On noninvertible mappings of the plane: eruptions. (English) Zbl 1055.37567

Summary: In this paper we are concerned with the dynamics of noninvertible transformations of the plane. Three examples are explored and possibly a new bifurcation, or “eruption,” is described. A fundamental role is played by the interactions of fixed points and singular curves. Other critical elements in the phase space include periodic points and an invariant line. The dynamics along the invariant line, in two of the examples, reduces to the one-dimensional Newton’s method which is conjugate to a degree two rational map. We also determine, computationally, the characteristic exponents for all of the systems. An unexpected coincidence is that the parameter range where the invariant line becomes neutrally stable, as measured by a zero Lyapunov exponent, coincides with the merging of a periodic point with a point on a singular curve.


37E99 Low-dimensional dynamical systems
65H10 Numerical computation of solutions to systems of equations
Full Text: DOI


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