摘要：Consider the class of planar systems of first-order rational difference equations 1´ X n + 1 = α 1 + β 1 x n + γ 1 y n A 1 + B 1 x n + C 1 y n Y n + 1 = α 1 + β 1 x n + γ 2 y n A 1 + B 2 x n + C 2 y n } , n = 0 , 1 , 2 , … , ( x 0 , y 0 ) ∈ R where R = { ( x , y ) ∈ [ 0 , ∞ ) 2 : A i + B i x + C i y ≠ 0 , i = 1 , 2 } , and the parameters are nonnegative and such that both terms in the right-hand side of (1′) are nonlinear. In this paper, we prove the following discretized Poincaré-Bendixson theorem for the class of systems (1′). If the map associated to system (1′) is bounded, then the following statements are true: (i) If both equilibrium curves of (1') are reducible conics, then every solution converges to one of up to four equilibria. (ii) If exactly one equilibrium curve of (1') is a reducible conic, then every solution converges to one of up to two equilibria. (iii) If both equilibrium curves of (1') are irreducible conics, then every solution converges to one of up to three equilibria or to a unique minimal period-two solution which occurs as the intersection of two elliptic curves. If both equilibrium curves of (1') are reducible conics, then every solution converges to one of up to four equilibria. If exactly one equilibrium curve of (1') is a reducible conic, then every solution converges to one of up to two equilibria. If both equilibrium curves of (1') are irreducible conics, then every solution converges to one of up to three equilibria or to a unique minimal period-two solution which occurs as the intersection of two elliptic curves. In particular, system (1′) cannot exhibit chaos when its associated map is bounded. Moreover, we show that if both equilibrium curves of (1′) are reducible conics and the map associated to system (1′) is unbounded, then every solution converges to one of up to infinitely many equilibria or to ( 0 , ∞ ) or ( ∞ , 0 ) . MSC:39A05, 39A11.
