DCDS
Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem
Jifeng Chu Pedro J. Torres Feng Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1921-1932 doi: 10.3934/dcds.2015.35.1921
For the Gylden-Meshcherskii-type problem with a periodically cha-nging gravitational parameter, we prove the existence of radially periodic solutions with high angular momentum, which are Lyapunov stable in the radial direction.
keywords: Gylden-Meshcherskii-type problem Radial stability twist. periodic solutions
DCDS
Lyapunov stability for regular equations and applications to the Liebau phenomenon
Feng Wang José Ángel Cid Mirosława Zima
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4657-4674 doi: 10.3934/dcds.2018204

We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order. As an application, we present some results on the existence and stability of $ T$-periodic solutions of a Liebau-type equation.

keywords: Lyapunov stability regular equation Liebau phenomenon KAM theory Moser twist theorem
DCDS
On the Cauchy problem for a higher-order μ-Camassa-Holm equation
Feng Wang Fengquan Li Zhijun Qiao
Discrete & Continuous Dynamical Systems - A 2018, 38(8): 4163-4187 doi: 10.3934/dcds.2018181

In this paper, we study the Cauchy problem of a higher-order μ-Camassa-Holm equation. We first establish the Green's function of $(μ-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in $H^{s}(\mathbb{S})$, $s≥ 4$. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.

keywords: Higher-order μ-Camassa-Holm equation global existence weak solutions non-uniformly continuous peakon solutions
DCDS-B
Prevalence of stable periodic solutions in the forced relativistic pendulum equation
Feng Wang Jifeng Chu Zaitao Liang
Discrete & Continuous Dynamical Systems - B 2018, 23(10): 4579-4594 doi: 10.3934/dcdsb.2018177

We study the prevalence of stable periodic solutions of the forced relativistic pendulum equation for external force which guarantees the existence of periodic solutions. We prove the results for a general planar system.

keywords: Prevalence stable periodic solutions forced relativistic pendulum equation planar system

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