The authors consider vector fields defined in \(\mathbb R^4\) which commute with the action of a group \(\Gamma\) of isometries of \(\mathbb R^4\): \[ \dot{x}=f (x),\text{ where }(\gamma x)=\gamma f(x)\text{ for all }\gamma \in \Gamma,\, \Gamma \subset O(4)\text{ is finite} \] and \(f\) is a smooth map in \(\mathbb R^4\). These kind of vector fields can possess robust heteroclinic cycles. The properties of these objects in the case of a vector field of \(\mathbb R^3\) has been already studied and a complete classification has been done. The dynamics in \(\mathbb R^4\) are more intricate. In an article [M. Krupa and I. Melbourne, Ergodic Theory Dyn. Syst. 15, No. 1, 121–147 (1995; Zbl 0818.58025)] a definition of simple robust heteroclinic cycle is provided. Let \(\Delta_j\) denote the isotropy subgroup of an equilibrium \(\xi_j\). The heteroclinic cycle is simple if

(i)

the fixed-point subspace of \(\Delta_j\), denoted \(\mathrm{Fix}(\Delta_j)\), is an axis;

(ii)

for each \(j\) , \(\xi_j\) is a saddle and \(\xi_{j+1}\) is a sink in an invariant plane \(\mathrm{Fix}(\Sigma_j)\) (hence \(\Sigma_j \subset \Delta_j \cap \Delta_{j+1}\));

(iii)

the isotypic decomposition for the action of \(\Delta_j\) in the tangent space at \(\xi_j\) only contains one-dimensional components.

In the aforementioned article, condition (iii) is omitted in the definition, although it is implicitly assumed. The heteroclinic cycles satisfying (i) and (ii) but not (iii) are called pseudo-simple.
The aim of the work is to investigate the dynamical properties of pseudo-simple heteroclinic cycles in \(\mathbb R^4\), such as stability and asymptotic stability. Some particular examples are analyzed.