los Alamos Report LA‐1940, May 1955;
reprinted in E. Fermi, Collected Works, U. Chicago, P. Chicago (1965), volume 2, page 978.
6.
N. J. Zabusky, in Proc. Conf. Mathematical Models in the Physical Sciences, S. Drobot, ed., Prentice–Hall, Englewood Cliffs, N.J. (1963), page 99.
7.
M. D. Kruskal, Proc. IBM Scientific Symposium—Large Scale Problems in Physics, 9–11 December 1963, IBM, Armonk, N.Y. (1965).
8.
R. M.
Miura
,
SIAM Rev.
18
,
412
(
1976
).
J. W.
Miles
,
J. Fluid Mech.
106
,
131
(
1981
).
9.
N. J.
Zabusky
,
M. D.
Kruskal
,
Phys. Rev. Lett.
15
,
240
(
1965
).
10.
M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM Philadelphia (1981).
11.
C. S.
Gardner
,
J. M.
Greene
,
M. D.
Kruskal
,
R. M.
Miura
,
Phys. Rev. Lett.
19
,
1095
(
1967
).
12.
M.
Toda
,
J. Phys. Soc. Japan
22
,
431
(
1967
);
M.
Toda
,
J. Phys. Soc. Japan Suppl
26
,
235
(
1969
);
M. Toda, Theory of Nonlinear Lattices, Springer‐Verlag, Berlin (1980) (Series in Solid State Sciences, volume 20).
13.
N. J. Zabusky, in The Significance of Nonlinearity in the Natural Sciences, B. Kursunoglu, A. Perlmutter, L. F. Scott, eds., Plenum, New York (1977), page 145.
14.
J. C. McWilliams, J. Fluid Mech., (1984) in press.
15.
N. J.
Zabusky
,
G. S.
Deem
,
J. Fluid Mech.
47
,
353
(
1971
).
16.
C.‐M.
Ho
,
P.
Huerre
,
Ann. Rev. Fluid Mech.
16
,
365
(
1984
).
F. P. Browand, C.‐M. Ho, J. Mécanique (1983) in press.
17.
J. P.
Christiansen
,
N. J.
Zabusky
,
J. Fluid Mech.
61
,
219
, (
1973
).
18.
N. J.
Zabusky
,
M. H.
Hughes
,
K. V.
Roberts
,
J. Comput. Phys.
30
,
96
(
1979
).
H.‐M.
Wu
,
E. A.
Overman
, II
,
N. J.
Zabusky
,
J. Comput. Phys
53
,
42
(
1984
).
19.
A. Koestler, The Act of Creation, Hutchison, London, 1964, page 454.
20.
M. L.
Norman
,
L. L.
Smarr
,
K.‐H. A.
Winkler
,
M. D.
Smith
,
Astron. Astrophys.
113
,
285
(
1982
);
M. L. Norman, K.‐H. A. Winkler, L. L. Smarr, in Astrophysical Jets, A. Ferrari, A. G. Pacholczyk, eds., Reidel, Boston (1983) 227;
M. L. Norman, L. L. Smarr, K.‐H. A. Winkler, in Numerical Astrophysics, a Festschrift in Honor of James R. Wilson, J. Centrella, R. Bowers, J. Leblanc, M. Leblanc, eds. Jones and Bartlett, Boston (1984);
M. L. Norman, K.‐H. A. Winkler, in Astrophysical Radiation Hydrodynamics, K.‐H. A. Winkler, M. L. Norman, eds., Reidel, Boston (1984).
21.
Figure 5 on supersonic jets and the cover figure have not been previously published and were provided by M. L. Norman, K.‐H. A. Winkler, and L. L. Smarr. A review of their work with 30 colored panels is in L. L. Smarr, M. L. Norman, K.‐H. A. Winkler, Physica D (1984), in press. Figure 6 is from work in progress by Norman, Winkler and Zabusky.
22.
For some vital thoughts on how things could be improved, see
P. J.
Davis
,
J. A.
Anderson
,
SIAM Rev.
21
,
112
(
1979
).
23.
E.
Chargaff
,
Science
172
,
637
(
1971
).
24.
C. J. Jung, W. Pauli, The Interpretation of Nature and the Psyche, Pantheon, New York (1955), page 151.
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