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Estimates for principal Lyapunov exponents: a survey. (English) Zbl 1316.34004
This survey is devoted to the estimate of principal Lyapunov exponents in time-dependent linear systems of differential equations and some of their generalizations.
One of the characteristic results is as follows. Consider a system \(T\)-periodic in \(t\) \[ u'=P(t)u \] and assume that the system is strongly cooperative (i.e., the off-diagonal entries of the matrix \(P(t)\) are positive for all \(t\)). Then there exist (and are uniquely determined) a number \(\rho>0\) and a positive solution \(v(t)\) with \(|v(0)|=1\) such that \(v(T)=\rho v(0)\). In this case, \[ \lim_{t\to\infty}(\log|u(t)|)/t=(\log\rho)/T \] for any positive solution \(u(t)\).
The survey contains a lot of results for second order parabolic PDEs, almost periodic linear systems of differential equations, systems with random coefficients, cooperative delay differential equations, and so on.
Concerning proofs, mostly hints are given.

MSC:
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
35B40 Asymptotic behavior of solutions to PDEs
37C65 Monotone flows as dynamical systems
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
34A30 Linear ordinary differential equations and systems
37C60 Nonautonomous smooth dynamical systems
34K06 Linear functional-differential equations
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