American Institute of Mathematical Sciences

DCDS

Rotation numbers and Lyapunov stability of elliptic periodic solutions

Jifeng Chu, Meirong Zhang

Using the relation between the Hill's equations and the Ermakov-Pinney equations established by Zhang [27], we will give some interesting lower bounds of rotation numbers of Hill's equations. Based on the Birkhoff normal forms and the Moser twist theorem, we will prove that two classes of nonlinear, scalar, time-periodic, Newtonian equations will have twist periodic solutions, one class being regular and another class being singular.

keywords: Ermakov-Pinney equation , twist coefficient. , periodic solution , Lyapunov stability , rotation number , Hill's equation

DCDS-B

Existence and stability of periodic oscillations of a rigid dumbbell satellite around its center of mass

Jifeng Chu, Zaitao Liang, Pedro J. Torres, Zhe Zhou

We study the existence and stability of periodic solutions of a differential equation that models the planar oscillations of a satellite in an elliptic orbit around its center of mass. The proof is based on a suitable version of Poincaré-Birkhoff theorem and the third order approximation method.

keywords: Satellite equation , twist periodic solutions , unstable periodic solutions , Poincaré-Birkhoff theorem , third order approximation

CPAA

Existence and stability of periodic solutions for relativistic singular equations

Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu

In this paper, we study the existence, multiplicity and stability of positive periodic solutions of relativistic singular differential equations. The proof of the existence and multiplicity is based on the continuation theorem of coincidence degree theory, and the proof of stability is based on a known connection between the index of a periodic solution and its stability.

keywords: Relativistic singular equations , stability , positive periodic solutions

DCDS-B

Lyapunov stability for conservative systems with lower degrees of freedom

Jifeng Chu, Jinzhi Lei, Meirong Zhang

It is a central theme to study the Lyapunov stability of periodic solutions of nonlinear differential equations or systems. For dissipative systems, the Lyapunov direct method is an important tool to study the stability. However, this method is not applicable to conservative systems such as Lagrangian equations and Hamiltonian systems. In the last decade, a method that is now known as the 'third order approximation' has been developed by Ortega, and has been applied to particular types of conservative systems including time periodic scalar Lagrangian equations (Ortega, J. Differential Equations, 128(1996), 491-518). This method is based on Moser's twist theorem, a prototype of the KAM theory. Latter, the twist coefficients were re-explained by Zhang in 2003 through the unique positive periodic solutions of the Ermakov-Pinney equation that is associated to the first order approximation (Zhang, J. London Math. Soc., 67(2003), 137-148). After that, Zhang and his collaborators have obtained some important twist criteria and applied the results to some interesting examples of time periodic scalar Lagrangian equations and planar Hamiltonian systems. In this survey, we will introduce the fundamental ideas in these works and will review recent progresses in this field, including applications to examples such as swing, the (relativistic) pendulum and singular equations. Some unsolved problems will be imposed for future study.

keywords: planar Hamiltonian system , Ermakov-Pinney equation , twist coefficient. , Hill equation , Lyapunov stability , periodic solution , Lagrangian equation

DCDS

Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero

Juntao Sun, Jifeng Chu, Zhaosheng Feng

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first order periodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we establish some new criteria to guarantee that Hamiltonian systems, with asymptotically quadratic terms and spectrum point zero, have at least one and infinitely many homoclinic orbits under certain conditions.

keywords: variational methods , Hölder inequality , Strongly indefinite problems. , Homoclinic orbits , Hamiltonian systems , asymptotically linear terms

DCDS

Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem

Jifeng Chu, Pedro J. Torres, Feng Wang

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

Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity

Hailong Zhu, Jifeng Chu, Weinian Zhang

In the setting of mean-square exponential dichotomies, we study the existence and uniqueness of mean-square almost automorphic solutions of non-autonomous linear and nonlinear stochastic differential equations.

keywords: Mean-square almost automorphic solutions , mean-square exponential dichotomy , stochastic differential equations

DCDS-B

Prevalence of stable periodic solutions in the forced relativistic pendulum equation

Feng Wang, Jifeng Chu, Zaitao Liang

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

DCDS

Periodic shadowing of vector fields

Jifeng Chu, Zhaosheng Feng, Ming Li

A vector field has the periodic shadowing property if for any $\varepsilon>0$ there is $d>0$ such that, for any periodic $d$-pseudo orbit $g$ there exists a periodic orbit or a singularity in which $g$ is $\varepsilon$-shadowed. In this paper, we show that a vector field is in the $C^1$ interior of the set of vector fields satisfying the periodic shadowing property if and only if it is $\Omega$-stable. More precisely, we prove that the $C^1$ interior of the set of vector fields satisfying the orbital periodic shadowing property is a subset of the set of $\Omega$-stable vector fields.

keywords: homoclinic connection , diffeomorphism , $\Omega$-stability. , periodic shadowing property , Vector fields , hyperbolic singularity

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