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Zh. Vychisl. Mat. Mat. Fiz., 2011, Volume 51, Number 4, Pages 620–630 (Mi zvmmf9230)  

On solutions of three-dimensional systems describing the transition from an unstable equilibrium to a stable cycle

S. E. Gorodetski, A. M. Ter-Krikorov

Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, Moscow oblast, 141700 Russia

Abstract: Given a three-dimensional dynamical system on the interval $t_0<t<+\infty$, the transition from the neighborhood of an unstable equilibrium to a stable limit cycle is studied. In the neighborhood of the equilibrium, the system is reduced to a normal form. The matrix of the linearized system is assumed to have a complex eigenvalue $\lambda=\varepsilon+i\beta$, with $\beta\gg\varepsilon>0$ and a real eigenvalue $\delta<0$ with $|\delta|\gg\varepsilon$. On the arbitrary interval $[t_0,+\infty)$, an approximate solution is sought as a polynomial $P_N(\varepsilon)$ in powers of the small parameter $\varepsilon$ with coefficients from Hölder function spaces. It is proved that there exist $\varepsilon_N$ and $C_N$ depending on the initial data such that, for $0<\varepsilon<\varepsilon_N$, the difference between the exact and approximate solutions does not exceed $C_{N^{\varepsilon^{N+1}}}$.

Key words: dynamical system, small parameter, transient process, unstable equilibrium, stable limit cycle.

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English version:
Computational Mathematics and Mathematical Physics, 2011, 51:4, 575–585

Bibliographic databases:

UDC: 519.624.2
Received: 07.10.2009

Citation: S. E. Gorodetski, A. M. Ter-Krikorov, “On solutions of three-dimensional systems describing the transition from an unstable equilibrium to a stable cycle”, Zh. Vychisl. Mat. Mat. Fiz., 51:4 (2011), 620–630; Comput. Math. Math. Phys., 51:4 (2011), 575–585

Citation in format AMSBIB
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\paper On solutions of three-dimensional systems describing the transition from an unstable equilibrium to a~stable cycle
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2011
\vol 51
\issue 4
\pages 620--630
\mathnet{http://mi.mathnet.ru/zvmmf9230}
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\vol 51
\issue 4
\pages 575--585
\crossref{https://doi.org/10.1134/S0965542511040166}
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