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Sufficient conditions for two-dimensional point dissipative nonlinear systems. (English) Zbl 0685.34054
Summary: A two-dimensional autonomous system \(\dot x=AX+(x^ TB^ 1x,x^ TB^ 2x)^ T\) of differential equations with quadratic nonlinearity is point dissipative, if there exists a positive number \(\gamma\) such that the symmetric matrices \(B^ 1\) and \(B^ 2\) are of the form \[ B^ 1=\begin{pmatrix} 0 & b^ 1_{12} \\ b^ 1_{12} & b^ 1_{22}\end{pmatrix},\quad B^ 2=-\gamma \begin{pmatrix} 2b^ 1_{12} & b^ 1_{22} \\ b^ 1_{22} & 0 \end{pmatrix} \] and \(b^ T\begin{pmatrix} \gamma&0\\ 0&1 \end{pmatrix} Ab<0\), where \(b^ T=(b^ 1_{22},-2b^ 1_{12})\).
34C99 Qualitative theory for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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