Selfinteracting fields with very strong selfinteraction can lead to chaotic dynamics.
If you want to know how chaotic self-interacting scalar fields look like,
and how methods from the theory of dynamical systems
can be used in quantum field theories,
you may look into the following paper:

  • C. Beck, Chaotic Strings and standard model parameters,
    Physica 171D, 72--106 (2002) (hep-th/0105152)

    You may also look into my book on this subject (World Scientific, 2002)

    For recent cosmological applications of this concept, see Chaotic scalar fields as models for dark energy Phys. Rev. 69D, 123515 (2004) (astro-ph/0310479)

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