Narrow Results

We study a geometrical formulation of the nonlinear second-order Riccati equation (SORE) in terms of the projective vector field equation on $S^1$, which in turn is related to the stability algebra of Virasoro orbit. Using Darboux integrability method, we obtain the first integral of the SORE and the results are applied to the study of its Lagrangian and Hamiltonian descriptions. Using these results, we show the existence of a Lagrangian description for SORE, and the Painlevé II equation is analyzed.

We characterize all generators of the exponential factors for the Rössler system by using weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations. Moreover, using Darboux polynomials and exponential factors we obtain the necessary and sufficient conditions in order that the Rössler system has a Darboux first integral.

We give the complete classification of irreducible invariant algebraic curves in quadratic systems from family $(I)$ of the Chinese classification, that is, of differential system

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.

We introduce a deterministic business cycle model and study its dynamics by determining the existence of nonconstant functions which can be obtained "by quadratures" of elementary functions and are constant on solution curves in some nonempty open set.

We characterize the Liouvillian and analytic first integrals for the polynomial differential systems of the form x′ = a - (b + 1)x + x²y, y′ = bx - x²y, with a, b ∈ ℝ, called the Brusselator differential systems.