Editing Lyapunov exponent (section)

==Lyapunov exponent for time-varying linearization==
To introduce Lyapunov exponent consider a fundamental matrix <math> X(t)</math> (e.g., for linearization along a stationary solution <math>x_0</math> in a continuous system), the fundamental matrix is <math> \exp\left( \left. \frac{ d f^t(x) }{dx} \right|_{x_0} t\right) </math> consisting of the linearly-independent solutions of  the first-order approximation of the system. The singular values <math>\{\alpha_j\big(X(t)\big)\}_{1}^{n}</math> of the matrix <math>X(t)</math> are the square roots of the eigenvalues of the matrix <math>X(t)^*X(t)</math>.
The largest Lyapunov exponent <math>\lambda_{\mathrm{max}}</math> is as follows<ref>{{cite book|last=Temam|first=R.|author-link=Roger Temam|title=Infinite Dimensional Dynamical Systems in Mechanics and Physics|publisher=Cambridge: Springer-Verlag|year=1988}}</ref>
<math display="block">
   \lambda_{\mathrm{max}}= \max\limits_{j}\limsup _{t \rightarrow \infty}\frac{1}{t}\ln\alpha_j\big(X(t)\big).
</math>
Lyapunov proved that if the system of the first approximation is regular (e.g., all systems with constant and periodic coefficients are regular) and its largest Lyapunov exponent is negative, then the solution of the original system is [[Lyapunov stability|asymptotically Lyapunov stable]].
Later, it was stated by O. Perron that the requirement of regularity of the first approximation is substantial.

===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 />