This article describes the Shilnikov bifurcation which is the basic nonlocal bifurcation leading to appearance of a complex behavior in dynamical system. Existence of a homoclinic orbit to a saddle focus with a positive saddle value is the well known now as the Shilnikov condition which guaranties chaotic behavior of a dynamical system and existence of a countable number of periodic orbits. The paper is well and clear written and extremely good illustrated. This article contains also many different examples which illustrate the Shilnikov bifurcation. I consider the text is aimed to be educational. In order to simplify formulas and without loss of generality, it is possible to assume that \gamma=1 and F_3=0. But that is only a recommendation.
Yes, one can do the time rescaling: t=(\gamma+F_3^*)t near the saddle-focus to make the "unstable" equation linear. Though this is mathematically sound, it makes sense to have natural eigenvalues in order to verify the Shilnikov condition in applications at once.
There are some misprints in the text like “…of of…“ and “…of is….”, which should be corrected.
Done, at two places, Hope there are no more.....