Integrability of vector fields versus inverse Jacobian multipliers and normalizers
Shiliang Weng Xiang Zhang
Discrete & Continuous Dynamical Systems - A 2016, 36(11): 6539-6555 doi: 10.3934/dcds.2016083
In this paper we provide characterization of integrablity of a system of vector fields via inverse Jacobian multipliers (matrix) and normalizers of smooth (or holomorphic) vector fields. These results improve and extend some well known ones, including the classical holomorphic Frobenius integrability theorem. Here we obtain the exact expression of an integrable system of vector fields acting on a smooth function via their known common first integrals. Moreover we characterize the relations between the integrability and the existence of normalizers for a system of vector fields. In the case of integrability of a system of vector fields we not only prove the existence of normalizers but also provide their exact expressions.
keywords: normalizers. integrability Vector fields first integral inverse Jacobian multiplier
Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems
Lijun Wei Xiang Zhang
Discrete & Continuous Dynamical Systems - A 2016, 36(5): 2803-2825 doi: 10.3934/dcds.2016.36.2803
This paper deals with the maximum number of limit cycles, which can be bifurcated from periodic orbits of planar piecewise smooth Hamiltonian systems, which are located in a neighborhood of a generalized homoclinic loop with a nilpotent saddle on a switch line. First we present asymptotic expressions of the Melnikov functions near the loop. Then using these expressions we study the number of limit cycles which are bifurcated from the periodic orbits near the homoclinic loop under small perturbations. Finally we provide two concrete examples showing applications of our main results.
keywords: Piecewise smooth system limit cycle bifurcation nilpotent saddle. generalized homoclinic loop Melnikov function
Center of planar quintic quasi--homogeneous polynomial differential systems
Yilei Tang Long Wang Xiang Zhang
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 2177-2191 doi: 10.3934/dcds.2015.35.2177
In this paper we first characterize all quasi--homogeneous but non--homogeneous planar polynomial differential systems of degree five, and then among which we classify all the ones having a center at the origin. Finally we present the topological phase portrait of the systems having a center at the origin.
keywords: generalized normal sectors. Quasi--homogeneous quintic polynomial systems blow up centers global phase portrait
Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds
Fei Liu Jaume Llibre Xiang Zhang
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1097-1111 doi: 10.3934/dcds.2011.29.1097
Let $\mathcal M$ be a smooth Riemannian manifold with the metric $(g_{ij})$ of dimension $n$, and let $H= 1/2 g^{ij}(q)p_ip_j+V(t,q)$ be a smooth Hamiltonian on $\mathcal M$, where $(g^{ij})$ is the inverse matrix of $(g_{ij})$. Under suitable assumptions we prove the existence of heteroclinic orbits of the induced Hamiltonian systems.
keywords: Heteroclinic orbit variational method Hamiltonian system Riemannian manifold.

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