Andronov–Witt theorem
A modification of Lyapunov's theorem (on the stability of a
periodic solution of a non-autonomous system of
differential equations) for the autonomous system
Let
be a periodic solution of the system
(1),
and let
be the corresponding system of variational equations which always has,
in the case here considered, one zero characteristic
exponent. The Andronov–Witt theorem is then valid: If
characteristic exponents of the system
(3)
have
negative real parts, a periodic solution
(2)
of the
system
(1)
is stable according to Lyapunov (cf.
Lyapunov characteristic exponent;
Lyapunov stability).
The Andronov–Witt theorem was first formulated by
A.A. Andronov
and
A.A. Witt
in
1930
and was proved by them in
1933
[1].
References| [1] |
A.A. Andronov,
"Collected works"
, Moscow
(1976)
(In Russian) | | [2] |
L.S. Pontryagin,
"Ordinary differential equations"
, Addison-Wesley
(1962)
pp. 264
(Translated from Russian) |
E.A. Leontovich-Andronova
CommentsThe Andronov–Witt theorem is usually found in the Western
literature under some heading like
"hyperbolic periodic attractorhyperbolic periodic attractor" .
Good additional general references are
[a1],
[a2],
[a3].
In
[a2]
the theorem under discussion occurs as a statement about
periodic attractors,
cf. pp. 277-278. The original Andronov–Witt paper is
[a4].
References| [a1] |
W. Hahn,
"Stability of motion"
, Springer
(1967)
pp. 422 | | [a2] |
M.W. Hirsch,
S. Smale,
"Differential equations, dynamic systems and linear algebra"
, Acad. Press
(1974) | | [a3] |
E.A. Coddington,
N. Levinson,
"Theory of ordinary differential equations"
, McGraw-Hill
(1955)
pp. 323 | | [a4] |
A.A. Andronov,
A. Witt,
"Zur Stabilität nach Liapounov"
Physikal. Z. Sowjetunion
, 4
(1933)
pp. 606–608 |
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
|