Eigen values and eigen vectors are characteristic to matrices which define linear operators. Are these concepts also extendible to the nonlinear systems
Yes, most nonlinear equations are expanded up to first order terms and then usually solved using the same methods as linear equations.
following links will be very useful
Computation of eigenvalues and eigenfunctions for the ...
dl.acm.org/citation.cfm?id=2260263
by X Ying - 1988 - Cited by 1 - Related articles
Dec 31, 1988 - Eigenvalues are roots of a nonlinear eigenequation and must be determined together with associated eigenfunctions prior to the use of the finite .... Modelling chemical systems I: Statistical modeling of fluid-solid reactions in porous media ... gradient method for iterative solution of linear systems
www.cds.caltech.edu/~murray/books/.../am06-complete_16Sep06.pdf
dl.acm.org/citation.cfm?id=2260263
arxiv.org/pdf/gr-qc/9904017
For Hamiltonian quantum mechanics of Kohn-Sham theory of many particle systems with corresponding nonlinear potentials , one on a regular basis finds energy and states
where what corresponds to eigenvalue problems in linear potentials is generalized in the meaning of optimization of Lagrange multipliers.
This is a great question. In addition to the other answers, Koopman operator theory concerns the "linearization" of nonlinear dynamical systems through the choice of a special coordinate system given by eigenfunctions of the Koopman operator.
The Koopman operator was defined by B. O. Koopman in 1931 for Hamiltonian systems. It has been studied recently in the context of data-driven models for nonlinear dynamical systems, such as in fluid dynamics.
Original paper: http://www.pnas.org/content/17/5/315.citation
Koopman eigenfunctions to linearize nonlinear dynamics: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0150171
Koopman in fluid dynamics:
http://dx.doi.org/10.1017/S0022112009992059
http://www.annualreviews.org/doi/abs/10.1146/annurev-fluid-011212-140652
Probably the most significant application of eigenvalues/eigenfunctions in nonlinear contexts appears in bifurcation theory: looking at the Implicit Function Theorem in infinite-dimensional contexts shows that lack of invertibility of the linearization (i.e., at an eigenvalue) indicates a qualitative change in the dynamics as a parameter varies. For a nice example see Kolodner's 1955 paper "Heavy rotating string—a nonlinear eigenvalue problem"
I would like to recommend a recent lovely book, "Nonlinear Perron-Frobenius theory," by L. Bas and R. Nussbaum (Cambridge University Press, 2012), ISBN 9780521898812.
It is an intellectual adventure to study this subject.
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