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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 $$b^ T=(b^ 1_{22},-2b^ 1_{12})$$.
##### MSC:
 34C99 Qualitative theory for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
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