Sil'nikov bifurcations in generic 4-unfoldings of a codimension-4 singularity
The authors consider singularities of vector fields on \(\mathbb{R}^3\) with linear part \(\dot x = Ax\), where \(A\) is conjugate to \(\left( \begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{smallmatrix} \right)\). Such singularities can be transformed to the form \[ \begin{aligned} \dot x & = y, \\ \dot y & = z, \\ \dot z & = a_{11} x^2 + a_{12} xy + a_{13} xz + a_{22} y^2 + a_3 x^3 + \text{ other cubic terms } + \dots \end{aligned} \] The authors assume \(a_{11} = 0\), so the singularity is of codimension 4. They study generic unfoldings of such singularities by means of a blowup of the spatial variables and the parameters that is reminiscent of the one used to study the Takens-Bogdanov bifurcation. The blown up vector field, with parameters \((\sigma_1, \varepsilon, \sigma_3, \sigma_4)\), has the form \[ \begin{aligned} \dot x & = y + o (\varepsilon) \\ \dot y & = z + x - {1 \over 2} x^2 + o (\varepsilon) \\ \dot z & = \sigma_1 x - z + \sigma_3 z + \sigma_4 x^2 + O (\varepsilon) \end{aligned} \] When \((\sigma_1, \varepsilon, \sigma_3, \sigma_4) = (0,0,0,0) \), the origin is a hyperbolic equilibrium with eigenvalues \(-1,-1\), \(1\), and a homoclinic orbit (the curve \(x^2 - y^2 - {x^3 \over 3} = 0\) in the plane \(z = 0)\). By using a variation of Melnikov's method, the authors show that the homoclinic orbit persists in a region of \((\sigma_1, \varepsilon, \sigma_3, \sigma_4)\) space where the eigenvalues of the equilibrium at the origin become \(- \rho \pm i \omega\) and \(\nu\) with \(o < \rho < \nu\). By a well-known result of L. P. Shil'nikov, there exist an infinite number of horseshoes near such a homoclinic orbit. According to \textit{L. Mora} and \textit{M. Viana} [Acta. Math. 171, 1-71 (1993; Zbl 0815.58016)], as the parameters near the Shil'nikov bifurcation point vary to form these horseshoes, strange attractors appear.
scientific article; zbMATH DE number 795277