Generalized periodic orbits of the time-periodically forced Kepler problem accumulating at the center and of circular and elliptic restricted three-body problems

In this paper, we consider a time-periodically forced Kepler problem in any dimension, with an external force which we only assume to be regular in a neighborhood of the attractive center. We prove that there exist infinitely many periodic orbits in this system, with possible double collisions with the center regularized, which accumulate at the attractive center. The result is obtained via a localization argument combined with a result on $$C^{1}$$ C 1 -persistence of closed orbits by a local homotopy-stretching argument. Consequently, by formulating the circular and elliptic restricted three-body problems of any dimension as time-periodically forced Kepler problems, we obtain that there exist infinitely many periodic orbits, with possible double collisions with the primaries regularized, accumulating at each of the primaries.

Dynamics of a discrete predator-prey model with Holling-II functional response

In this paper, we use a semidiscretization method to derive a discrete predator–prey model with Holling type II, whose continuous version is stated in [F. Wu and Y. J. Jiao, Stability and Hopf bifurcation of a predator-prey model, Bound. Value Probl. 129(2019) 1–11]. First, the existence and local stability of fixed points of the system are investigated by employing a key lemma. Then we obtain the sufficient conditions for the occurrence of the transcritical bifurcation and Neimark–Sacker bifurcation and the stability of the closed orbits bifurcated by using the Center Manifold theorem and bifurcation theory. Finally, we present numerical simulations to verify corresponding theoretical results and reveal some new dynamics.

Orbit Growth of Shift Spaces Induced by Bouquet Graphs and Dyck Shifts

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

Semiclassical study on photodetachment of hydrogen negative ion in a harmonic potential confined by a quantum well

We have studied the photodetachment dynamics of the H− ion in a harmonic potential confined in a quantum well for the first time. The closed orbits of the detached electron in a confined harmonic potential are found and the photodetachment spectra of this system are calculated. It is interesting to find that the photodetachment spectra depend sensitively on the size of the quantum well and the harmonic frequency. For smaller size of the quantum well, the harmonic potential can be considered as a perturbation, the interference effect between the returning electron wave bounced back by the quantum well and the initial outgoing wave is very strong, which makes the photodetachment spectra exhibits an irregular saw-tooth structure. With the increase of the size of the quantum well, the photodetachment spectra oscillates complicatedly in the higher energy region. For very large size of the quantum well, the photodetachment spectra approach to the case in a free harmonic potential, which is a regular saw-tooth structure. In addition, the harmonic frequency can also affect the photodetachment spectra of this system greatly. Our work provides a new method for the study of spatially confined low-dimensional systems and may guide the future experimental research for the photodetachment dynamics in the ion trap.

Refinements of topological invariants of flows

<p style='text-indent:20px;'>We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces. In particular, the abstract weak orbit spaces of flows on topological spaces are refinements of Morse graphs of flows on compact metric spaces, Reeb graphs of Hamiltonian flows with finitely many singular points on surfaces, and the CW decompositions which consist of the unstable manifolds of singular points for Morse flows on closed manifolds. Though the CW decomposition of a Morse flow is finite, the intersection of the unstable manifold and the stable manifold of closed orbits need not consist of finitely many connected components. Therefore we study the finiteness. Moreover, we consider when the time-one map reconstructs the topology of the original flow. We show that the orbit space of a Hamiltonian flow with finitely many singular points on a compact surface is homeomorphic to the abstract weak orbit space of the time-one map by taking an arbitrarily small reparametrization and that the abstract weak orbit spaces of a Morse flow on a compact manifold and the time-one map are homeomorphic. In addition, we state examples whose Morse graphs are singletons but whose abstract weak orbit spaces are not singletons.</p>

Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones

<p style='text-indent:20px;'>In this paper, we consider the nonsmooth bifurcation around a class of critical crossing cycles, which are codimension-2 closed orbits composed of tangency singularities and regular orbits, for a two-parameter family of planar piecewise smooth system with two zones. By the construction of suitable displacement function (equivalently, Poincar<inline-formula><tex-math id="M1">\begin{document}$ {\rm\acute{e}} $\end{document}</tex-math></inline-formula> map), the stability and the existence of periodic solutions under the variation of the parameters inside this system are characterized. More precisely, we obtain some parameter regions on the existence of crossing cycles and sliding cycles near those loops. As applications, several examples are given to illustrate our main conclusions.</p>