For a dynamical system arising from ${\mathbb{Z}}^k$-action on a higher rank graph with finite vertex set, we show that the semidirect product of the asymptotic equivalence relation groupoid is essentially principal if and only if the $k$-graph satisfies the aperiodic condition. Then we show that the corresponding asymptotic Ruelle algebra is simple if the graph is primitive with the aperiodic condition.

The bifurcation problem is one of the most important subjects in dynamical systems. Motivated by M. Li et al. who used compound matrices to judge the stability of matrices and the existence of Hopf bifurcations in continuous dynamical systems, we obtained some effective methods to judge the Schur stability of matrices on the base of the spectral property of compound matrices, which can be used to judge the asymptotical stability and the existence of Hopf bifurcations of discrete dynamical systems.

Let $ℂ$ be the field of all complex numbers, $M_n$ the space of all $n\times n$ matrices over $ℂ$, and $S_n$ the subspace of $M_n$ consisting of all symmetric matrices. The map $\varphi :S_n\to M_n$ satisfies that $A-\lambda B$ is $k$-potent in $S_n$ implying that $\varphi (A)-\lambda \varphi (B)$ is $k$-potent in $M_n$, where $\lambda \in \mathrm{ℂ,}$ then there exist an invertible matrix $P\in M_n$ and $ϵ\in ℂ$ with ${ϵ}^k=ϵ$ such that $\varphi (X)=ϵP^{-1}(X)P$ for every $X\in S_n$. Moreover, the inductive method used in this paper can be used to characterise similar maps from $M_n$ to $M_n$.

We study colorings of a tree induced from isometries of the hyperbolic plane given an ideal tessellation. We show that, for a given tessellation of the hyperbolic plane by ideal polygons, a coloring can be associated with any element of Isom(${ℍ}^2$), and the element is a commensurator of $\mathrm{\Gamma }$ if and only if its associated coloring is periodic, generalizing a result of Hedlund and Morse.

Considering the impact of the nonlinear stiffness, a 2-DOF vibration model with cubic terms was established according to the structural feature and nonlinear behavior. Ignoring the impact of nonlinear terms, the system was linearly analyzed. In the case of primary resonance and 1 : 1 internal resonance, a multiscale method was used to obtain a first-order approximate solution. Taking the parameters of one tracked ambulance for instance, the approximate solution was corroborated and the influence of the parameters on damping effect was investigated. Finally, motion stability of the damping system was analyzed with singularity theory. The theoretical bases for improving efficiency of the damping system were provided.

We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form $(g(S^{n}y))$ for a measure preserving system $(Y,S)$ and $g\in L^{\infty }(Y)$, avoiding in the latter case the problem of finding the full measure set of appropriate points $y$.

For circular restricted $N+1$-body problems, we study the motion of a sufficiently small mass point (called the zero mass point) in the plane of $N$ equal masses located at the vertices of a regular polygon. By using variational minimizing methods, for some $N$, we prove the existence of the noncollision periodic solution for the zero mass point with some fixed wingding number.

Based on the works of Perko and Walter, Moeckel and Simo, and Zhang and Zhou, we study the necessary conditions and suffcient conditions for the uniformly rotating planar nested regular polygonal periodic solutions for the $2N$-body problems.

The dynamics of a 2-dimensional neural network model in neutral form are investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. Finally, some numerical simulations are carried out to support the analytic results.

Let $X$ be an infinite dimensional Banach space, and $\mathrm{\Phi }:B(X)\to B(X)$ is a nonlinear Lie derivation. Then Φ is the form $\delta +\tau$ where $\delta$ is an additive derivation of $B(X)$ and $\tau$ is a map from $B(X)$ into its center $Z_{B(X)}$, which maps commutators into the zero.

We show the existence of spatial central configurations for the $N+2p+1$-body problems. In the $N+2p+1$-body problems, N bodies are at the vertices of a regular N-gon T ; 2p bodies are symmetric with respect to the center of T, and located on the straight line which is perpendicular to the regular N-gon T and passes through the center of T; the $N+2p+1$th is located at the the center of T. The masses located on the vertices of the regular N-gon are assumed to be equal; the masses located on the same line and symmetric with respect to the center of T are equal.