\(e^ z\): Dynamics and bifurcations. (English) Zbl 0874.58066

Devaney, Robert L.

Int. J. Bifurcation Chaos Appl. Sci. Eng. 1, No. 2, 287-308 (1991).

Summary: We describe some of the dynamical behavior of the complex exponential \(\lambda\exp z\). For various values of \(\lambda\), this family exhibits chaotic behavior on the entire complex plane. For other values, the dynamics are relatively tame. We show how to analyze this behavior via symbolic dynamics and investigate the structural instability at various parameter values. Finally, we describe the relationship between the parameter space for the exponential family and the related families of polynomials given by \(\lambda(1+{z\over d})^d\).

MSC:

37F99	Dynamical systems over complex numbers
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37E99	Low-dimensional dynamical systems
30D05	Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable

Keywords:

complex exponential; chaotic behaviour; complex plane; symbolic dynamics; structural instability; polynomials