Invariant Sets and Attractors of Quadratic Mapping of Plane: Computer Experiment and Analytical Treatment

Consider a dynamical system generated by a quadratic mapping of the plane (x, y) 7−→ (λx + xy, μy + x) with parameters λ, μ ∈ R. This 2-parameter family arises as a finite-difference approximation of an ODE system with special quadratic nonlinearity and also as a leading part of a map on a center manifold (in some problems of intersections of bifurcations). Computer experiments on attractors with varying λ and μ gives rise to finding and deriving some analytical results. We determine several interesting invariant manifolds and sets (invariant straight lines, invariant crosses (line-line and line-parabola) and circle) that exist for definite values of parameters. Next we point out the invariant measure (density) on the invariant disc bounded by the invariant circle.