Localization and construction of cycles in Hopf's bifurcation at infinity (Q1363276)

Consider the \(n\)-dimensional autonomous differential system \[ dx/dt= A(\lambda)x+ f(x,\lambda)\tag{\(*\)} \] under the following assumptions: (i) \(\lambda\) is a scalar parameter, (ii) \(A:\mathbb{R}\to L(\mathbb{R}^n,\mathbb{R}^n)\) is continuous, \(A(\lambda_0)\) has a pair of simple pure imaginary eigenvalues \(\pm i\omega_0\), \(\omega_0>0\); \(0\) and \(\pm k\omega_0 i\) \((k=2,3,\dots)\) are no eigenvalues of \(A(\lambda_0)\). (iii) \(\lim_{|x|\to\infty}\max_{|\lambda-\lambda_0|\leq 1}{|f(x,\lambda)|\over|x|}= 0\). The authors study Hopf bifurcation from infinity by means of an equivalent integral equation. To approximate the bifurcating cycle, the method of functionalization of parameters and the Newton-Kantorovič method is used.