Symmetric functions and exact Lyapunov exponents

Newton's method applied to the elementary symmetric functions of polynomials generates a class of dynamical systems. These systems have invariant lines on which the Lyapunov exponents can be found analytically, thus predicting the exact parameter values for which these structures are chaotic attractors (in the sense of Milnor) and precisely when bifurcations, such as blowout, destroy their stability. Often, blowout bifurcations lead to a behavior called on-off intermittency. We also present evidence that the bursting mechanism of on-off intermittency frequently occurs in neighborhoods focal points, a common feature of maps that have singularities. Finally, because of an embedding property, this new class of examples can be extended to construct systems having this bifurcation in higher dimensions.