The Gursey model is a 4D conformally invariant pure fermionic model with a nonlinear spinor self-coupled term. Gursey proposed his model as a possible basis for a unitary description of elementary particles following the "Heisenberg dream." In this paper, we consider the system of Gursey nonlinear differential equations (GNDEs) formed by using the Heisenberg ansatz. We use it to understand how the behavior of spinor-type Gursey instantons can be affected by excitations. For this, the regular and chaotic numerical solutions of forced GNDEs are investigated by constructing their Poincar, sections in phase space. A hierarchical cluster analysis method for investigating the forced GNDEs is also presented.