Narrow Results

In this paper we study the nonchaotic and chaotic behavior of all 3D conservative quadratic ODE systems with five terms on the right-hand side and one nonlinear term (5-1 systems). We prove a theorem which provides sufficient conditions for solutions in 3D autonomous systems being nonchaotic. We show that all but five of these systems: (15a, 15b), (18b), (41)(A = ∓1), (43b), and (49a, 49b) are nonchaotic. Numerical simulations show that only one of the five systems, (43b), really appears to be chaotic. If proved to be true, it will be the simplest ODE system having chaos.

The Maxwell–Bloch system arises naturally from laser models, which describes the dynamics of a two-state quantum system coupled to the basic electromagnetic resonator model. In this paper, we study its integrability and give a complete classification of the irreducible Darboux polynomials, which can help us find all polynomial and rational first integrals of the Maxwell–Bloch system. To this end, in the case that the loss rate of field is zero, we use the characteristic curve method and weight homogeneous polynomials to investigate the integrability of the Maxwell–Bloch system, and in the case that the loss rate of field is nonzero, we provide a linear scaling of variables to transform the Maxwell–Bloch system into the Lorenz system and obtain its integrability from some corresponding results of the Lorenz system. Our results show that although the Maxwell–Bloch system has chaotic behaviors for a large range of its parameters, it is nonchaotic when its parameters satisfy one of seven conditions, which may help us understand the complex and rich dynamics of nonlinear laser models.