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In dynamical systems people often classify (groups of) dynamical behaviours with the terminology of conic sections. Hyperbolic, elliptic and parabolic dynamics all presumably have some kind of meaning (even if it is not super precise). Hyperbolic dynamical systems are particularly well studied and are characterised by the existence of "expanding and contracting directions of the derivative". However it is not clear why we call this behaviour hyperbolic? If it is just terminology, fine - but if there is a more concise geometric meaning then I would like to know.

Finally, and to the main point of the question - how are the other classifications defined? e.g. parabolic and elliptic dynamics. There seems to be little information about these classes in their own right, meanwhile there is a lot of introductory information about the hyperbolic case. Are they not interesting? Or overly exotic? Can we provide a simple example for each?

algae
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    in my field at least (holomorphic dynamics), elliptic means conjugated to a rotation, and parabolic means convergence to a neutral fixed point. in particular, neither is chaotic, and arguably less interesting than hyperbolic systems (there are still some very deep and fascinating questions related to (local) elliptic or parabolic behaviour) – Albert Mar 05 '21 at 22:58
  • Very interesting. Could you point me in the direction of a paper trail? – algae Mar 05 '21 at 23:00
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    I think the origin of the terminology comes from the local classification of a zero of a vector field in the plane : either you have sink or source (boring), or hyperbolic fixed point (where trajectories locally look like hyperbolas), or an elliptic fixed point (both eigenvalues are purely imaginary and trajectories locally look like ellipses). parabolic singularities must fit in there somewhere – Albert Mar 05 '21 at 23:03
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    There is also a link with elliptic/parabolic/hyperbolic (or loxodromic) subgroups of PSL2(R). Note that the associated vector fields may not have fixed points, so the relationship with the classification of fixed points is not obvious (although, sometimes, with well-chosen Poincaré sections). For instance, in this setting, the geodesic flow is hyperbolic, while the horocyclic flow is parabolic. – D. Thomine Mar 05 '21 at 23:48
  • Wikipedia: Elliptic partial differential equation , Hyperbolic partial differential equation , Parabolic partial differential equation . – Han de Bruijn Mar 09 '21 at 10:55
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    This may help you a lot : https://arxiv.org/pdf/1503.07859.pdf – nmasanta Mar 12 '21 at 08:10

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