Narrow Results

In this paper, we study two classes of three-dimensional differential systems. By using the Poincaré map and the averaging method, we give sufficient conditions for the existence of invariant surfaces consisting of periodic orbits for the systems. Illustrative examples are given for our main results.

In this paper, we classify parabolic revolution surfaces in the three-dimensional simply isotropic space ${𝕀}^3$ under the condition

Consider a three-dimensional system having an invariant surface. By using bifurcation techniques and analyzing the solutions of bifurcation equations, the authors study the spacial bifurcation phenomena of a k multiple closed orbit in the invariant surface. The sufficient conditions of the existence of many closed orbits bifurcate from the k multiple closed orbit are obtained.

A competitive system on the n-rectangle: {x ∈ Rⁿ: 0 ≤ x_i ≤ l_i, i = 1, …, n} was considered, each species of which, in isolation, admits logistic growth with the hyperbolic structure saturation. It has an (n - 1)-dimensional invariant surface called carrying simplex Σ as a globe attractor, hence the long term dynamics of the system is completely determined by the dynamics on Σ. For the three-dimensional system, the whole dynamical behavior was presented. It has a unique positive equilibrium point and any limit set is either an equilibrium point or a limit cycle. The system is permanent and it is proved that the number of periodic orbits is finite and non-periodic oscillation the May–Leonard phenomenon does not exist. A criterion for the positive equilibrium to be globally asymptotically stable is provided. Whether there exist limit cycles or not remains open.