A construction of three-dimensional vector fields which have a codimension two heteroclinic loop at Glendinning-Sparrow T-point
The author gives the following result: Consider the three-dimensional vector field given by the autonomous differential system (1) \(\dot x=y\), \(\dot y=2x(x-1)\) \((x-2)+ \varepsilon z+\mu(x-2)^ 2-{8 \over 15} \kappa (x-2)\), \(\dot z=({3 \over 2} x-1)z+ \varepsilon + \kappa y\). For any small \(\delta>0\) there exists a triple \((\varepsilon_ 0,\mu_ 0,\kappa_ 0)\) with \(\varepsilon_ 0>0\) and \(| \varepsilon_ 0 |+| \mu_ 0 |+| \kappa_ 0 | <\delta\) such that the vector field (1) with \((\varepsilon,\mu,\kappa) =(\varepsilon_ 0, \mu_ 0, \kappa_ 0)\) possesses a heteroclinic loop at a (Glendinning- Sparrow's) \(T\)-point. Here a heteroclinic loop of length 2 is considered, consisting of only two heteroclinic orbits each of which connects hyperbolic equilibria \(O^{(1)}\) and \(O^{(2)}\). This heteroclinic loop is said to be the one at a \(T\)-point if: (i) One of the equilibria (say, \(O^{(1)})\) is of real saddle type and the other one (say, \(O^{(2)})\) is of saddle-focus type; (ii) The unstable manifold of \(O^{(2)}\) and the stable manifold of \(O^{(1)}\) intersect transversely along the heteroclinic orbit from \(O^{(2)}\) to \(O^{(1)}\), and, on the other hand, the heteroclinic orbit from \(O^{(1)}\) to \(O^{(2)}\) is of codimension 2. The proof of the result is based on the Melnikov-like perturbation method. A relation among the Shil'nikov bifurcation, the fishhook structure and the chaotic dynamics is discussed too.
scientific article; zbMATH DE number 409593
scientific article