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Non-oscillation principle for eventually competitive and cooperative systems

This work is partially supported by NSF of China No.11825106, 11771414 and Wu Wen-Tsun Key Laboratory

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  • A nonlinear dynamical system is called eventually competitive (or cooperative) provided that it preserves a partial order in backward (or forward) time only after some reasonable initial transient. We present in this paper the Non-oscillation Principle for eventually competitive or cooperative systems, by which the non-ordering of (both $ \omega $- and $ \alpha $-) limit sets is obtained for such systems; and moreover, we established the Poincaré-Bendixson Theorem and structural stability for three-dimensional eventually competitive and cooperative systems.

    Mathematics Subject Classification: Primary: 34C12, 37C65; Secondary: 58F09, 92A15.


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  • Figure 1.  The intervals $ I_{n} = [t_{0}+nA, b+t_{*}+nA] = [t_n, t_n+E] $ for $ n\in \mathbb{N} $, where $ E = \vert I_n\vert = \vert I_0\vert $ and $ [a, b] $ is the rising interval

    Figure 2.  Case (Ⅰ) with $ n_{0}D_{0}\in [A-E, A] $, where $ c_{n_{0}}\in [t_{n_{0}(k_{0}+1)-1}, t_{n_{0}(k_{0}+1)-1}+E] = I_{n_0(k_0+1)-1} $ with $ n_{0} = 3, k_{0} = 1, l_{*} = 3 $ and $ n_{*} = 5 $

    Figure 3.  Case (Ⅱ) with $ n_{0}D_{0}>A $, where $ c_{n_{0}} = t_{n_0(k_0+1)-1}-D_1\in (t_{n_{0}(k_{0}+1)-2}+E, t_{n_{0}(k_{0}+1)-1}) $ and $ c_{n\cdot n_0} = t_{n\cdot[n_0(k_0+1)-1]}-nD_1 $, with $ n_{0} = n = 2 $, $ k_{0} = 1 $ and $ D_1 = n_0D_0-A $

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