Bifurcations of the Riccati Quadratic Polynomial Differential Systems

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system [Formula: see text] with [Formula: see text], [Formula: see text] nonzero (otherwise the system is a Bernoulli differential system), [Formula: see text] (otherwise the system is a Liénard differential system), [Formula: see text] a polynomial of degree at most [Formula: see text], [Formula: see text] and [Formula: see text] polynomials of degree at most 2, and the maximum of the degrees of [Formula: see text] and [Formula: see text] is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of [Formula: see text] adding the circle [Formula: see text] of the infinity) modulo topological equivalence.

On the Generalized Reflective Function of the Three-Degree Polynomial Differential System

The structure of the generalized reflective function of three-degree polynomial differential systems is considered in this paper. The generated results are used for discussing the existence of periodic solutions of these systems.

The Liouvillian Integrability of Several Oscillators

In this paper, we characterize all the Liouvillian first integrals of a cubic polynomial differential system that contains the van der Pol and the Duffing oscillators. It is also shown that the centers correspond to the Liouville integrable cases.

Periodic Orbits Bifurcating from a Nonisolated Zero–Hopf Equilibrium of Three-Dimensional Differential Systems Revisited

In this paper, we study the periodic solutions bifurcating from a nonisolated zero–Hopf equilibrium in a polynomial differential system of degree two in [Formula: see text]. More specifically, we use recent results of averaging theory to improve the conditions for the existence of one or two periodic solutions bifurcating from such a zero–Hopf equilibrium. This new result is applied for studying the periodic solutions of differential systems in [Formula: see text] having [Formula: see text]-scroll chaotic attractors.