Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

Abstract

We investigate a dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analyzed and yield the transcritical bifurcation as the codimension-one bifurcation while the saddle-node–transcritical interaction and the Hopf–transcritical interactions as the codimension-two bifurcations. The unfolding of this degeneracy is also analyzed and reveal global bifurcations such as homoclinic and heteroclinic bifurcations. We apply our results to a modified Lotka–Volterra model and to an infection model in HIV diseases.

Saddle-node--transcritical interactions in a stressed predator-prey-nutrient system

“…The normal form for the single zero SNTC interaction proposed by Saputra et al [2010] and Saputra [2015]…”

“…At such a codimension two point, curves of transcritical bifurcations and saddle-node bifurcations are tangent. This case can be realised in a model with a single degree of freedom and was investigated in detail by Saputra et al [2010] and Saputra [2015]. When one only considers nonnegative equilibria, the unfolding looks similar to that of the generalised Hopf bifurcation, which explains why this singularity is sometimes called the generalised transcritical bifurcation.…”

“…The corresponding eigenvector lies in the invariant plane, while the generalised eigenvector is transversal to it. Normal forms and unfoldings for the double zero SNTC interaction were presented by Saputra et al [2010] and Saputra [2015] and involve, in addition to the saddle-node and transcritical bifurcations, a Hopf bifurcation and a curve along which a periodic orbit disappears. The latter can be either a heteroclinic bifurcation or a homoclinic to a saddlenode.…”

“…This transcritical-Hopf case was investigated in some detail in the literature, starting from Langford [1979], who presented several one-parameter unfoldings near the singularity. Later, Jiang & Wang [2010] and Saputra [2015] presented four distinct two-parameter unfoldings. In the vicinity of this point, one can find periodic orbits inside and outside the invariant plane, as well as quasi-periodic motion.…”

“…Kooi et al [2008] showed that, around this point, stable equilibrium and periodic solutions exist with zero or positive predator density. From the layout of the Hopf and transcritical bifurcations, we can tell that the unfolding corresponds to case III of Saputra [2015].…”