A degenerate Takens--Bogdanov bifurcation in a normal form of Lorenz's equations

TL;DR

The paper investigates the dynamics of a Lorenz-normal-form unfolding near a $\text{triple-zero}$ singularity to understand double-zero bifurcations. It employs center-manifold reductions and higher-order normal forms to derive a Takens–Bogdanov-type structure and, at a codimension-three point $\varepsilon_3=2/B$, a fifth-order reduction yielding a three-parameter unfolding that organizes various degenerate bifurcations. Using AUTO continuation, it maps bifurcation curves including Hopf, homoclinic, and heteroclinic connections, identifying degenerate points that seed infinite families of homoclinic orbits and, in some parameter regimes, chaotic attractors. The results provide a concrete, partial portrait of the Lorenz system near its triple-zero singularity and illustrate how analytical and numerical tools together illuminate complex organizing centers in nonlinear dynamics.

Abstract

In this work we consider an unfolding of a normal form of the Lorenz system near a triple-zero singularity. We are interested in the analysis of double-zero bifurcations emerging from that singularity. Their local study provide partial results that are extended by means of numerical continuation methods. Specifically, a curve of heteroclinic connections is detected. It has a degenerate point from which infinitely many homoclinic connections emerge. In this way, we can partially understand the dynamics near the triple-zero singularity.

Figures (4)

• Figure 1: For $\varepsilon_2=-1, B=-0.1, D=0.01$ partial bifurcation set in a neighborhood of the point $\mathtt {DZ}$: (a) in the fourth quadrant; (b) in the second quadrant. (c) Projection onto the $(x,z)$-plane of the heteroclinic cycle $\mathtt {He}$ of panel (a) that exists when $(\varepsilon_1,\varepsilon_3) \approx (0.2, -0.0043175)$. (d) Projection onto the $(x,z)$-plane of the heteroclinic cycle $\mathtt {He}$ of panel (b) that exists when $(\varepsilon_1,\varepsilon_3) \approx (-0.2, 0.0037414)$.
• Figure 2: For $\varepsilon_2=-1, B=-0.1, D=0.01$ partial bifurcation set: (a) in the second quadrant; (b) zoom of panel (a) in the vicinity of the points $\mathtt {DHe^{1}}$ (upper panel) and $\mathtt {TB}$ (lower panel).
• Figure 3: For $\varepsilon_2=-1$, $B=-0.1$, $D=0.01$: (a) partial bifurcation set in the second quadrant. (b) Zoom of panel (a). Projection onto the $(x,z)$-plane of T-point heteroclinic loops connecting $E_2$ and the equilibria $E_{3,4}$ (note that because of the symmetry a pair of the corresponding orbits exists): (c) principal T-point for $(\varepsilon_1,\varepsilon_3)\approx(-8.3738877, 0.0841835)$; (d) secondary T-point when $(\varepsilon_1,\varepsilon_3)\approx(-8.4159326, 0.0850368)$.
• Figure 4: For $\varepsilon_2=-1, \varepsilon_3=0.085, B=-0.1, D=0.01$ chaotic attractor when: (a) $\varepsilon_1=-8$, with initial conditions $(x_0,y_0,z_0)=(0,0.1,-8)$. (b) $\varepsilon_1=-6.3$, with initial conditions $(x_0,y_0,z_0)=(0,0.1,-7)$.