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Given a dynamical system:

  1. Is there a set of necessary and sufficient conditions for it to have an attractor?
  2. Is there a way to test the nature of the attractor? whether it is a strange attractor or a fixed point or a limit circle?
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The question is much too broad to have a useful answer. Regarding 2: fixed points are relatively easy to find, and testing them for being attractors is not hard either. – user53153 Jan 27 '13 at 21:57
Depends on the class of dynamical systems: whether the state space is discrete, continuous or hybrid, whether the time (semi)group is continuous or discrete and specifics of each of these. For example in smooth ODEs in the plane, strange attractors are precluded (the 2D topology prevents orbits from crossing over each other). – alancalvitti Jan 27 '13 at 23:52

about question 2: you can reconstruct the phase space of any discrete dynamic system and then recognize type of the attractor. if you have continious system with equations, you should make a autonomus equation respect to time and then draw picture of equation.

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