Chengzhi Li

Integrability and linearizability of the Lotka–Volterra systems

Abstract Integrability and linearizability of the Lotka–Volterra systems are studied. We prove sufficient conditions for integrable but not linearizable systems for any rational resonance ratio. We give new sufficient conditions for linearizable Lotka–Volterra systems. Sufficient conditions for integrable Lotka–Volterra systems with 3:−q resonance are given. In the particular cases of 3:−5 and 3:−4 resonances, necessary and sufficient conditions for integrable systems are given.

Uniqueness of periodic orbits of some vector fields with codimension two singularities

Abstract We consider the question of uniqueness of limit cycles of some normal form equations of vector fields with singularities of codimension two. By using suitable choices of scaling on the normal form equations, we give simple proofs on uniqueness of periodic orbits.

Cubic Liénard equations with quadratic damping having two antisaddles

AbstractThe paper deals with differential equations of the form

Perturbations from an Elliptic Hamiltonian of Degree Four: I. Saddle Loop and Two Saddle Cycle

\dot x = y, \dot y = - g(x) - f(x)y

Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops

withf andg polynomials of degree respectively two and three. They are called cubic Liénard equations with quadratic damping. Attention goes to the case in which the equations have three singular points of which one is a saddle and two are antisaddles. A lot of results are given on the upper-bound for the number of limit cycles, depending on the relative position of the zeros off andg on thex-axis.