This interesting paper studies the influence of (deterministic and random) external forcing on a saddle-node bifurcation pattern for interval maps. The maps under consideration are given in skew product form \[ f:\Theta\times X\to\Theta\times X,\quad (\theta,x)\mapsto(\omega(\theta),f_\theta(x)), \] where the base space \(\Theta\) is either a topological space (deterministic forcing) or a measure space (random forcing), while \(X\) (mostly) denotes a compact interval \([a,b]\). The fixed points as bifurcating objects in the unperturbed case are replaced by invariant graphs, namely measurable functions \(\phi:\Theta\to[a,b]\) satisfying \(f_\theta(\phi(\theta))=\phi(\omega(\theta))\) for almost all \(\theta\in\Theta\). On this basis, direct analogs to the classical autonomous result are obtained: Provided the fiber maps \(f_\theta\) are monotone and convex, there exists a critical parameter value distinguishing two invariant graphs from the situation of no such graph. At the critical parameter itself, two alternatives exist: In addition to the existence of a unique invariant graph, there might exist a pair of so-called pinched invariant graphs. In the quasi-periodic case they are often called “strange non-chaotic attractors”.
The proof relies on a result due to G.Keller [Fundam. Math.151, No. 2, 139–148 (1996; Zbl 0899.58033)]. Moreover, the authors extend contributions by R.Sturman and J.Stark [Nonlinearity 13, No. 1, 113–143 (2000; Zbl 1005.37016)] on the structure of invariant sets, as well as C. Núñez and R. Obaya [Discrete Contin.Dyn.Syst., Ser.B 9, No.  3–4, 701–730 (2008; Zbl 1151.37021)].
Finally, extensions to continuous time systems and various illustrative examples are given.