Rigid centres on the center manifold of tridimensional differential systems

Motivated by the definition of rigid centres for planar differential systems, we introduce the study of rigid centres on the center manifolds of differential systems on $\mathbb {R}^{3}$ . On the plane, these centres have been extensively studied and several interesting results have been obtained. We present results that characterize the rigid systems on $\mathbb {R}^{3}$ and solve the centre-focus problem for several families of rigid systems.

Bifurcation Analysis in Delayed Nicholson Equation With Harvest Term

In this paper, we investigate a delayed Nicholson equation with delay harvesting term which was proposed in open problems and conjectures formulated by Berezansky et al. (Applied Mathematical Modelling 34 (2010) 1405). The stability switching curves by taking two delays as parameters are obtained via the method introduced by An et al.(J. Differential Equations 266 (2019) 7073). The existence of Hopf singularity on a two-parameter plane is determined by the varying direction of two parameters. Furthermore, the normal form near the Hopf singularity is derived via applying the center manifolds theory and normal forms method of FDEs. Finally, some numerical simulations are carried out to illustrate the theoretical conclusions.

Integrability and bifurcation of a three-dimensional circuit differential system

<p style='text-indent:20px;'>We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus.</p>