On the asymptotic behaviour of nonlinear systems of ordinary differential equations (original) (raw)
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The present paper deals with the following system:\begin{align*}x^{\prime}&=-e(t)x+f(t)\phi_{p^*}(y),\\y^{\prime}&=-(p-1)g(t)\phi_p(x)-(p-1)h(t)y,\end{align*}wherepandp*are positive numbers satisfying 1/p+ 1/p*= 1, andϕq(z) = |z|q−2zforq=porq=p*. This system is referred to as a half-linear system. We herein establish conditions on time-varying coefficientse(t),f(t),g(t) andh(t) for the zero solution to be uniformly globally asymptotically stable. If (e(t),f(t)) ≡ (h(t),g(t)), then the half-linear system is integrable. We consider two cases: the integrable case (e(t),f(t)) ≡ (h(t),g(t)) and the non-integrable case (e(t),f(t)) ≢ (h(t),g(t)). Finally, some simple examples are presented to illustrate our results.