Trajectory structure rule of a third-order nonlinear difference equation

• Main Authors: PAN Zhikang, LI Xianyi
• Format: Article
• Language: English
• Published: Academic Journals Center of Shanghai Normal University 2020-06-01
• Series: Journal of Shanghai Normal University (Natural Sciences)
• Subjects: nonlinear difference equation; nontrivial solution; cycle length; oscillation and nonoscillation; global asymptotic stability
• Online Access: http://qktg.shnu.edu.cn/zrb/shsfqkszrb/ch/reader/view_abstract.aspx?file_no=20200302

In this paper the dynamics for a third-order nonlinear difference equation is considered in detail. By utilizing some beautiful mathematical skills, we describe the rule for the trajectory structure of this equation clearly and completely. The successive lengths of positive and negative semicycles of any nontrivial solutions of this equation occur periodically with prime period 7; the rule is 3⁺, 2^-, 1⁺, 1^- in a period. Using the rule, we verify that the positive equilibrium point of the equation is globally asymptotically stable.