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Localization and construction of cycles in Hopf’s bifurcation at infinity. (English. Russian original) Zbl 0880.34038

Dokl. Math. 52, No. 2, 223-226 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 344, No. 4, 446-449 (1995).
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.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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