Finitely determined singularities of functions generated by unsolved integration (Q1022263)

The author considers the solution of a problem \(F(\dot x,t)= 0\), \(x(0)= 0\), where \(F(\dot x,t)\) \((F(0,0)= 0)\) is a function of a vector and scalar arguments belonging to some class of smoothness. The above equation can be reduced to (after the change \(\dot x(t)= y)\): \[ F(y, t)= 0.\tag{1} \] The main purpose of the paper is to discover the mechanism of generation of singularity of \(x(t)\) at zero for a smooth function \(F\). We formulate the main result from Section 1. Theorem 1. Let \(y_1(t)\) and \(y_2(t)\) be different simple solutions of (1) with asymptotic orders \(l_1\) and \(l_2\), respectively. Let \(\overline y(t)\) be a solution of equation \(F_1(y,t)= 0\), where \(F_1\) is \(C^{r+1}\), \(F_1\in j^r_0(F)\). Then the relations \(\overline y(t)= y_1(t)+ o_1(|t|^{l_1})\), \(\overline y(t)= y_2(t)+ o_2(|t|^{l_2})\) cannot hold simultaneously. In Section 2 the author gives a sufficient criterion of \(r\)-determination of a simple small solution. In Section 3 the relationship between the finite determination of an equation and the bifurcation multiplicity of the mapping \(F\) is considered. In Section 4 a criteria of finite determination of an equation in a more general case is formulated. In Section 5 four corollaries are given.