Explosions in completely unstable flows. I. Preventing explosions

Author:
Zbigniew Nitecki

Journal:
Trans. Amer. Math. Soc. **245** (1978), 43-61

MSC:
Primary 58F10

DOI:
https://doi.org/10.1090/S0002-9947-1978-0511399-2

MathSciNet review:
511399

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Several conditions are equivalent to the property that a flow (on an open manifold) and its perturbations have only wandering points. These conditions are: (i) there exists a strong Liapunov function; (ii) there are no generalized recurrent points in the sense of Auslander; (iii) there are no chain recurrent points, in the sense of Conley; (iv) there exists a fine sequence of filtrations; (v) relative to some metric; the flow is the gradient flow of a function without critical points. We establish these equivalences, and consider a few questions related to structural stability when all orbits wander.

**[1]**J. Auslander,*Generalized recurrence in dynamical systems*, Contributions to Differential Equations**3**(1964), 65-74. MR**0162238 (28:5437)****[2]**J. Auslander and P. Seibert,*Prolongations and stability in dynamical systems*, Ann. Inst. Fourier (Grenoble)**14**(1964), 237-268. MR**0176180 (31:455)****[3]**N. P. Bhatia and G. P. Szegö,*Stability theory of dynamical systems*, Springer, New York, 1970. MR**0289890 (44:7077)****[4]**C. Conley,*The gradient structure of a flow*. I, unpublished IBM Research Report, Yorktown Heights.**[5]**J. Dugundji and H. A. Antosiewicz,*Parallelizable flows and Lyapunov's second method*, Ann. of Math. (2)**73**(1961), 543-555. MR**0123064 (23:A395)****[6]**M. Krych,*Note on structural stability of open manifolds*, Bull. Acad. Polon. Sci.**22**(1974), 1033-1038. MR**0397792 (53:1650)****[7]**P. Mendes,*On stability of dynamical systems on open manifolds*, J. Differential Equations**16**(1974), 144-167. MR**0345137 (49:9876)****[8]**V. V. Nemytskii and V. V. Stepanov,*Qualitative theory of differential equations*, Princeton Univ. Press, Princeton, N. J., 1960. MR**0121520 (22:12258)****[9]**Z. Nitecki,*Explosions in completely unstable dynamical systems, in Dynamical systems*, Proc. Univ. Florida Internat. Sympos., Academic Press, New York, 1977.**[10]**-,*Explosions in completely unstable flows*. II.*Some examples*, Trans. Amer. Math. Soc.**245**(1978), 63-88. MR**511400 (81e:58031)****[11]**Z. Nitecki and M. Shub,*Filtrations, decompositions, and explosions*, Amer. J. Math.**97**(1976), 1029-1047. MR**0394762 (52:15561)****[12]**J. Palis,*A note on -stability*, Proc. Sympos. Pure Math., Vol. 14, Amer. Math. Soc., Providence, R. I., 1970, pp. 221-222. MR**0270387 (42:5276)****[13]**M. Shub and S. Smale,*Beyond hyperbolicity*, Ann. of Math. (2)**96**(1972), 587-591. MR**0312001 (47:563)****[14]**F. Takens and W. White,*Vector fields with no nonwandering points*, Amer. J. Math.**98**(1976), 415-425. MR**0418163 (54:6205)****[15]**F. W. Wilson,*Smoothing derivatives of functions and applications*, Trans. Amer. Math. Soc.**139**(1969), 413-428. MR**0251747 (40:4974)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F10

Retrieve articles in all journals with MSC: 58F10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0511399-2

Keywords:
Completely unstable flow,
-explosion,
chain recurrence,
generalized recurrence,
prolongation,
gradient flow,
Liapunov function,
filtration,
structural stability

Article copyright:
© Copyright 1978
American Mathematical Society