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This article is cited in 1 scientific paper (total in 1 paper)
Bifurcation of the Point of Equilibrium in Systems with Zero Roots of the Characteristic Equation
V. V. Basov Saint-Petersburg State University
Abstract:
We consider the following real autonomous system of $2d$ differential equations with a small positive parameter $\varepsilon $:
$$
\dot x_i=x_{i+d}+X_i^{(n+1)}(x,\varepsilon ),\qquad
\dot x_{i+d}=-x_i^{2n-1}+X_{i+d}^{(2n)}(x,\varepsilon ),\qquad i=1,\dots,d,
$$
where $d\ge 2$, $n\ge 2$, and the $X_j^{(k)}$ are continuous functions continuously differentiable with respect to $x$ and $\varepsilon $ the required number of times in the neighborhood of zero; their expansion begins with order $k$ if we assume that the variables $x_i$ are of first order of smallness, $\varepsilon $ is of second order, and the variables $x_{i+d}$ are of order $n$. We write out a finite number of explicit conditions on the coefficients of the lower terms in the expansion of the right-hand side of this system guaranteeing that for any sufficiently small $\varepsilon > 0$ the system has one or several $d$-dimensional invariant tori with infinitely small frequencies of motions on them.
Received: 24.12.2002 Revised: 11.06.2003
Citation:
V. V. Basov, “Bifurcation of the Point of Equilibrium in Systems with Zero Roots of the Characteristic Equation”, Mat. Zametki, 75:3 (2004), 323–341; Math. Notes, 75:3 (2004), 297–314
Linking options:
https://www.mathnet.ru/eng/mzm35https://doi.org/10.4213/mzm35 https://www.mathnet.ru/eng/mzm/v75/i3/p323
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