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Local analysis of singularities of an invertible system of ordinary differential equations. (English. Russian original) Zbl 0862.34021
Russ. Math. Surv. 50, No. 6, 1258-1259 (1995); translation from Usp. Mat. Nauk 50, No. 6, 169-170 (1995).
The authors study the four-dimensional system \(x'= (L+M) x+\varphi (x)\), where \(L\) is a Jordan block with zero eigenvalue, \(M\) is the matrix of small parameters, and the Taylor series of \(\varphi\) are free of constant and linear terms. It is assumed that the system is reversible, i.e., invariant with respect to the transformation \((t,x_1, x_2, x_3,x_4) \to (-t,x_1,-x_2, x_3,-x_4)\). They apply the theory of normal forms to study local bifurcations of the system. For example, they establish explicit conditions for the existence of periodic and quasiperiodic solutions. No proofs are given.

MSC:
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
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