Optimal packings of two to four equal circles on any flat torus

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.     

Chaos in the softening duffing system under multi-frequency periodic forces

The chaotic dynamics of the softening-spring Duffing system with multi-frequency external periodic forces is studied. It is found that the mechanism for chaos is the transverse heteroclinic tori. The Poincare map, the stable and the unstable manifolds of the system under two incommensurate periodic forces were set up on a two-dimensional torus. Utilizing a global perturbation technique of Melnikov the criterion for the transverse interaction of the stable and the unstable manifolds was given. The system under more but finite incommensurate periodic forces was also studied. The Melnikov‘s global perturbation technique was therefore generalized to higher dimensional systems. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies will increase the area in parameter space where chaotic behavior can occur.     

参数激励耦合系统的复杂动力学行为分析

分析了耦合van der Pol振子参数共振条件下的复杂动力学行为．基于平均方程，得到了参数平面上的转迁集，这些转迁集将参数平面划分为不同的区域，在各个不同的区域对应于系统不同的解．随着参数的变化，从平衡点分岔出两类不同的周期解，根据不同的分岔特性，这两类周期解失稳后，将产生概周期解或3—D环面解，它们都会随参数的变化进一步导致混吨．发现在系统的混沌区域中，其混吨吸引子随参数的变化会突然发生变化，分解为两个对称的混吨吸引子．值得注意的是，系统首先是由于2—D环面解破裂产生混吨，该混吨吸引子破裂后演变为新的混吨吸引子，却由倒倍周期分岔走向3—D环面解，也即存在两条通向混沌的道路：倍周期分岔和环面破裂，而这两种道路产生的混吨吸引子在一定参数条件下会相互转换．     

A torus bifurcation theorem with symmetry

A general theory for the study of degenerate Hopf bifurcation in the presence of symmetry has been carried out only in situations where the normal form equations decouple into phase/amplitude equations. In this paper we prove a theorem showing that in general we expect such degeneracies to lead to secondary torus bifurcations. We then apply this theorem to the case of degenerate Hopf bifurcation with triangular (D₃) symmetry, proving that in codimension two there exist regions of parameter space where two branches of asymptotically stable 2-tori coexist but where no stable periodic solutions are present. Although this study does not lead to a theory for degenerate Hopf bifurcations in the presence of symmetry, it does present examples that would have to be accounted for by any such general theory.     

Stochastic Hopf bifurcation in quasi-integrable-Hamiltonian systems

A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system‘s energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system‘s parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions.     

本刊英文版Vol.30(2014),No.9论文摘要(英文)

<正>Least Number of Periodic Points of Self-maps of Lie Groups Jerzy JEZIERSKI Abstract There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M→M of a compact smooth manifold of dimension at least 3:NF_n(f)=min{#Fix(g~n);g~f;g is continuous}and NJD_n(f)=min{#Fix(g~n);g~f;g is smooth}.In general,NJD_n(f)may be much greater than NF_n(f).If M is a torus,then the invariants are     

Classification of extensions of torus algebra II

We use extension theory and algebraic methods to give a complete characterization of extensions of torus algebra by stable Cuntz algebras,and prove certain classification theorems of these extension algebras.     

Analytic Hardy spaces on the quantum torus

Analytic Hardy and BMO spaces on the quantum torus are introduced. Some basic properties of these spaces are presented. In particular, the associated H 1-BMO duality theorem is proved. Finally, we discuss some possible extensions of the obtained results.     

K6‐minors in triangulations and complete quadrangulations

In this paper, we shall prove that a projective‐planar (resp., toroidal) triangulation G has K₆ as a minor if and only if G has no quadrangulation isomorphic to K₄ (resp., K₅ ) as a subgraph. As an application of the theorems, we can prove that Hadwiger's conjecture is true for projective‐planar and toroidal triangulations. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 302‐312, 2009