Lyapunov
exponents sign inversions (Perron effects) and chaos:
Linearization,
stability and instability by the first approximation
In 1892, the general problem
of stability by the first approximation was considered by Lyapunov. He proved
that if the system of the first approximation is regular and all its Lyapunov exponents (or characteristic
exponents) are negative, then the solution of the original system is
asymptotically Lyapunov stable. In 1930, it was stated by O. Perron that the
requirement of regularity of the first approximation is substantial. He
constructed an example of the second-order system of the first approximation,
which has negative characteristic exponents along a zero solution of the original system but, at the same time, this zero solution of
original system is Lyapunov unstable. Furthermore, in a certain neighborhood of
this zero solution almost all solutions of original system have positive
characteristic exponents. The effect of a sign reversal of characteristic
exponents of solutions of the original system and the system of first
approximation with the same initial data was subsequently called the Perron effect. The counterexample of
Perron impressed on the contemporaries and gave an idea of the difficulties
arising in the justification of the first approximation theory for
nonautonomous and nonperiodic linearizations.
Later, Persidskii [1947], Massera
[1957], Malkin [1966], and Chetaev
[1955], obtained sufficient conditions of stability by the first approximation
for nonregular linearizations generalizing the Lyapunov theorem. At the same time,
according to Malkin (Theory of Motion Stability,
1966):
The counterexample of Perron shows that the
negativeness of largest Lyapunov exponents (or characteristic exponents) is not
a sufficient condition of stability by the first approximation. In the general
case necessary and sufficient conditions of stability by the first
approximation are not obtained.
In the 1940s Chetaev [1948] published the criterion of instability by
the first approximation for regular linearizations. However, in the proof of
these criteria a flaw was discovered and, at present, the complete proof of Chetaev theorems is given for a more weak condition in
comparison with that for instability by Lyapunov, namely, for instability by Krasovsky only.
The discovery of
strange attractors was made obvious with the study of instability by the first
approximation. Nowadays the problem of the justification of the nonstationary
linearizations for complicated nonperiodic motions on strange attractors bears
a striking resemblance to the situation that occurred 120 years ago. The
founders of the automatic control theory D.K. Maxwell [1868], and A.I. Vyschegradskii [1877] courageously used a linearization in
a neighborhood of stationary motions, leaving the justification of such
linearization to A. Poincare [1886] and A.M. Lyapunov [1892].
At present, many
specialists in chaotic dynamics believe that the positiveness of the largest Lyapunov
exponent of a linear system of the first approximation implies the instability
of solutions of the original system. Moreover, there are a number of computer
experiments, in which the various numerical methods for calculating the
characteristic exponents and the Lyapunov exponents of linear systems of the
first approximation are used. As a rule, authors ignore the justification of
the linearization procedure and use the numerical values of exponents so
obtained to construct various numerical characteristics of attractors of the
original nonlinear systems (Lyapunov dimensions, metric entropies, and so on).
Sometimes, computer experiments serve as arguments for the partial
justification of the linearization procedure.
But positive largest Lyapunov exponent doesn't, in
general, indicate chaos and negative largest Lyapunov exponent doesn't, in
general, indicate stability!
Publications below
show the contemporary state of the art of the problem of the justification of
nonstationary linearizations. Here for the discrete and continuous systems the
results of stability by the first approximation for regular and nonregular
linearizations are given, the Perron effects are considered, the criteria of
the stability and instability of the flow and cascade of solutions and the
criteria of instability by Lyapunov and Krasovsky are
obtained.
Publications
- G.A. Leonov, N.V. Kuznetsov, Time-Varying
Linearization and the Perron effects,
International
Journal of Bifurcation and Chaos, Vol. 17, No. 4, 2007,
pp. 1079-1107 [PDF]
[DOI]
- Leonov G.A. Strange Attractors and Classical
Stability Theory. St. Petersburg University Press, 2009
- Kuznetsov N.V., Stability and
Oscillations of Dynamical Systems: Theory and Applications.
Jyväskylä University Printing House, 2008 (ISBN:
978-951-39-3428-6)
- Kuznetsov N.V., Leonov G.A., On stability
by the first approximation for discrete systems, 2005 International Conference
on Physics and Control, PhysCon 2005, Proceedings
Volume 2005, 2005, art. num. 1514053, pp. 596-599 [DOI]
- Kuznetsov N.V., Leonov G.A., Criteria of
stability by the first approximation for discrete nonlinear systems,
Vestnik St. Petersburg University. Mathematics.
Vol. 38, No. 3, pp. 21-30, 2005 (Allerton Press,
USA)
- Kuznetsov N.V., Leonov G.A., Criterion of
stability to first approximation of nonlinear discrete systems,
Vestnik St. Petersburg University. Mathematics.
Vol. 38, No. 2, pp. 52-60, 2005 (Allerton Press,
USA)
- Kuznetsov N.V., Leonov G.A., Stability by
the first approximation for discrete systems, Vestnik St.Petersburg University. Mathematics. Vol.
36, No. 1, 2003, pp. 21-27, (Allerton Press,
USA)
- Kuznetsov N.V., Leonov G.A., Counterexample
of Perron in the Discrete Case,
Izv. RAEN, Diff. Uravn.
5, pp. 71