|
The dynamical systems approach to differential equations
Author:
Morris W. Hirsch
Journal:
Bull. Amer. Math. Soc. 11 (1984), 1-64
MSC (1980):
Primary 00A25, 00A99, 01A45, 01A55, 01A60, 34A40, 34C35, 34D10, 34D30, 35B05, 35B35, 35B40, 35B50, 35B30, 35K55, 54H20, 58F25, 58F40, 58F12, 58F10, 58D25, 58D07, 46A40, 46E10, 47H07, 47H20, 92A15, 92A17; Secondary 06F30, 35B65, 58F13, 35J60, 46E05, 92-03, 90-03, 90A16
MathSciNet review:
741723
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-
R. Abraham, 1971 Predictions for the future of differential equations, Lecture Notes in Math., vol. 206 (D. Chillingworth, ed.), Springer-Verlag.
-
R. Abraham and J. Marsden, 1967 Foundation of mechanics, Benjamin, New York.
-
Ralph
Abraham and Jerrold
E. Marsden, Foundations of mechanics, Benjamin/Cummings
Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978. Second
edition, revised and enlarged; With the assistance of Tudor Raţiu
and Richard Cushman. MR 515141
(81e:58025)
-
Ralph
H. Abraham and Christopher
D. Shaw, Dynamics—the geometry of behavior. Part 1,
Visual Mathematics Library: Vismath, vol. 1, Aerial Press Inc., Santa
Cruz, Calif., 1982. Periodic behavior. MR 669376
(84m:58001)
-
Herbert
Amann, Fixed point equations and nonlinear eigenvalue problems in
ordered Banach spaces, SIAM Rev. 18 (1976),
no. 4, 620–709. MR 0415432
(54 #3519)
-
A. A. Andronov and L. Pontryagin, 1937 Systèmes grossiers, Dokl. Akad. Nauk. SSSR 14, 247-251.
-
D.
V. Anosov, Roughness of geodesic flows on compact Riemannian
manifolds of negative curvature, Dokl. Akad. Nauk SSSR
145 (1962), 707–709 (Russian). MR 0143156
(26 #716)
-
D.
V. Anosov, Geodesic flows on closed Riemannian manifolds of
negative curvature, Trudy Mat. Inst. Steklov. 90
(1967), 209 (Russian). MR 0224110
(36 #7157)
-
V.
I. Arnol′d, Geometrical methods in the theory of ordinary
differential equations, Grundlehren der Mathematischen Wissenschaften
[Fundamental Principles of Mathematical Science], vol. 250,
Springer-Verlag, New York, 1983. Translated from the Russian by Joseph
Szücs; Translation edited by Mark Levi. MR 695786
(84d:58023)
-
Gunnar
Aronsson and Ingvar
Mellander, A deterministic model in biomathematics. Asymptotic
behavior and threshold conditions, Math. Biosci. 49
(1980), no. 3-4, 207–222. MR 573187
(81i:92016), http://dx.doi.org/10.1016/0025-5564(80)90079-6
-
D. Berlinski, 1976 On systems analysis, MIT Press, Cambridge, Mass.
-
G. D. Birkhoff, 1920 Recent advances in dynamics, Science (N.S.) 51, 51-55 and Collected mathematical papers, vol. 2, Amer. Math. Soc., Providence, R. I., pp. 106-110.
-
George
D. Birkhoff, Dynamical systems, With an addendum by Jurgen
Moser. American Mathematical Society Colloquium Publications, Vol. IX,
American Mathematical Society, Providence, R.I., 1966. MR 0209095
(35 #1)
-
George
D. Birkhoff, On the periodic motions of dynamical systems,
Acta Math. 50 (1927), no. 1, 359–379. MR
1555257, http://dx.doi.org/10.1007/BF02421325
-
G. D. Birkhoff, 1932 Sur l’existence de régions d’instabilité en dynamique, Ann. Inst. H. Poincaré 2, 369-386 and Collected mathematical papers, vol. 2, op. cit., pp. 444-461.
-
George
D. Birkhoff, Sur le problème restreint des trois corps
(premier mémoire), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2)
4 (1935), no. 3, 267–306 (French). MR
1556760
-
George
D. Birkhoff, Some unsolved problems of theoretical dynamics,
Science 94 (1941), 598–600. MR 0006260
(3,279j)
-
G. D. Birkhoff, 1935a Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Sci. Nov. Lyn. 1, 85-216 and Collected mathematical papers, vol. 2, op. cit., pp. 530-661.
-
George
D. Birkhoff, The mathematical nature of physical theories,
American Scientist 31 (1943), 281–310. MR 0009000
(5,86f)
-
G. D. Birkhoff, 1950 Collected mathematical papers, vols. 1, 2, 3, Amer. Math. Soc., Providence, R. I.
-
G. D. Birkhoff and B. O. Koopman, 1932 Recent contributions to ergodic theory, Proc. Nat. Acad. Sci. U.S.A. 18, 279-282 and Collected mathematical papers, op. cit., vol. 3, pp. 462-465.
-
L. E. J. Brouwer, 1907 On the foundation of mathematics, Collected Mathematical Works, Vol. 1, American Elsevier, New York.
-
L. E. J. Brouwer, 1913 Intuitionism and formalism, Bull. Amer. Math. Soc. 20, 81-96; Collected works, Vol. 1, op. cit., pp. 123-138.
-
L.
E. J. Brouwer, Collected works. Vol. 1, North-Holland
Publishing Co., Amsterdam, 1975. Philosophy and foundations of mathematics;
Edited by A. Heyting. MR 0532661
(58 #27140)
-
John
L. Casti, Recent developments and future perspectives in nonlinear
system theory, SIAM Rev. 24 (1982), no. 3,
301–331. MR
667511 (84h:93035), http://dx.doi.org/10.1137/1024065
-
T. M. Cherry, 1938 Analytic quasi-periodic curves of discontinuous type on a torus, Proc. London Math. Soc. (2) 44, 175-215.
-
T.
M. Cherry, Asymptotic solutions of analytic Hamiltonian
systems, J. Differential Equations 4 (1968),
142–159. MR 0223678
(36 #6726)
-
W.
A. Coppel, Stability and asymptotic behavior of differential
equations, D. C. Heath and Co., Boston, Mass., 1965. MR 0190463
(32 #7875)
-
J.
Coste, J.
Peyraud, and P.
Coullet, Asymptotic behaviors in the dynamics of competing
species, SIAM J. Appl. Math. 36 (1979), no. 3,
516–543. MR
531612 (80d:92022), http://dx.doi.org/10.1137/0136039
-
D´Alembert and Diderot (Editors), 1754 Encyclopédie, Vol. 4, Briasson, David, LeBreton, Durand, Paris.
-
Darwin, 1892 Life of Charles Darwin (F. Darwin, ed.), John Murray, London.
-
Darwin, 1936 The origin of species and The descent of man, Modern Library, New York.
-
Arnaud
Denjoy, Sur les courbes définies par les équations
différentielles, Advancement in Math. 4
(1958), 161–187 (Chinese). MR 0101369
(21 #180)
-
Stillman
Drake, Galileo at work, University of Chicago Press, Chicago,
Ill., 1978. His scientific biography. MR 531510
(83m:01018)
-
A. S. Eddington, 1927 The nature of the physical world, Cambridge Univ. Press, Cambridge, England.
-
A. Einstein, 1930 On the occasion of the three hundredth anniversary of Kepler’s death, Frankfurter Zeitung, November 9, 1930 and Ideas and opinions, Crown, New York, 1954.
-
A. Einstein, 1931 On the one hundreth anniversary of Maxwell’s birth, James Clerk Maxwell: A Commemorative Volume, Cambridge Univ. Press, Cambridge, England, and Ideas and opinions, op. cit.
-
John
M. Franks, Homology and dynamical systems, CBMS Regional
Conference Series in Mathematics, vol. 49, Published for the
Conference Board of the Mathematical Sciences, Washington, D.C., 1982. MR 669378
(84f:58067)
-
Herbert
I. Freedman, Deterministic mathematical models in population
ecology, Monographs and Textbooks in Pure and Applied Mathematics,
vol. 57, Marcel Dekker Inc., New York, 1980. MR 586941
(83h:92043)
-
H.
I. Freedman and Paul
Waltman, Persistence in models of three interacting predator-prey
populations, Math. Biosci. 68 (1984), no. 2,
213–231. MR
738903 (85h:92037), http://dx.doi.org/10.1016/0025-5564(84)90032-4
-
N. Georgescu-Roegen, 1966 Analytical economics: Issues and problems, Harvard Univ. Press, Cambridge, Mass.
-
Charles
Coulston Gillispie, The edge of objectivity, Princeton
University Press, Princeton, NJ, 1990. An essay in the history of
scientific ideas; Reprint of the 1960 original. MR 1145737
(92j:01078)
-
Walter
Helbig Gottschalk and Gustav
Arnold Hedlund, Topological dynamics, American Mathematical
Society Colloquium Publications, Vol. 36, American Mathematical Society,
Providence, R. I., 1955. MR 0074810
(17,650e)
-
Stephen
Grossberg, Competition, decision, and consensus, J. Math.
Anal. Appl. 66 (1978), no. 2, 470–493. MR 515909
(84c:92003), http://dx.doi.org/10.1016/0022-247X(78)90249-4
-
J.
E. Marsden and M.
McCracken, The Hopf bifurcation and its applications,
Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G.
Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O.
Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S.
Smale; Applied Mathematical Sciences, Vol. 19. MR 0494309
(58 #13209)
-
J. Guckenheimer, 1982 Noise in chaotic systems, Nature 298, 358-361.
-
John
Guckenheimer and George
Buzyna, Dimensional measurements for geostrophic turbulence,
Phys. Rev. Lett. 51 (1983), no. 16, 1438–1441.
MR 718790
(84k:58146), http://dx.doi.org/10.1103/PhysRevLett.51.1438
-
John
Guckenheimer and Philip
Holmes, Nonlinear oscillations, dynamical systems, and bifurcations
of vector fields, Applied Mathematical Sciences, vol. 42,
Springer-Verlag, New York, 1983. MR 709768
(85f:58002)
-
J. Hadamard, 1912 L’oeuvres mathématique de Henri Poincaré, Acta Math. 38, 203-287; and Oeuvres de Jacques Hadamard4, CNRS, Paris, pp. 1921-2005.
-
J. Hadamard, 1912a Henri Poincaré et le problème de trois corps, Rev. Mois 16, 385-418 and Oeuvres de Jacques Hadamard, op. cit., pp. 2007-2041.
-
J. Hadamard, 1968 Oeuvres de Jacques Hadamard4, CNRS, Paris.
-
Jack
K. Hale, Oscillations in nonlinear systems, McGraw-Hill Book
Co., Inc., New York, 1963. MR 0150402
(27 #401)
-
Thomas
L. Hankins, Sir William Rowan Hamilton, Johns Hopkins
University Press, Baltimore, Md., 1980. MR 618834
(82h:01051)
-
Brian
D. Hassard, Nicholas
D. Kazarinoff, and Yieh
Hei Wan, Theory and applications of Hopf bifurcation, London
Mathematical Society Lecture Note Series, vol. 41, Cambridge
University Press, Cambridge, 1981. MR 603442
(82j:58089)
-
R. Heilbroner, 1953 The worldly philosphers, Simon and Schuster, New York.
-
Daniel
Henry, Geometric theory of semilinear parabolic equations,
Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981.
MR 610244
(83j:35084)
-
H. W. Hethcote, 1973 Asymptotic behavior in a deterministic epidemic model, J. Math. Biol. 35, 607-614.
-
H. Hethcote, A. Nold and L. Yorke, 1982 Gonorrhea modelling: a comparison of control methods, Math. Biosci. 58, 93-109.
-
Morris
W. Hirsch, Differential equations and convergence almost everywhere
in strongly monotone semiflows, Nonlinear partial differential
equations (Durham, N.H., 1982) Contemp. Math., vol. 17, Amer. Math.
Soc., Providence, R.I., 1983, pp. 267–285. MR 706104
(84h:34095)
-
M. W. Hirsch, 1983a Systems of differential equations which are competitive or cooperative. II: Convergence almost everywhere, SIAM J. Math. Anal. (in press); preprint, Center for Pure and Appl. Math., Univ. of California, Berkeley.
-
Morris
W. Hirsch, Systems of differential equations which are competitive
or cooperative. I. Limit sets, SIAM J. Math. Anal. 13
(1982), no. 2, 167–179. MR 647119
(83i:58081), http://dx.doi.org/10.1137/0513013
-
S.
B. Hsu, S.
P. Hubbell, and Paul
Waltman, Competing predators, SIAM J. Appl. Math.
35 (1978), no. 4, 617–625. MR 512172
(80a:92047), http://dx.doi.org/10.1137/0135051
-
E.
Kamke, Zur Theorie der Systeme gewöhnlicher
Differentialgleichungen. II, Acta Math. 58 (1932),
no. 1, 57–85 (German). MR
1555344, http://dx.doi.org/10.1007/BF02547774
-
F. Klein, 1911 Lectures on mathematics (Proc. Colloq., Evanston, 1893), Amer. Math. Soc., Providence, R. I.
-
A. Kolmogorov, 1936 Sulla teoria di Volterra della lotta per l´esistenzia, Giorn. Ist. Ital. Attuari 7, 47-80.
-
Ivan
Kupka, Contribution à la théorie des champs
génériques, Contributions to Differential Equations
2 (1963), 457–484 (French). MR 0165536
(29 #2818a)
-
Ana
Lajmanovich and James
A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous
population, Math. Biosci. 28 (1976), no. 3/4,
221–236. MR 0403726
(53 #7537)
-
Robert
M. May and Warren
J. Leonard, Nonlinear aspects of competition between three
species, SIAM J. Appl. Math. 29 (1975), no. 2,
243–253. Special issue on mathematics and the social and biological
sciences. MR
0392035 (52 #12853)
-
S. Lie, 1895 Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung, Leipziger Berichte 47, 53-128 and Sophus Lie Gesammelte Abhandlungen (F. Engel, ed.), Bd. 4, Teubner, Leipzig, 1929, pp. 320-386.
-
E. N. Lorenz, 1963 Deterministic non-periodic flow, J. Atmospheric Sci. 20, 130-141.
-
A. Lotka, 1956 Principles of mathematical biology, Dover, New York.
-
J.
E. Marsden and M.
McCracken, The Hopf bifurcation and its applications,
Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G.
Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O.
Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S.
Smale; Applied Mathematical Sciences, Vol. 19. MR 0494309
(58 #13209)
-
Xavier
Mora, Semilinear parabolic problems define
semiflows on 𝐶^{𝑘}\
spaces, Trans. Amer. Math. Soc. 278 (1983), no. 1, 21–55. MR 697059
(84e:35077), http://dx.doi.org/10.1090/S0002-9947-1983-0697059-8
-
Robert
Edouard Moritz, On mathematics and mathematicians. (Formerly
titled: Memorabilia mathematica or the philomath’s
quotation-book.), Dover Publications Inc., New York, 1958. MR 0104557
(21 #3310)
-
Jürgen
Moser, Stable and random motions in dynamical systems,
Princeton University Press, Princeton, N. J., 1973. With special emphasis
on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced
Study, Princeton, N. J; Annals of Mathematics Studies, No. 77. MR 0442980
(56 #1355)
-
Sheldon
E. Newhouse, Nondensity of axiom 𝐴(𝑎) on
𝑆², Global Analysis (Proc. Sympos. Pure Math., Vol. XIV,
Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970,
pp. 191–202. MR 0277005
(43 #2742)
-
Sheldon
E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth
stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ.
Math. 50 (1979), 101–151. MR 556584
(82e:58067)
-
Sheldon
E. Newhouse, Lectures on dynamical systems, Dynamical systems
(C.I.M.E. Summer School, Bressanone, 1978) Progr. Math., vol. 8,
Birkhäuser Boston, Mass., 1980, pp. 1–114. MR 589590
(81m:58028)
-
H.
G. Othmer, The qualitative dynamics of a class of biochemical
control circuits, J. Math. Biol. 3 (1976),
no. 1, 53–78. MR 0406568
(53 #10355)
-
Jacob
Palis Jr. and Welington
de Melo, Geometric theory of dynamical systems,
Springer-Verlag, New York, 1982. An introduction; Translated from the
Portuguese by A. K. Manning. MR 669541
(84a:58004)
-
J.
Palis and S.
Smale, Structural stability theorems, Global Analysis (Proc.
Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc.,
Providence, R.I., 1970, pp. 223–231. MR 0267603
(42 #2505)
-
S. Peirce, 1955 The philosophical writings of C. S. Peirce, Dover, New York.
-
M.
M. Peixoto, Structural stability on two-dimensional manifolds,
Topology 1 (1962), 101–120. MR 0142859
(26 #426)
-
M.
M. Peixoto, On the classification of flows on 2-manifolds,
Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic
Press, New York, 1973, pp. 389–419. MR 0334289
(48 #12608)
-
H. Poincaré, 1881 Mémoire sur les courbes définies par une équation différentielle, J. Math. Pures Appl. 7, 375-422 and Oeuvres, Vol. 1, Gauthier-Villars, Paris.
-
H. Poincaré, 1880-1890 Sur les courbes définies par les équations différentielles. I-VI, Oeuvres, Vol. 1, op. cit.
-
H. Poincaré, 1890a Sur les équations de la dynamique et le problème de trois corps, Acta. Math. 13, 1-270.
-
H. Poincaré, 1892, 1893, 1899 Les méthodes nouvelles de la mécanique céleste, Vols. I, II, III, Gauthier-Villars, Paris.
-
H. Poincaré, 1908 Atti IV Congr. Internaz. Mat. (Roma, 1908), Vol. I, Acad. dei Lincei, Rome, 1909.
-
H. Poincaré, 1914 La valeur de la science, Flammarion, Paris.
-
H. Poincaré, 1916-1956 Oeuvres de Henri Poincaré, Vols. 1-11, Gauthier-Villars, Paris.
-
Analyse des travaux scientifiques, Acta Math. 38
(1921), no. 1, 3–135 (French). Henri poincaré faite par
lui-m\cflex eme. MR
1555105, http://dx.doi.org/10.1007/BF02392063
-
G. Pólya And S. Szegö, 1971 Problems and theorems in analysis, Springer-Verlag.
-
Murray
H. Protter and Hans
F. Weinberger, Maximum principles in differential equations,
Prentice-Hall Inc., Englewood Cliffs, N.J., 1967. MR 0219861
(36 #2935)
-
A. Rescigno and I. Richardson, 1967 The struggle for life. I: Two species, Bull. Math. Biophys. 29, 377-388.
-
J. Robinson, 1956 The accumulation of capital, Macmillan, London.
-
David
Ruelle and Floris
Takens, On the nature of turbulence, Comm. Math. Phys.
20 (1971), 167–192. MR 0284067
(44 #1297)
-
M. Ruse, 1979 The Darwinian revolution, Univ. of Chicago Press, Chicago, Ill.
-
B. Russell, 1948 Human knowledge, its scope and limits, Simon and Schuster, New York.
-
B. Russell, 1955 Basic writings, 1903-1959 (R. Egner and L. Denner, eds.), Simon and Schuster, New York.
-
James
F. Selgrade, Mathematical analysis of a cellular control process
with positive feedback, SIAM J. Appl. Math. 36
(1979), no. 2, 219–229. MR 524498
(80g:34046), http://dx.doi.org/10.1137/0136019
-
James
F. Selgrade, Asymptotic behavior of solutions to single loop
positive feedback systems, J. Differential Equations
38 (1980), no. 1, 80–103. MR 592869
(82a:34038), http://dx.doi.org/10.1016/0022-0396(80)90026-1
-
L. Silk, 1976 The economists, Basic Books, New York.
-
Stephen
Smale, On gradient dynamical systems, Ann. of Math. (2)
74 (1961), 199–206. MR 0133139
(24 #A2973)
-
S.
Smale, Dynamical systems and the topological conjugacy problem for
diffeomorphisms, Proc. Internat. Congr. Mathematicians (Stockholm,
1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 49–496. MR 0176487
(31 #759)
-
S.
Smale, A structurally stable differentiable homeomorphism with an
infinite number of periodic points, Qualitative methods in the theory
of non-linear vibrations (Proc. Internat. Sympos. Non-linear Vibrations,
Vol. II, 1961), Izdat. Akad. Nauk Ukrain. SSR, Kiev, 1963,
pp. 365–366. MR 0160220
(28 #3434)
-
Stephen
Smale, Diffeomorphisms with many periodic points, Differential
and Combinatorial Topology (A Symposium in Honor of Marston Morse),
Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR 0182020
(31 #6244)
-
S.
Smale, Structurally stable systems are not dense, Amer. J.
Math. 88 (1966), 491–496. MR 0196725
(33 #4911)
-
S.
Smale, Differentiable dynamical
systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 0228014
(37 #3598), http://dx.doi.org/10.1090/S0002-9904-1967-11798-1
-
S. Smale, 1967a Dynamical systems on n-dimensional manifolds, Sympos. Differential Equations and Dynamical Systems (Puerto Rico), Academic Press, New York.
-
Steve
Smale, The mathematics of time, Springer-Verlag, New York,
1980. Essays on dynamical systems, economic processes, and related topics.
MR 607330
(83a:01068)
-
S.
Smale, On the differential equations of species in
competition, J. Math. Biol. 3 (1976), no. 1,
5–7. MR
0406579 (53 #10366)
-
Steve
Smale, The mathematics of time, Springer-Verlag, New York,
1980. Essays on dynamical systems, economic processes, and related topics.
MR 607330
(83a:01068)
-
M. Spivak, 1970 A comprehensive introduction to differential geometry, Vol. 1, Publish or Perish, Waltham, Mass.
-
S. Sternberg, 1969 Celestial mechanics, Part 1, Benjamin, Reading, Mass.
-
J. Strachey, 1956 Contemporary capitalism, Random House, New York.
-
J. Sylvester, 1877 Appendix to Address on Commemoration Day at Johns Hopkins University, in Collected mathematical papers of James Joseph Sylvester, Vol. 3, Cambridge Univ. Press, Cambridge, England, 1908, pp. 85-87.
-
G.
Temple, Linearization and delinearization, Proc. Internat.
Congress Math. 1958, Cambridge Univ. Press, New York, 1960,
pp. 233–247. MR 0117376
(22 #8156)
-
René
Thom, Structural stability and morphogenesis, W. A. Benjamin,
Inc., Reading, Mass.-London-Amsterdam, 1976. An outline of a general theory
of models; Translated from the French by D. H. Fowler; With a foreword by
C. H. Waddington; Second printing. MR 0488156
(58 #7722b)
-
Wolfgang
Walter, Differential and integral inequalities, Translated
from the German by Lisa Rosenblatt and Lawrence Shampine. Ergebnisse der
Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York,
1970. MR
0271508 (42 #6391)
-
John
Andrew Walker, Dynamical systems and evolution equations,
Mathematical Concepts and Methods in Science and Engineering, vol. 20,
Plenum Press, New York, 1980. Theory and applications. MR 561511
(81f:34061)
-
Hermann
Weyl, The classical groups, Princeton Landmarks in
Mathematics, Princeton University Press, Princeton, NJ, 1997. Their
invariants and representations; Fifteenth printing; Princeton Paperbacks.
MR
1488158 (98k:01049)
-
R.
F. Williams, The structure of Lorenz attractors, Inst. Hautes
Études Sci. Publ. Math. 50 (1979), 73–99. MR 556583
(82b:58055b)
-
R. F. Williams, 1983 Review of “Dynamical Systems on Surfaces” by C. Godbillon, Amer. Math. Monthly (in press).
-
A. Winfree, 1979 The geometric theory of biological time, Springer-Verlag.
-
J. Z. Young, 1951 Doubt and certainty in science, Oxford Univ. Press, New York.
- R. Abraham, 1971 Predictions for the future of differential equations, Lecture Notes in Math., vol. 206 (D. Chillingworth, ed.), Springer-Verlag.
- R. Abraham and J. Marsden, 1967 Foundation of mechanics, Benjamin, New York.
- R. Abraham and J. Marsden, 1978 Foundation of mechanics, 2nd ed., Benjamin/Cummings, Reading, Mass. MR 515141
- R. Abraham and C. Shaw, 1982 Dynamics--The geometry of behavior, Aerial Press, Santa Cruz, Calif. MR 669376
- M. Amman, 1976 Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18, 620-709. MR 415432
- A. A. Andronov and L. Pontryagin, 1937 Systèmes grossiers, Dokl. Akad. Nauk. SSSR 14, 247-251.
- D. V. Anosov, 1962 Roughness of geodesic flows on compact Riemannian manifolds of negative curvature, Soviet Math. Dokl. 3, 1068-1070. MR 143156
- D. V. Anosov, 1967 Geodesic flows on closed Riemannian manifolds with negative curvature, Trudy Mat. Inst. Steklov 90, 3-209; English transl. in Proc. Steklov. Inst. Math. (I. G. Petrovskiĭ and S. M. Nikol´skiĭ, eds.), Amer. Math. Soc., Providence, R. I., 1969. MR 224110
- V. I. Arnold, 1983 Geometrical methods in the theory of ordinary differential equations, Grundlehren Math. Wiss., vol. 250, Springer-Verlag. MR 695786
- G. Aronsson and I. Mellander, 1980 A deterministic model in biomathematics: asymptotic behavior and threshold conditions, Math. Biosci. 49, 207-222. MR 573187
- D. Berlinski, 1976 On systems analysis, MIT Press, Cambridge, Mass.
- G. D. Birkhoff, 1920 Recent advances in dynamics, Science (N.S.) 51, 51-55 and Collected mathematical papers, vol. 2, Amer. Math. Soc., Providence, R. I., pp. 106-110.
- G. D. Birkhoff, 1927 Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math. Soc., Providence, R. I.; reprinted 1966. MR 209095
- G. D. Birkhoff, 1927a On the periodic motion of dynamical systems, Acta Math. 50, 359-379 and Collected mathematical papers, vol. 2, op. cit., pp. 333-353. MR 1555257
- G. D. Birkhoff, 1932 Sur l’existence de régions d’instabilité en dynamique, Ann. Inst. H. Poincaré 2, 369-386 and Collected mathematical papers, vol. 2, op. cit., pp. 444-461.
- G. D. Birkhoff, 1935 Sur le problème restreint des trois corps (premiere mèmoire), Ann. Scuola Norm. Sup. Pisa (2) 4, 267-306 and Collected mathematical papers, vol. 2, op. cit., pp. 466-306. MR 1556760
- G. D. Birkhoff, 1941 Some unsolved problems of theoretical dynamics, Science 94, 598-600 and Collected mathematical papers, vol. 2, op. cit., pp. 710-712. MR 6260
- G. D. Birkhoff, 1935a Nouvelles recherches sur les systèmes dynamiques, Mem. Pont. Acad. Sci. Nov. Lyn. 1, 85-216 and Collected mathematical papers, vol. 2, op. cit., pp. 530-661.
- G. D. Birkhoff, 1943 The mathematical nature of physical theories, Amer. Sci. 31, 281-310 and Collected mathematical papers, vol. 2, op. cit., pp. 890-919. MR 9000
- G. D. Birkhoff, 1950 Collected mathematical papers, vols. 1, 2, 3, Amer. Math. Soc., Providence, R. I.
- G. D. Birkhoff and B. O. Koopman, 1932 Recent contributions to ergodic theory, Proc. Nat. Acad. Sci. U.S.A. 18, 279-282 and Collected mathematical papers, op. cit., vol. 3, pp. 462-465.
- L. E. J. Brouwer, 1907 On the foundation of mathematics, Collected Mathematical Works, Vol. 1, American Elsevier, New York.
- L. E. J. Brouwer, 1913 Intuitionism and formalism, Bull. Amer. Math. Soc. 20, 81-96; Collected works, Vol. 1, op. cit., pp. 123-138.
- L. E. J. Brouwer, 1975 Collected works, Vol. 1 (A. Heyting, ed.), American Elsevier, New York. MR 532661
- J. L. Casti, 1982 Recent developments and future perspectives in nonlinear systems theory, SIAM Rev. 24, 301-331. MR 667511
- T. M. Cherry, 1938 Analytic quasi-periodic curves of discontinuous type on a torus, Proc. London Math. Soc. (2) 44, 175-215.
- T. M. Cherry, 1968 Asymptotic solutions of analytic Hamiltonian systems, J. Differential Equations 4, 142-159. MR 223678
- W. A. Coppel, 1965 Stability and asymptotic behavior of differential equations, Heath, Boston. MR 190463
- J. Coste, J. Peyraud and P. Coullet, 1979 Asymptotic behaviors in the dynamics of competing species, SIAM J. Appl. Math. 36 (1979), 516-543. MR 531612
- D´Alembert and Diderot (Editors), 1754 Encyclopédie, Vol. 4, Briasson, David, LeBreton, Durand, Paris.
- Darwin, 1892 Life of Charles Darwin (F. Darwin, ed.), John Murray, London.
- Darwin, 1936 The origin of species and The descent of man, Modern Library, New York.
- A. Denjoy, 1932 Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl. (4) 17, 333-375. MR 101369
- S. Drake, 1978 Galileo at work, Univ. of Chicago Press, Chicago, Ill. MR 531510
- A. S. Eddington, 1927 The nature of the physical world, Cambridge Univ. Press, Cambridge, England.
- A. Einstein, 1930 On the occasion of the three hundredth anniversary of Kepler’s death, Frankfurter Zeitung, November 9, 1930 and Ideas and opinions, Crown, New York, 1954.
- A. Einstein, 1931 On the one hundreth anniversary of Maxwell’s birth, James Clerk Maxwell: A Commemorative Volume, Cambridge Univ. Press, Cambridge, England, and Ideas and opinions, op. cit.
- J. M. Franks, 1982 Homology and dynamical systems, CBMS Regional Conf. Ser. in Math., Vol. 49, Amer. Math. Soc., Providence, R. I. MR 669378
- H. I. Freedman, 1980 Deterministic mathematical models in population ecology, Dekker, New York. MR 586941
- H. I. Freedman and P. Waltman, 1984 Persistence in models of three interacting species, Math. Biosci. (in press). MR 738903
- N. Georgescu-Roegen, 1966 Analytical economics: Issues and problems, Harvard Univ. Press, Cambridge, Mass.
- C. C. Gillispie, 1960 The edge of objectivity, Princeton Univ. Press, Princeton, N. J. MR 1145737
- W. H. Gottschalk and G. A. Hedlund, 1955 Topological dynamics, Amer. Math. Soc. Colloq. Publ., vol. 36, Amer. Math. Soc., Providence, R. I. MR 74810
- S. Grossberg, 1978 Competition, decision and consensus, J. Math. Anal. Appl. 66, 470-493. MR 515909
- J. Guckenheimer, 1976 A strange strange attractor, The Hopf Bifurcation and Its Applications (J. E. Marsden and M. McCracken, eds.), Springer-Verlag, pp. 368-381. MR 494309
- J. Guckenheimer, 1982 Noise in chaotic systems, Nature 298, 358-361.
- J. Guckenheimer and G. Buzyna, 1983 Dimension measurements for geostrophic turbulence, Phys. Rev. Lett. 51, 1438-1441. MR 718790
- J. Guckenheimer And P. Holmes, 1983 Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Math. Sci., Vol. 42, Springer-Verlag. MR 709768
- J. Hadamard, 1912 L’oeuvres mathématique de Henri Poincaré, Acta Math. 38, 203-287; and Oeuvres de Jacques Hadamard4, CNRS, Paris, pp. 1921-2005.
- J. Hadamard, 1912a Henri Poincaré et le problème de trois corps, Rev. Mois 16, 385-418 and Oeuvres de Jacques Hadamard, op. cit., pp. 2007-2041.
- J. Hadamard, 1968 Oeuvres de Jacques Hadamard4, CNRS, Paris.
- J. K. Hale, 1963 Oscillations in nonlinear systems, McGraw-Hill, New York. MR 150402
- T. L. Hankins, 1980 Sir William Rowan Hamilton, Johns Hopkins Univ. Press, Baltimore. MR 618834
- B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, 1980 Theory and application of the Hopf bifurcation, Cambridge Univ. Press, Cambridge, England. MR 603442
- R. Heilbroner, 1953 The worldly philosphers, Simon and Schuster, New York.
- D. Henry, 1981 Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, Springer-Verlag. MR 610244
- H. W. Hethcote, 1973 Asymptotic behavior in a deterministic epidemic model, J. Math. Biol. 35, 607-614.
- H. Hethcote, A. Nold and L. Yorke, 1982 Gonorrhea modelling: a comparison of control methods, Math. Biosci. 58, 93-109.
- M. W. Hirsch, 1983 Differential equations and convergence almost everywhere in strongly monotone flows, Contemporary Math., vol. 17, Amer. Math. Soc., Providence, R. I., pp. 267-285. MR 706104
- M. W. Hirsch, 1983a Systems of differential equations which are competitive or cooperative. II: Convergence almost everywhere, SIAM J. Math. Anal. (in press); preprint, Center for Pure and Appl. Math., Univ. of California, Berkeley.
- M. W. Hirsch, 1982 Systems of differential equations which are competitive or cooperative. I: Limit sets, SIAM J. Math. Anal. 13, 167-179. MR 647119
- S. B. Hsu, S. P. Hubbell And P. Waltman, 1978 Competing predators, SIAM J. Appl. Math. 35, 617-625. MR 512172
- E. Kamke, 1932 Zur Theorie der Systeme gewöhnlicher differential-gleichungen. II, Acta Math. 58, 57-85. MR 1555344
- F. Klein, 1911 Lectures on mathematics (Proc. Colloq., Evanston, 1893), Amer. Math. Soc., Providence, R. I.
- A. Kolmogorov, 1936 Sulla teoria di Volterra della lotta per l´esistenzia, Giorn. Ist. Ital. Attuari 7, 47-80.
- I. Kupka, 1963 Contribution à la théorie des champs génériques, Contributions to Differential Equations 2, 457-484. MR 165536
- A. Lajmanovich and J. Yorke, 1976 A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28, 221-236. MR 403726
- W. Leonard and R. May, 1975 Nonlinear aspects of competition between species, SIAM J. Appl. Math. 29, 243-275. MR 392035
- S. Lie, 1895 Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung, Leipziger Berichte 47, 53-128 and Sophus Lie Gesammelte Abhandlungen (F. Engel, ed.), Bd. 4, Teubner, Leipzig, 1929, pp. 320-386.
- E. N. Lorenz, 1963 Deterministic non-periodic flow, J. Atmospheric Sci. 20, 130-141.
- A. Lotka, 1956 Principles of mathematical biology, Dover, New York.
- J. E. Marsden and M. McCracken (Editors), 1976 The Hopf bifurcation and its applications, Springer-Verlag. MR 494309
- X. Mora, 1983 Semilinear problems define semiflows in C, Trans. Amer. Math. Soc. 278, 21-55. MR 697059
- R. E. Moritz, 1914 Memorabilia mathematica, Macmillan, New York; reprinted as On mathematics and mathematicians, Dover, New York, 1958. MR 104557
- J. Moser, 1973 Stable and random motions in dynamical systems, Princeton Univ. Press, Princeton, N. J. MR 442980
- S. Newhouse, 1971 Nondensity of axiom A(a), Global Analysis, Proc. Sympos. Pure Math, vol. 14 (S.-S. Chern and S. Smale, eds.), Amer. Math. Soc., Providence, R. I., pp. 191-203. MR 277005
- S. Newhouse, 1979 The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50, 101-151. MR 556584
- S. Newhouse, 1980 Lectures on dynamical systems, Dynamical Systems (CIME Summer School, Bressanone, 1978), Birkhäuser, Boston, pp. 1-114. MR 589590
- H. G. Othmer, 1976 The qualitative dynamics of a class of biochemical control circuits, J. Math. Biol. 3, 53-78. MR 406568
- J. Palis and W. Demelo, 1982 Geometric theory of dynamical systems: An introduction, Springer-Verlag. MR 669541
- J. PaLis and S. Smale, 1970 Structural stability theorems, Global Analysis, Proc. Sympos, Pure Math., vol. 14 (S.-S. Chern and S. Smale, eds.), Amer. Math. Soc., Providence, R. I. MR 267603
- S. Peirce, 1955 The philosophical writings of C. S. Peirce, Dover, New York.
- M. Peixoto, 1962 Structural stability on 2-dimensional manifolds, Topology 1, 101-120. MR 142859
- M. Peixoto, 1973 On the classification of flows on two-manifolds, Dynamical Systems (M. Peixoto, ed.), Academic Press, New York. MR 334289
- H. Poincaré, 1881 Mémoire sur les courbes définies par une équation différentielle, J. Math. Pures Appl. 7, 375-422 and Oeuvres, Vol. 1, Gauthier-Villars, Paris.
- H. Poincaré, 1880-1890 Sur les courbes définies par les équations différentielles. I-VI, Oeuvres, Vol. 1, op. cit.
- H. Poincaré, 1890a Sur les équations de la dynamique et le problème de trois corps, Acta. Math. 13, 1-270.
- H. Poincaré, 1892, 1893, 1899 Les méthodes nouvelles de la mécanique céleste, Vols. I, II, III, Gauthier-Villars, Paris.
- H. Poincaré, 1908 Atti IV Congr. Internaz. Mat. (Roma, 1908), Vol. I, Acad. dei Lincei, Rome, 1909.
- H. Poincaré, 1914 La valeur de la science, Flammarion, Paris.
- H. Poincaré, 1916-1956 Oeuvres de Henri Poincaré, Vols. 1-11, Gauthier-Villars, Paris.
- H. Poincaré, 1921 Analyse des travaux scientifiques de Henri Poincaré faites par lui-même, Acta Math. 38, 1-135 and Oeuvres, Vol. 1, op. cit. MR 1555105
- G. Pólya And S. Szegö, 1971 Problems and theorems in analysis, Springer-Verlag.
- M. Protter and H. Weinberger, 1967 Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, N. J. MR 219861
- A. Rescigno and I. Richardson, 1967 The struggle for life. I: Two species, Bull. Math. Biophys. 29, 377-388.
- J. Robinson, 1956 The accumulation of capital, Macmillan, London.
- D. Ruelle and F. Takens, 1971 On the nature of turbulence, Comm. Math. Phys. 20, 167-192; ibid. 23, 343-344. MR 284067
- M. Ruse, 1979 The Darwinian revolution, Univ. of Chicago Press, Chicago, Ill.
- B. Russell, 1948 Human knowledge, its scope and limits, Simon and Schuster, New York.
- B. Russell, 1955 Basic writings, 1903-1959 (R. Egner and L. Denner, eds.), Simon and Schuster, New York.
- J. F. Selgrade, 1979 Mathematical analysis of a cellular control process with positive feedback, SIAM J. Appl. Math. 36, 219-229. MR 524498
- J. F. Selgrade, 1980 Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38, 80-103. MR 592869
- L. Silk, 1976 The economists, Basic Books, New York.
- S. Smale, 1961 On gradient dynamical systems, Ann. of Math. (2) 74, 199-206. MR 133139
- S. Smale, 1962 Dynamical systems and the topological conjugacy problem for diffeomorphisms, Proc. Internat. Congress Math. (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 440-496. MR 176487
- S. Smale, 1963 A structurally stable differentiable homeomorphism with an infinite number of periodic points, Proc. Internat. Sympos. Nonlinear Vibrations, Izdat. Akad. Nauk. Ukrain. SSR, Kiev. MR 160220
- S. Smale, 1965 Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (S. Cairns, ed.), Princeton Univ. Press, Princeton, N. J., pp. 63-80. MR 182020
- S. Smale, 1966 Structurally stable systems are not dense, Amer. J. Math. 88, 491-495. MR 196725
- S. Smale, 1967 Differentiable dynamical systems, Bull. Amer. Math. Soc. 73, 747-817. MR 228014
- S. Smale, 1967a Dynamical systems on n-dimensional manifolds, Sympos. Differential Equations and Dynamical Systems (Puerto Rico), Academic Press, New York.
- S. Smale, 1967b Stability and genericity of dynamical systems, Sem. Bourbaki, no. 374, 1969-1970 and The mathematics of time, Springer-Verlag, 1980. MR 607330
- S. Smale, 1976 On the differential equations of species in competition, J. Math. Biol. 3, 5-7. MR 406579
- S. Smale, 1980 The mathematics of time, Springer-Verlag. MR 607330
- M. Spivak, 1970 A comprehensive introduction to differential geometry, Vol. 1, Publish or Perish, Waltham, Mass.
- S. Sternberg, 1969 Celestial mechanics, Part 1, Benjamin, Reading, Mass.
- J. Strachey, 1956 Contemporary capitalism, Random House, New York.
- J. Sylvester, 1877 Appendix to Address on Commemoration Day at Johns Hopkins University, in Collected mathematical papers of James Joseph Sylvester, Vol. 3, Cambridge Univ. Press, Cambridge, England, 1908, pp. 85-87.
- G. Temple, 1958 Linearization and delinearization, Proc. Internat. Congress Math. (Edinburgh, 1958), Cambridge Univ. Press, Cambridge, England, pp. 233-247. MR 117376
- R. Thom, 1975 Structural stability and morphogenesis, Benjamin, Reading, Mass.; original ed., Paris, 1972. MR 488156
- W. Walter, 1970 Differential and integral inequalities, Springer-Verlag. MR 271508
- J. A. Walker, 1980 Dynamical systems and evolution equations, Plenum Press, New York. MR 561511
- H. Weyl, 1939 The classical groups: Their invariants and representations, Princeton Univ. Press, Princeton, N. J. MR 1488158
- R. F. Williams, 1979 The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50, 101-152. MR 556583
- R. F. Williams, 1983 Review of “Dynamical Systems on Surfaces” by C. Godbillon, Amer. Math. Monthly (in press).
- A. Winfree, 1979 The geometric theory of biological time, Springer-Verlag.
- J. Z. Young, 1951 Doubt and certainty in science, Oxford Univ. Press, New York.
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DOI:
http://dx.doi.org/10.1090/S0273-0979-1984-15236-4
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S 0273-0979(1984)15236-4
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