Using the relation between the Hill's equations and the
Ermakov-Pinney equations established by Zhang , 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.
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.
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.
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.
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.