How can one define and calculate (analytically or numerically) Lyapunov exponents for partial differential equations? Do there exist examples of nonlinear PDEs for which Lyapunov exponents can be calculated analytically?
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$\begingroup$ en.wikipedia.org/wiki/C0-semigroup#Exponential_stability This is quite comprehensive for the linear pde case. $\endgroup$– András BátkaiCommented Apr 4, 2013 at 20:57
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$\begingroup$ math.stackexchange.com/questions/3347935/… $\endgroup$– Charlie ChangCommented Mar 8, 2022 at 12:57
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There is at least one paper I am aware of, concerning the calculation of Lyapunov exponents for a non-linear, Partial Differential Equation (namely, the Kuramoto-Sivashinsky equation). The reference is:
Shibata, Hiroshi. "Lyapunov exponent of partial differential equation", Physica A 264 (1999) 226-233
What Shibata calculates, is the mean and the local Lyapunov exponent, and roughly the procedure is to make the discretization of the equation (in terms of finite differences) and then form a Jacobi matrix, from which one can study the evolution of the (now) linearized system.
Having said that, it would help if you could clarify if the non-linearity comes from the function multiplying one of the partial derivatives (the case above), or, if it is coupled system of two (or more) PDEs (multiphysics, for example). If the later is the case, perhaps it would come of handy the following paper, concerning the decoupling of the KdV equation, rendering a system of ordinary differential equations (ODEs). From here, I think that in principle it should then be possible to apply the methods of calculating the Lyapunov exponents known for ODEs:
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$\begingroup$ @Semen Dziuba. I have both papers, in case you want them to be sent directly to you, just tell me how. This goes for anybody else, of course. $\endgroup$ Commented Jul 4, 2013 at 19:40
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1$\begingroup$ Predrag Cvitanovic has also looked at stability of Kuramoto-Sivashinsky, this time with a Fourier basis. cns.gatech.edu/~predrag/papers/preprints.html $\endgroup$– user25199Commented Jul 5, 2013 at 6:39
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I am aware of a few "applied" papers where this is done: http://pre.aps.org/abstract/PRE/v75/i4/e045203 http://pre.aps.org/abstract/PRE/v85/i4/e046201
answered Aug 3, 2013 at 19:13