Zero-Hopf calculations for neutral differential equations (Q6652558)

The paper focus on the necessary conditions to guarantee the existence of the zero-Hopf singularity for differential equations of neutral type. Consider a neutral functional differential equation \N\[\N\dot{z(t)} +E \dot{z}(t-\tau)= A(\epsilon)z(t)+B(\epsilon)z(t-\tau)+F(z(t),z(t-\tau),\epsilon)\tag{1}\N\]\Nwhere \( z\in \mathbb{R}^n, \epsilon \in \mathbb{R}^m,\) and \( E \) is \( n \times n \) real matrix. \( A(\epsilon), B(\epsilon) \in C^2(\mathbb{R}^m, \textit{M}_{n\times n}(\mathbb{R})) \) and \(F\in C^3(\mathbb{R}^{2n+m}, \mathbb{R}^n ) \) and satisfies\N\[\NF(0,0,\epsilon)= \frac{\partial F}{\partial x}(0,0,\epsilon)=\frac{\partial F}{\partial y}(0,0,\epsilon)=0\N\]\NIn the literature, zero-Hopf singularity studies have been discussed a lot for different types of systems. The paper discussed for delayed differential equations of general type, by applying the normal form theory and the theorem of reduction on the center manifold, reduced the considered neutral differential equation to an ordinary differential equation form.