Stability and chaos in 2-D discrete systems

An Envelope Surface Method for Determining Oscillation of a Delay 2-D Discrete Convection System

“…Combining case (1) with case (2) implies that the characteristic equation (15) does not have any positive roots if and only if p ≥ 0, q ≥ 0 and r ≥ 0 or p > 0, q < 0 and…”

“…Note that when x > 0 and y < 0, we have z(x, y) > 0, ∂ 2 z/∂ x 2 > 0, ∂ 2 z/∂ y 2 > 0 and ∂ 2 z/∂ x 2 · ∂ 2 z/∂ y 2 − (∂ 2 z/∂ x∂ y) 2 on the positive coordinate axis x = 0, that is, p > 0, q ≥ 0 and r ≥ 0, there cannot be any tangent plane of the envelope S which also passes through this point, and when the point ( p, q, r ) is vertically above the envelope S in the forth octant, that is, p > 0, q < 0 and r > (−1) τ +1 τ τ q τ +1 /(τ + 1) τ +1 p τ , there cannot exist any tangent plane of the envelope S which also passes through this point, while the point ( p, q, r ) is situated elsewhere, such a tangent plane can be drawn. Since (15) is identical with (17) for the existence of positive solutions, it follows from Lemma 4 that (15) does not have any positive roots if and only if p > 0, q ≥ 0 and r ≥ 0 or p > 0, q < 0 and…”

“…In fact, system (2) is a convection system with a source term and two delays. It is well known that it is a formidable task to derive exact analytical solutions of system (2) in practice. Therefore, the study of system (1) may provide some useful information for analyzing the continuous system (2).…”

“…It is well known that it is a formidable task to derive exact analytical solutions of system (2) in practice. Therefore, the study of system (1) may provide some useful information for analyzing the continuous system (2). In this paper, we focus on oscillation behavior of system (1).…”

Chaotification for Partial Difference Equations via Controllers

“…Chen and Liu studied the chaos for (1) in R 3 by constructing spatial periodic orbits in 2003 [3]. Chen et al [4] reformulated (1) to a discrete system:…”