Normal forms for neutral functional differential equations (Q2738691)

The author discusses neutral functional-differential equations of the form NEWLINE\[NEWLINE \frac{d}{dt}\Big[Dx_t-G(x_t)\Big]=Lx_t+F(x_t), NEWLINE\]NEWLINE where \(x_t\in C=C([-r, 0], \mathbb{R}^n), x_t(\theta)=x(t+\theta)\), and \(D, L\) are bounded linear operators from \(C\) to \(\mathbb{R}^n\). More specificly, \(D\varphi =\varphi (0)-\int_{-r}^0d[\mu (\theta)] \varphi (\theta), L\varphi =\int_{-r}^0d[\eta (\theta)] \varphi (\theta)\), where \(\mu\) and \( \eta\) are matrix-valued functions of bounded variation that are continuous from the left on \((-r, 0)\) and such that \(\eta (0)=\mu(0)=0\) and \(\mu\) is non-atomic at zero. The functions \(F\) and \(G\) are \(C^N\)-smooth, \(N\geq 2, F(0)=G(0)=0\) and \(F'(0)=G'(0)=0\). For normal forms of retarded functional-differential equations, the main idea is to write the RFDE as an abstract ODE in an appropriate phase space and to extend the methods known for finite-dimensional ODEs to the infinite-dimensional case. The author extends that idea to the above NFDE and the idea presented here allows the normal forms of the NFDE to be directly computed without first finding the equation on the finite-dimensional manifold. Finally, a Bogdanov-type singularity is considered as an example.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00044].