Symmetry－breaking Subharmonic Bifurcationsof Periodicaly Parameter－excited Systems

Ｓｙｍｍｅｔｒｙ－ｂｒｅａｋｉｎｇＳｕｂｈａｒｍｏｎｉｃＢｉｆｕｒｃａｔｉｏｎｓｏｆＰｅｒｉｏｄｉｃａｌｙＰａｒａｍｅｔｅｒ－ｅｘｃｉｔｅｄＳｙｓｔｅｍｓ１１ＴｈｅｐａｐｅｒｗａｓｒｅｃｅｉｖｅｄｏｎＡｕｇ．３０ｔｈ，１９９６ＧｕｏｗｅｉＨＥ１Ｌａｂ...     

A Method for Solving Exact Solution to Two－dimensional Korteweg－de Vires－Burgers Equation

ＡＭｅｔｈｏｄｆｏｒＳｏｌｖｉｎｇＥｘａｃｔＳｏｌｕｔｉｏｎｔｏＴｗｏ－ｄｉｍｅｎｓｉｏｎａｌＫｏｒｔｅｗｅｇ－ｄｅＶｉｒｅｓ－ＢｕｒｇｅｒｓＥｑｕａｔｉｏｎＪｉａｎｇＷｅｉｌｉｎ（姜伟林）ＺｈａｎｇＪｉｅｆａｎｇ（张解放）（ＨｅｎａｎＵｎｉｖｅｒｓ...     

k－spaces and Products of Spaces with σ－hereditarily Closure－Preserving k－networks

ｋ－ｓｐａｃｅｓａｎｄＰｒｏｄｕｃｔｓｏｆＳｐａｃｅｓｗｉｔｈσ－ｈｅｒｅｄｉｔａｒｉｌｙＣｌｏｓｕｒｅ－Ｐｒｅｓｅｒｖｉｎｇｋ－ｎｅｔｗｏｒｋｓ￥ＤａｉＭｕｍｉｎ（戴牧民）ＬｉｕＣｈｕａｎ（戴牧民）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓＧ...     

U1－sr条件

本文讨论Ｕ１－ｓｒ条件，这一条件有益于计算环的Ｋ１群．得到主要结果为；（１）完全确定满足Ｕ１－ｓｒ条件的半局部环：（２）给出使ＥｎｄＲ（Ｍ）满足Ｕ１－ｓｒ条件的一个刻划；（３）引进比Ｕ１－ｓｒ更强的一个条件ＳＵ１－ｓｒ，利用上述结果证明了：若Ｒ∈ＳＵ１－ｓｒ，则Ｍｎ（Ｒ）∈Ｕ１－ｓｒ；（４）证明了对于满足ＳＵ１－ｓｒ的环Ｒ，Ｋ１Ｒ＝ＧＬ１（Ｒ）ａｂ．     

ConcerningWeightedStirling－typePairs

ＣｏｎｃｅｒｎｉｎｇＷｅｉｇｈｔｅｄＳｔｉｒｌｉｎｇ－ｔｙｐｅＰａｉｒｓＬｅｅｔｓｃｈＣ．ＨｓｕＹｕＨｏｎｇｑｕａｎ（ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｒｙ，Ｄａｌｉａｎ１１６０２４）ＣｏｎｃｅｒｎｉｎｇＷｅｉｇｈｔｅｄＳｔｉｒｌ...     

The Nonlinear Convection－Reaction－DifusionEquation

ＴｈｅＮｏｎｌｉｎｅａｒＣｏｎｖｅｃｔｉｏｎ－Ｒｅａｃｔｉｏｎ－ＤｉｆｕｓｉｏｎＥｑｕａｔｉｏｎＳｈｉｍｉｎＴＡＮＧ，ＪｉａｎｇｈａｎｇＷＵａｎｄＭａｏｃｈａｎｇＣＵＩｆ（ＤｅｐａｒｔｍｅｎｔｏｆＭｅｃｈａｎｉｃｓ，ＰｅｋｉｎｇＵｎｉｖｅｒｓｉｔｙ，...     

一个与G－分次环和G－集的Smash积有关的Maschke－Type定理

对任意群Ｇ，［１］研究了有单位元１的Ｇ－分次环与有限可迁Ｇ－集的Ｓｍａｓｈ积．在本文中，我们对任意可迁Ｇ－集Ａ讨论了具有局部单位元的Ｇ－分次环与Ｇ－集Ａ的Ｓｍａｓｈ积，证明了有关的一个Ｍａｓｃｈｋｅ－ｔｙＰｅ定理．推广了［２］［３］中的一些重要结果．     

sinx＝－sinx

ｓｉｎｘ＝－ｓｉｎｘ０６３６０２河北省乐亭县新寨中学于永平题目：已知／（Ｓｉｌｌｌｌ）一ＣＯＳＳ（ＳＥＲ），求／（ＣＯＳＳ）的表达式．诡辩揭底：对任意ｘＥＲ都有ｓｉｎ（。一ｘ）所以／（ｓｉｎｘ）—一ｃｏｓｘ成立．而解法ｌ，：只是分别求得问题的一个解，...     

Integral Solution Operators for the Cauchy－Riemann Equations on an Open Set with Piecewise C~1－Boundary

ＩｎｔｅｇｒａｌＳｏｌｕｔｉｏｎＯｐｅｒａｔｏｒｓｆｏｒｔｈｅＣａｕｃｈｙ－ＲｉｅｍａｎｎＥｑｕａｔｉｏｎｓｏｎａｎＯｐｅｎＳｅｔｗｉｔｈＰｉｅｃｅｗｉｓｅＣ￣１－ＢｏｕｎｄａｒｙＭａＺｈｏｎｇｔａｉ（马忠泰）（Ｄｅｐｔ．ｏｆＭａｔｈ．Ｓｈａｎｄｏｎ...     

QUASI－CONVEX MULTIOBJECTIVE GAME－SOLUTION CONCEPTS，EXISTENCE AND SCALARIZATION

ＱＵＡＳＩ－ＣＯＮＶＥＸＭＵＬＴＩＯＢＪＥＣＴＩＶＥＧＡＭＥ－ＳＯＬＵＴＩＯＮＣＯＮＣＥＰＴＳ，ＥＸＩＳＴＥＮＣＥＡＮＤＳＣＡＬＡＲＩＺＡＴＩＯＮ￥ＬＩＹＵＡＮＸＩＡｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｓｏｌｕｔｉｏｎｃｏｎｃｅ...     

Statistical cluster points and turnpike *

In this paper we study an asymptotic behaviour of optimal paths of a difference inclusion. The turnpike property in some wording [5,8, and so on] provided that there is a certain stationary point and optimal paths converge to that point. In this case only a finite number terms of the path (sequence) remain on the outside of every neighbourhood of that point

In the present paper a statistical cluster point introduced in [1] instead of the usual concept of limit point is considered and tue turnpiKe tueorem is proved, Mere it is es-ablished that there exists a stationary point which is a statistical cluster point for the all optimal paths. In this case not only a finite number but also infinite number terms of the path may remain on the outside of every small neighbourhood of the stationary point, but the number of these terms in comparison with the number of terms in the neighbourhood is so small that we can say:the path “almost” remains in this neighbourhood

Note that the main results are obtained under certain assumptions which are essentially weaker than the usual convexity assumption. These assumptions first were introduced for continuous systems in [6]     

Hopf-flip bifurcations of vibratory systems with impacts

Two vibro-impact systems are considered. The period n single-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. Stability and local bifurcations of single-impact periodic motions are analyzed by using the Poincaré maps. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. It is found that near the point of codim 2 bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Period doubling bifurcation of period one single-impact motion is commonly existent near the point of codim 2 bifurcation. However, no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of Hopf bifurcation associated with period two fixed points. The results from simulation shows that there exists an interest torus doubling bifurcation occurring near the value of Hopf-flip bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transit to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems.     

Singular Points Near an X_0-breaking Double Singular Fold Point in Z_2-symmetric Nonlinear Equations

ＳｉｎｇｕｌａｒＰｏｉｎｔｓＮｅａｒａｎＸ_０－ｂｒｅａｋｉｎｇＤｏｕｂｌｅＳｉｎｇｕｌａｒＦｏｌｄＰｏｉｎｔｉｎＺ_２－ｓｙｍｍｅｔｒｉｃＮｏｎｌｉｎｅａｒＥｑｕａｔｉｏｎｓＳｕＹｉ（苏毅）ａｎｄＷｕｗｅｉ（吴微）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅ...     

Nonlinear behavior for nanoscale electrostatic actuators with Casimir force

The influence of Casimir force on the nonlinear behavior of nanoscale electrostatic actuators is studied in this paper. A one degree of freedom mass-spring model is adopted and the bifurcation properties of the actuators are obtained. With the change of the geometrical dimensions, the number of equilibrium point varies from zero to two. Stability analysis shows that one equilibrium point is Hopf point and the other is unstable saddle point when there are two equilibrium points. We also obtain the phase portraits, in which the periodic orbits exist around the Hopf point, and a homoclinic orbit passes through the unstable saddle point.     

Turing Instabilities at Hopf Bifurcation

Turing–Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this paper we suggest a way for deriving asymptotic expansions to the limit cycle solutions due to a Hopf bifurcation in two-dimensional reaction systems and we use them to build convenient normal modes for the analysis of Turing instabilities of the limit cycle. They extend the Fourier modes for the steady state in the classical Turing approach, as they include time-periodic fluctuations induced by the limit cycle. Diffusive instabilities can be properly considered because of the non-catastrophic loss of stability that the steady state shows while the limit cycle appears. Moreover, we shall see that instabilities may appear even though the diffusion coefficients are equal. The obtained normal modes suggest that there are two possible ways, one weak and the other strong, in which the limit cycle generates oscillatory Turing instabilities near a Turing–Hopf bifurcation point. In the first case slight oscillations superpose over a dominant steady inhomogeneous pattern. In the second, the unstable modes show an intermittent switching between complementary spatial patterns, producing the effect known as twinkling patterns.     

Hopf bifurcation near a double singular point

The nonlinear equation f(x,λ,) = 0, f:X × R²→X, where X is a Banach space and f satisfies a Z₂-symmetry relation is considered. Interest centres on a certain type of double singular point, where the solution x is symmetric and f_x has a double zero eigenvalue, with one eigenvector symmetric and one antisymmetric.

We show that under certain nondegeneracy conditions, which are stated both algebraically and geometrically, there exists a path of Hopf bifurcations or imaginary Hopf bifurcations passing through the double singular point, and for which x is not symmetric except at the double singular point. An easy geometrical test is found to decide which type of phenomenon occurs. A biproduct of the analysis is that explicit expressions are obtained for quantities which help to provide a reliable numerical method to compute these paths. A pseudo-spectral method was used to obtain numerical results for the Brusselator equations to illustrate the theory.     

Local analysis near a folded saddle-node singularity

Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory—the blow-up technique—and from delayed Hopf bifurcation theory—complex time path analysis—to analyze the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. To derive these canard results, we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths. This enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost. Furthermore, we define a way-in/way-out function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity. Branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical (slow) manifold make our analysis significantly different from the classical delayed Hopf bifurcation analysis where these eigenvalues are complex only.     

Zero‐Hopf bifurcation in the FitzHugh–Nagumo system

We characterize the values of the parameters for which a zero‐Hopf equilibrium point takes place at the singular points, namely, O (the origin), P₊, and P_? in the FitzHugh–Nagumo system. We find two two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at the origin is a zero‐Hopf equilibrium. For these two families, we prove the existence of a periodic orbit bifurcating from the zero‐Hopf equilibrium point O. We prove that there exist three two‐parameter families of the FitzHugh–Nagumo system for which the equilibrium point at P₊ and at P_? is a zero‐Hopf equilibrium point. For one of these families, we prove the existence of one, two, or three periodic orbits starting at P₊ and P_?. Copyright © 2014 John Wiley & Sons, Ltd.     

Restrictions and unfolding of double Hopf bifurcation in functional differential equations

The normal form of a vector field generated by scalar delay-differential equations at nonresonant double Hopf bifurcation points is investigated. Using the methods developed by Faria and Magalhães (J. Differential Equations 122 (1995) 181) we show that (1) there exists linearly independent unfolding parameters of classes of delay-differential equations for a double Hopf point which generically map to linearly independent unfolding parameters of the normal form equations (ordinary differential equations), (2) there are generically no restrictions on the possible flows near a double Hopf point for both general and -symmetric first-order scalar equations with two delays in the nonlinearity, and (3) there always are restrictions on the possible flows near a double Hopf point for first-order scalar delay-differential equations with one delay in the nonlinearity, and in nth-order scalar delay-differential equations (n?2) with one delay feedback.     

Hopf bifurcation and the centers on center manifold for a class of three-dimensional Circuit system

In this paper, Hopf bifurcation and center problem for a generic three-dimensional Chua's circuit system are studied. Applying the formal series method of computing singular point quantities to investigate the two cases of the generic circuit system, we find necessary conditions for the existence of centers on a local center manifold for the systems, then Darboux method is applied to show the sufficiency. Further, we determine the maximum number of limit cycles that can bifurcate from the corresponding equilibrium via Hopf bifurcation.