Global behaviour of quaternion Riccati rational difference equation

Abstract

Since non-commutative algebraic structure of quaternions, it is difficult to study the forbidden sets and asymptotic stability of quaternion rational difference equations. In this paper, by generalized Niven's algorithm and its related properties, the representations of general solutions for the third order linear quaternion difference equation are given through the roots of the corresponding quaternion characteristic equation, based on which, the corresponding forbidden sets of the Riccati quaternion rational difference equation are computed. Moreover, we obtain the quaternion curves of initial points located in two-dimensional quaternion space outside which we determine the convergence, divergence and global asymptotic stability of the solutions for the second order Riccati quaternion rational difference equations, some sufficient and necessary conditions for global asymptotic stability are established. Finally, several examples are illustrated to show the feasibility of our obtained results.