Bifurcations

This chapter discusses the study of the quadratic family of functions Q_c(x) = x² + c where c is a constant. It introduces two of the most important types of bifurcations that occur in dynamics. Bifurcation means a division in two, a splitting apart, and that is exactly what has happened to the fixed points of Q_c. The chapter considers a one-parameter family of functions F_λ. Bifurcations occur in a one-parameter family of functions when there is a change in the fixed or periodic point structure as λ passes through some particular parameter value. Among the most important bifurcations is the saddle-node or tangent bifurcation. A saddle-node or tangent bifurcation occurs if the functions Fλ have no fixed points in an interval I for λ-values slightly less than λ₀, exactly one fixed point in I when λ = λ₀, and exactly two fixed points in I for λ slightly larger than λ₀.