The autonomous planar half-linear differential system
x′=ax+bϕp∗(y),y′=cϕp(x)+dy\documentclass[12pt]{minimal}
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\begin{document}$$x' = a x + b \phi_{p^*}(y), \quad y' = c \phi_p(x) + d y$$\end{document}is considered, where a, b, c and d are real constants, p and p∗\documentclass[12pt]{minimal}
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\begin{document}$${p^*}$$\end{document} are positive numbers with 1/p+1/p∗=1\documentclass[12pt]{minimal}
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\begin{document}$${1/p + 1/p^* = 1}$$\end{document}, and ϕq(s)=|s|q-2s\documentclass[12pt]{minimal}
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\begin{document}$${\phi_q(s)=|s|^{q-2}s}$$\end{document} for s≠0\documentclass[12pt]{minimal}
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\begin{document}$${s \ne 0}$$\end{document} and ϕq(0)=0,q>1\documentclass[12pt]{minimal}
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\begin{document}$${\phi_q(0) = 0, q > 1}$$\end{document}. When p=2\documentclass[12pt]{minimal}
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\begin{document}$${p = 2}$$\end{document}, this system is reduced to the linear system
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\begin{document}$$x' = a x + b y, \quad y' = c x + d y,$$\end{document}which can be solved by eigenvalues of the matrixabcd\documentclass[12pt]{minimal}
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\begin{document}$$
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$$\end{document}, that is, roots of the characteristic equation (λ-a)(λ-d)-bc=0\documentclass[12pt]{minimal}
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\begin{document}$${(\lambda - a)(\lambda - d) - bc = 0}$$\end{document}. In this paper, the characteristic equation for the autonomous planar half-linear differential system is introduced, and the asymptotic behavior of its solutions is established by roots of the characteristic equation.
