Editing Lyapunov exponent (section)

===Perron effects of largest Lyapunov exponent sign inversion===
In 1930 [[Oskar Perron|O. Perron]] constructed an example of a second-order system, where the first approximation has negative Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of the original nonlinear system is Lyapunov unstable. Furthermore, in a certain neighborhood of this zero solution almost all solutions of original system have positive Lyapunov exponents. Also, it is possible to construct a reverse example in which the first approximation has positive Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of original nonlinear system
is Lyapunov stable.<ref name = 2005-IEEE-Discrete-system-stability-Lyapunov-exponent>{{cite book
 |author1=N.V. Kuznetsov |author2=G.A. Leonov |title=Proceedings. 2005 International Conference Physics and Control, 2005 |chapter=On stability by the first approximation for discrete systems | year = 2005
 | volume = Proceedings Volume 2005
 | pages = 596–599
 | url = http://www.math.spbu.ru/user/nk/PDF/2005-IEEE-Discrete-system-stability-Lyapunov-exponent.pdf
 | doi = 10.1109/PHYCON.2005.1514053
|isbn=978-0-7803-9235-9 |s2cid=31746738 }}
</ref><ref name = 2007-IJBC-Lyapunov-exponent>{{cite journal
 |author1=G.A. Leonov |author2=N.V. Kuznetsov | year = 2007
 | title = Time-Varying Linearization and the Perron effects
 | journal = International Journal of Bifurcation and Chaos
 | volume = 17
 | issue = 4
 | pages = 1079–1107
 | url = http://www.math.spbu.ru/user/nk/PDF/2007_IJBC_Lyapunov_exponent_Linearization_Chaos_Perron_effects.pdf
 | doi = 10.1142/S0218127407017732
| bibcode= 2007IJBC...17.1079L|citeseerx=10.1.1.660.43 }}
</ref>
The effect of sign inversion of Lyapunov exponents of solutions of the original system and the system of first approximation with the same initial data was subsequently
called the Perron effect.<ref name = 2005-IEEE-Discrete-system-stability-Lyapunov-exponent /><ref name = 2007-IJBC-Lyapunov-exponent />

Perron's counterexample shows that a negative largest Lyapunov exponent does not, in general, indicate stability, and that
a positive largest Lyapunov exponent does not, in general, indicate chaos.

Therefore, time-varying linearization requires additional justification.<ref name = 2007-IJBC-Lyapunov-exponent />