Homoclinic saddle to saddle-focus transitions in 4D systems

Abstract

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is 3-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3-dimensional stable leading eigenspace. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that the standard Belyakov case. We also give an example of this bifurcation in the wild case occuring in a perturbed Lorenz-Stenflo 4D ODE model.

Complex dynamics in an unexplored simple model of the peroxidase–oxidase reaction

“…37 Furthermore, the saddle quantity σ 0 (sum of leading stable and unstable eigenvalues) was found to be positive throughout the domains of the 2D isospike stability diagrams shown below, which suggests the possibility of Shilnikov chaos. 38 Fig. 1 shows three 2D isospike stability diagrams for parameter pairs of k 8 against k 1 , k 2 , or k 3 , respectively.…”

Operon dynamics with state dependent transcription and/or translation delays

“…15 Configuration of leading negative real and complex-conjugate eigenvalues as v min M varies. A 3dimensional transition (Kalia et al 2019) occurs on the middle branch of unstable steady states as in Fig. 11 the steady state along a direction that is tangential to the dominant stable manifold before leaving along a direction that is tangential to the unstable manifold.…”

“…15, for a very nearby value of v min M , the leading negative real eigenvalue λ 2 and complex-conjugate eigenvalues λ 3,4 exchange order. Such a transition in a system also having one real positive eigenvalue is called a 3-dimensional or 3DL transition and is associated with rich Shilnikov homoclinic bifurcation structures (Kalia et al 2019).…”

Global and local bifurcations, three-dimensional Henon maps and discrete Lorenz attractors

“…In continuous time such a homoclinic bifurcation was studied recently in Ref. 38, where it was called the 3DL-bifurcation. Under small perturbations, manifold W ss appears with alternating direction and dimension, namely in the case when |λ 1 | < λ 2 it is one-dimensional and tangent to the λ 1 eigendirection, and when |λ 1 | > λ 2 , strong stable manifold W ss is two-dimensional and tangent to the eigendirections corresponding to complex eigenvalues λ 2 e ±iϕ .…”