Intertwined basins of attraction of dynamical systems (Q1030243)

The article deals with a modification of a notion of interwined attractors for the planar dynamical system \[ \dot x = X(x,y), \quad \dot y = Y(x,y). \] The authors say that the dynamical system (1.1) has interwined basins of attraction beside a point \(p\) if there exists a small sector \(S\), which \(p\) is the vertex, such that for any \(\epsilon > 0\) and any radius \(L\) in the sector \(S\) such that both \[ L \cap B(A_1) \cap D(p,\epsilon) \neq \emptyset \quad \text{and} \quad L \cap B(A_2) \cap D(p,\epsilon) \neq \emptyset \] hold for two different attractors \(A_1\) and \(A_2\) (\(B(A)\) is the basin of attraction of an attractor \(A\), \(D(p,\epsilon)\) is not defined). Further, two theorems are presented. The first of them gives a test when the system (1.1) has the interwining property (this property is not also defined). The second states conditions when the system (1.1) with the interwining property has the nonempty \(\alpha\)-limit set for some \(q\) near a saddle point. As an example the following system \[ \dot x = y \cdot (x^2 + y^2 - 3), \quad \dot y = (x - x^3 - \phi(x^2 + y^2) \cdot y)(x^2 + y^2 - 3) \] is considered.