The non-transverse Shil'nikov-Hopf bifurcation ; uncoupling of homoclinic orbits and homoclinic tangencies

It is known that in a neighbourhood of a codimension-two Shil'nikov-Hopf bifurcation, primary periodic orbits lie on a single wiggly curve in period--parameter space, with accumulation points at parameter values of a pair of homoclinic tangencies to a periodic orbit. In contrast, it has recently been shown by Hirschberg and Laing that primary periodic orbits lie on an infinity of isolas in a neighbourhood of certain degenerate homoclinic tangency to a periodic orbit. This paper analyses the codim 3 bifurcation caused by a non-transverse (that is, degenerately parametrically unfolded) Shil'nikov-Hopf bifurcation, which contains nearby dynamics akin to both degeneracies. Two cases are classified as being downward pointing or upward pointing depending on whether the variation of a third parameter causes either the annihilation of a locus of saddle-focus homoclinic orbits to equilibria, or the uncoupling of this locus from the locus of Hopf bifurcations. We undertake a heuristic analysis of the unfolding, showing that in both cases it contains codimension-two non-transversal homoclinic orbits to equilibria and non-transversal homoclinic tangencies to periodic orbits. Unfolding the former non-transverse orbit, is shown to cause two wiggly curves to coalesce and leave finitely many isolas of periodic orbits. Unfolding the latter causes two wiggly curves to coalesce into first infinitely many and then finitely many isolas. Asymptotic expressions are given for the accumulation of two types of isola forming bifurcations. The implications of $Z_2$-equivariance on the unfolding is discussed. Finally, numerical evidence is presented for both upward and downward pointing non-transverse Shil'nikov-Hopf bifurcations occuring in a model of an autonomous nonlinear electronic circuit, both with respect to a $Z_2$-equivariant and a $Z_2$-non-equivariant equilibrium. Numerical computation of curves of periodic orbits, homoclinic orbits and homoclinic tangencies to periodic orbits are shown to agree broadly with the theory but to uncover extra complications.