Bulletin of the American Mathematical Society

MathSciNet review: 1567253
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Book Information:

Author: V. I. Arnold
Title: Ordinary differential equations
Additional book information: translated from the Russian by Richard A. Silverman, MIT Press, Cambridge, Massachusetts, 1978, x + 280 pp., $8.95.

Stephen Bancroft, Jack K. Hale, and Daniel Sweet, Alternative problems for nonlinear functional equations, J. Differential Equations 4 (1968), 40–56. MR 220118, DOI 10.1016/0022-0396(68)90047-8

N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974/75), 17–37. MR 440205, DOI 10.1080/00036817408839081

Nathaniel Chafee, Behavior of solutions leaving the neighborhood of a saddle point for a nonlinear evolution equation, J. Math. Anal. Appl. 58 (1977), no. 2, 312–325. MR 445080, DOI 10.1016/0022-247X(77)90209-8

C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35–61. MR 279830, DOI 10.1090/S0002-9947-1971-0279830-1

Robert Easton, Regularization of vector fields by surgery, J. Differential Equations 10 (1971), 92–99. MR 315741, DOI 10.1016/0022-0396(71)90098-2

Harry Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. MR 498471, DOI 10.1007/BF02813304

H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math. 34 (1978), 61–85 (1979). MR 531271, DOI 10.1007/BF02790008

H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math. 34 (1978), 275–291 (1979). MR 531279, DOI 10.1007/BF02790016

Jack K. Hale, Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969), 39–59. MR 244582, DOI 10.1016/0022-247X(69)90175-9

Joseph P. LaSalle, An invariance principle in the theory of stability, Differential Equations and Dynamical Systems (Proc. Internat. Sympos., Mayaguez, P.R., 1965) Academic Press, New York, 1967, pp. 277–286. MR 0226132

11.

A. Liapunov, Problème général de la stabilité du mouvement, Ann. Sci. Toulouse 2 (1907), 203-474.

B. J. Matkowsky, A simple nonlinear dynamic stability problem, Bull. Amer. Math. Soc. 76 (1970), 620–625. MR 257544, DOI 10.1090/S0002-9904-1970-12461-2

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math. 29 (1976), no. 6, 724–747. MR 426052, DOI 10.1002/cpa.3160290613

Charles C. Pugh, On a theorem of P. Hartman, Amer. J. Math. 91 (1969), 363–367. MR 257533, DOI 10.2307/2373513

Jacques Roels, An extension to resonant cases of Liapunov’s theorem concerning the periodic solutions near a Hamiltonian equilibrium, J. Differential Equations 9 (1971), 300–324. MR 273155, DOI 10.1016/0022-0396(71)90084-2

J. Roels, Families of periodic solutions near a Hamiltonian equilibrium when the ratio of two eigenvalues is $3$, J. Differential Equations 10 (1971), 431–447. MR 294789, DOI 10.1016/0022-0396(71)90005-2

Dieter S. Schmidt and Daniel Sweet, A unifying theory in determining periodic families for Hamiltonian systems at resonance, J. Differential Equations 14 (1973), 597–609. MR 328221, DOI 10.1016/0022-0396(73)90070-3

Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Die Grundlehren der mathematischen Wissenschaften, Band 187, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme. MR 0502448

Alan Weinstein, Lagrangian submanifolds and Hamiltonian systems, Ann. of Math. (2) 98 (1973), 377–410. MR 331428, DOI 10.2307/1970911

Alan Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math. 20 (1973), 47–57. MR 328222, DOI 10.1007/BF01405263

1.

S. Bancroft, J. K. Hale and D. Sweet, Alternative problems for nonlinear functional equations, J. Differential Equations 4 (1968), 40-56. MR 0220118

2.

N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, C.D.S. Tech. Rep. 74-5, Lefschetz Center for Dynamical Systems, Brown Univ., 1974. MR 440205

3.

N. Chafee, Behavior of solutions leaving the neighborhood of a saddle point for a nonlinear evolution equation, J. Math. Anal. Appl. 58 (1977), 312-325. MR 445080

4.

C. C. Conley and R. Easton, Isolated invariant sets and isolating blocks. Trans. Amer. Math. Soc. 158 (1971), 35-61. MR 279830

5.

R. Easton, Regularization of vector fields by surgery, J. Differential Equations 10 (1971) 92-99. MR 315741

6.

H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204-256. MR 498471

7.

H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math. 34 (1978), 61-85. MR 531271

8.

H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyze Math. 34 (1978), 275-291. MR 531279

9.

J. K. Hale, Dynamical systems and stability, J. Math. Anal. Appl. 26 (1969), 39-59. MR 244582

10.

J. P. LaSalle, An invariance principle in the theory of stability, Int. Symp. on Diff. Eqs. and Dyn. Systems, J. K. Hale and J. P. LaSalle (eds.), Academic Press, New York, 1967, p. 277. MR 226132

11.

A. Liapunov, Problème général de la stabilité du mouvement, Ann. Sci. Toulouse 2 (1907), 203-474.

12.

B. J. Matkowsky, A simple nonlinear dynamic stability problem, Bull. Amer. Math. Soc. 76 (1970), 620-625. MR 257544

13.

J. Moser, Periodic orbits near an equilibrium and a theorem by A. Weinstein, Comm. Pure Appl. Math. 29 (1976), 727-747. MR 426052

14.

C. C. Pugh, On a theorem of P. Hartman, Amer. J. Math. 91 (1969), 363-369. MR 257533

15.

J. Roels, An extension to resonant case of Liapunov's theorem concerning the periodic solutions near a Hamiltonian equilibrium, J. Differential Equations 9 (1971), 300-324. MR 273155

16.

J. Roels, Families of periodic solutions near a Hamiltonian equilibrium when the ratio of 2 eigenvalues is 3, J. Differential Equations 10 (1971), 431-447. MR 294789

17.

D. S. Schmidt and D. Sweet, A unifying theory in determining periodic families for Hamiltonian systems at resonance, Tech. Rep. TR 73-3, Univ. of Maryland, 1973. MR 328221

18.

C. L. Siegel and J. Moser, Lectures on celestial mechanics, Springer-Verlag, New York, 1971. MR 502448

19.

A. Weinstein, Lagrangian submanifolds and Hamiltonian systems, Ann. Math. 98 (1973), 377-410. MR 331428

20.

A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math. 20 (1973), 47-57. MR 328222