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Reduction of periodic motions to steady-state motions. (Russian) Zbl 0617.34031

The system \[ (1)\quad \dot x_ k=\lambda_ kx_ k+\alpha_ kx_{k- 1}+\sum a_ k^{(m_ 1,...,m_ n)}(t)x_ 1^{m_ 1}...x_ n^{m_ n}+\phi_ k(x_ 1,...,x_ n,t) \] (k\(=1,...,n\); \(2\leq m_ 1+...+m_ n\leq N)\), where \(a_ k^{(m_ 1,...,m_ n)}(t)\) are T- periodic with respect to t, \(\alpha_ k\) are equal to 0 or 1 and \(\lambda_ k\) may satisfy the condition \(\sum^{n}_{\sigma =1}m_{\sigma}\lambda_{\sigma}-\lambda_ k=(2\pi /T)iN_ k^{(m_ 1,...,m_ n)},\) \(N_ k^{(m_ 1,...,m_ n)}=0,\pm 1,\pm 2,...,m_ k=0,1,2,...\), is considered.
Periodic transformations of equation (1) to the form \(\dot y_ k=\lambda_ ky_ k+\alpha_ ky_{k-1}+\sum c_ k^{(m_ 1,...,m_ n)}y_ 1^{m_ 1}...y_ n^{m_ n}+\psi_ k(y_ 1,...,y_ n,t)\) \((k=1,...,n\); \(2\leq m_ 1+...+m_ n\leq N)\), where \(c_ k^{(m_ 1,...,m_ n)}\) are constants, \(\psi_ k(y_ 1,...,y_ n,t)\) are analytic functions of \(y_ 1,...,y_ n\) and its decomposition has no terms with order less than \((N+1)\) are given.
Reviewer: V.Kozobrod

MSC:

34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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