Dynamical system generated by two-parametric family of quadratic mappings of plane is under consideration. Computer experiment on attractors gives prompts for finding and deriving of some analytical results. We determine several interesting invariant manifolds and sets (invariant straight lines, invariant crosses (line-line and line-parabola) and circle) existing for definite values of parameters. We point out the invariant measure (density) on invariant disc bounded by the invariant circle. Taking the special cases mentioned above as a base of perturbation method it is possible to understand some interesting generic situations.
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