On the preservation of properties of differential equations under discretization (Q1375000)

Consider the system \[ \dot x=F(x,\alpha), \tag{1} \] where \(x\) is an \(n\)-dimensional vector, \(n\geq 1\), \(F(x,\alpha)\) is a sufficiently smooth or analytic vector-function of \((x,\alpha)\), \(F(0,0)=0\), and \(\alpha\) is a parameter. Let, under discretization, system (1) be replaced by the discrete systems \[ \begin{aligned} \overline{x}&= x+h \sum_{i=1}^s b_iF(Y_i,\alpha), \tag{2}\\ \overline{x}&= x+hF (\overline{x},\alpha), \tag{3}\end{aligned} \] which are constructed by the \(s\)-stage \(p\)th-order Runge-Kutta method and by the implicit Euler method, where \[ Y_i=x+h \sum_{j=1}^s a_{ij} F(Y_j,\alpha), \qquad 1\leq i\leq s, \] \(a_{ij}\) and \(b_j\) are coefficients of the Runge-Kutta method, and \[ \sum_{i=1}^sb_i=1, \qquad s\geq p\geq 1,\quad h>0. \] We consider the problem of the preservation of the properties of equilibrium states of system (1), when it is replaced by the systems (2) and (3), if equilibrium states of system (1) have one zero root. We give sufficient conditions for the preservation of multiplicity, stability and bifurcations of equilibrium states of system (1) in the case \(n=1,2\) and \(n\geq 3\) for \(h\) being not necessarily sufficiently small. We present an example.