Classification of potential flows under renormalization group transformation (Q896623)

The paper studies the system \[ \frac{dx_i}{dt}=x^TA^{(i)}x, \quad i=1,2,\dots,n, \eqno(1) \] where \(x=(x_1,\dots,x_n)^T\in R^n\), \(A^{(i)}=(a_{ij}^k)\) is a symmetric \(n\times n\) matrix satisfying the totally symmetric conditions \[ a^{(i)}_{jk}=a^{(j)}_{ik}=a^{(k)}_{ij},\quad i,j,k = 1,2,\dots,n. \] The initial condition \(x(0)\) is supposed to satisfy \(x(0)\not\in E\), where \(E\) is the set of equilibria of (1). The authors consider system (1) in the presence of a renormalization-group potential and give a sufficient condition on the inital conditions for the blow-up behaviour of solutions. They also derive the polar equation for (1). Finally, they prove that the blow-up rate of the solution \(x_i\) is \((t-T)^{-1}\), where \(T\) is the blow-up time. The last section of the paper is devoted to the discussion.