Jack Hale had important work in generalizing concepts of dynamical systems to the infinite dimensional setting, see his monographs:

[Asymptotic behavior of dissipative systems][1]

It turns out that, under compactness assumtions, it is possible to generalize a lot of the finite dimensional concepts. The work of Temam, Foiaș and their group connected to the Navier-Stokes equations is also extremely important, see

[Infinite dymensional dynamical systems][2]

On more recent work of the chaotic behavior (as mentioned b Uwe in his comment) you should consult the excellent monograph

[Linear Chaos][3]

where there is an account on recent results.

But of course the main problem is, as remarked by Uwe, that we are still struggling with the proper notions of generalization of some of the concepts mentioned. [1]: http://books.google.hu/books?id=fSzscCu37ygC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false [2]: http://books.google.hu/books?id=0VZMYKSgkqMC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false [3]: http://books.google.hu/books?id=UwsBihOIDkoC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false

Jack Hale had important work in generalizing concepts of dynamical systems to the infinite dimensional setting, see his monographs:

Asymptotic behavior of dissipative systems

It turns out that, under compactness assumtions, it is possible to generalize a lot of the finite dimensional concepts. The work of Temam, Foiaș and their group connected to the Navier-Stokes equations is also extremely important, see

Infinite dymensional dynamical systems

On more recent work of the chaotic behavior (as mentioned b Uwe in his comment) you should consult the excellent monograph

Linear Chaos

where there is an account on recent results.

But of course the main problem is, as remarked by Uwe, that we are still struggling with the proper notions of generalization of some of the concepts mentioned.