Weakly structural stability of cubic homogeneous parameterized vector fields
The authors study the weak structural stability of the vector field \(X\) with the property \[ j_3(X(x, y))_{(0,0)}= X_3(x,y),\quad j_2(X(x, y))_{(0,0)}= 0, \] where \(J_k(X(x, y))_{(0,0)}\) denotes the \(k\)-jet of \(X\) near the origin, and where \(X_3\) is of the form \[ X_3(x,y)= (a_1 x^3+ a_2 x^2y+ a_3 xy^2, b_2x^2y+ b_3xy^2+ b_4 y^3). \] They apply the theorems and technique of Takens to find sufficient conditions on the parameter space such that \(X\) is weakly structurally stable. Specifically, they prove that \(X\) is weakly structurally stable under one of the following conditions: \[ b_4\neq a_3,\quad b_3\neq a_2,\quad b_2\neq a_1;\quad b_4\neq a_3,\quad b_3= a_2,\quad b_2\neq a_1. \]
cubic homogeneous parameterized vector fields
scientific article; zbMATH DE number 1542473