ABUNDANCE OF UNIMODAL MAPS WITH DENSE CRITICAL ORBIT AND PREFIXED CRITICAL ORBIT

ＩｎｔｒｏｄｕｃｔｉｏｎＡｍａｊｏｒｐｒｏｂｌｅｍｏｆｏｎｅｄｉｍｅｎｓｉｏｎａｌｄｙｎａｍｉｃｓｉｓｔｏｓｔｕｄｙｔｈｅａｔｔｒａｃｔｏｒｓｏｆｉｎｔｅｒｖａｌｍａｐｓ,ｉ.ｅ.ｔｈｅｃｌａｓｓｉｆｉｃａｔｉｏｎａｎｄｓｔｒｕｃｔｕｒｅｏｆａｔｔｒａｃｔｏｒｓ.ＦｏｌｌｏｗｉｎｇＭｉｌｎｏｒ’ｓｄｅｆｉｎｉｔｉｏｎｏｎａｔｔｒａｃｔｏｒｓ[1],ｉｆｆｉｓａｕｎｉｍｏｄａｌｍａｐｗｉｔｈｎｅｇａｔｉｖｅＳｃｈｗａｒｚｉａｎｄｅｒｉｖａｔｉｖｅ ,ｔｈｅｎｂｙＧｕｃｋｅｎｈｅｉｍｅｒ’ｓｔｈｅｏｒｅｍ[2 ]…     

TheC k instability of navier-Stokes equation appending polynomials of unknown functions

Applying the theory of stratification, the solution space structure about a class of deformed Navier-Stokes equation isdetermined. It is proved that such kind of equation has no C^k (k≥2) stable solution by the fact that the strate transversale is a null set. Foundation item: the National Natural Science Foundation of China (19971054), Shanghai Natural Science Foundation (99ZA14034) Biography: HE You-hua (1960-)     

Twisted bifurcations and stability of homoclinic loop with higher dimensions

1　IntroductionandHypothesesInrecentyears,withthedevelopmentofnonlinearsciencesandthedeepstudyofchaoticphenomena,thebifurcationproblemsofhomoclinicloopsforhigherdimensionalsystemswerestudiedextensivelyandalotofresultswereobtained (seeRefs.[1～ 8] ) .Especially ,Refs.[3 ,4]discussedthehomoclinicloopbifurcationswithcodimension 2 .Refs.[5,6]consideredthedegeneratedhomoclinicbifurcations.Inthispaper,westudythebifurcationsoftwistedhomoclinicloopsandthestabilityinhigherdimensionalspace .Considerth…     

Chaotic behavior of logistic map in superior orbit and an improved chaos-based traffic control model

With a vital role of discrete chaos, standard logistic map has found a celebrated place in the dynamics of chaos theory and in various applications of science, such as a discrete traffic flow model, image encryption in cryptography, secure communication, and weather forecasting. Traditionally, this discrete chaos is controlled by one parameter \(\lambda \) using Picard orbit, a one-step feedback procedure. This article presents a one-step forward, applying Mann orbit (superior orbit) the chaotic properties such as period-doubling, period-3 window, and Lyapunov exponent of the standard logistic map is investigated. The results are illustrated analytically and experimentally followed by concluding remarks and a few counter examples. Due to the extra degree of freedom in parameter \(\lambda \), the map provides improved chaotic properties that increases the performance of dynamical phenomena. Moreover, this study describes an improved chaos-based discrete traffic control model. Surprisingly, added new parameter \(\alpha \) in Mann orbit works as control variable that increases the stability performance of the traffic model.     

Dynamical properties of 2-torus parabolic maps

In this paper, a class of linear maps on the 2-torus are discussed. Discussions are focused on the case that the maps are parabolic. It is shown that the maximal invariant set for a 2-torus parabolic map is indeed invariant, and is almost closed, and the Lebesgue measure restricted to a maximal invariant set is invariant. Under this invariant measure, all Lyapunov exponents of a parabolic map are zero. In certain simple cases, the Lebesgue measure of the maximal invariant sets are computed and estimated. For the case the maps are invertible, it is shown that the inverse of a non-horocyclic parabolic map is no longer a parabolic map. Interesting properties of the conjugation of invertible parabolic maps by automorphisms of the torus are characterized, and a conjugation invariant for such maps are obtained. And it is proven that all these maps can be reduced to a family of one parameter rigid rotations. Mathematics Subject Classification: 37C15, 37D50     

Structure of nonnegative nontrivial and positive solutions of singularly perturbed p -laplace equations

Structure of nonnegative nontrivial and positive solutions was precisely studied for some singularly perturbed p-Laplace equations. By virtue of sub- and supersolution method, it is shown that there are many nonnegative nontrivial spike-layer solutions and positive intermediate spike-layer solutions. Moreover, the upper and lower bound on the measure of each spike-layer solutions. Moreover, the upper and lower bound on the measure of each spike-layer were estimated when the parameter is sufficiently small. Foundation items: the National Key Basic Research Development Project Program of China (1999032801); the National Natural Science Foundation of China (10371095) Biography: ZHANG Zheng-ce (1976≈)     

ON THE CRITICAL POINTS OF THE MAP Fp:X→‖AXB-C‖pp

Suppose A,B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A^- such that (AA^-)^*=AA^- and B has a generalized inverse B^- such that (B^-B)^*=B^-B,the general characteristic forms for the critical points of the map F_p:X^→‖AXB-C‖_p^p(1p=2. Similarly, the same question has been discussed for several operators.     

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincaré map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion. The project supported by the National Natural Science Foundation of Chine (10082003)     

Generalized fatigue constant life curve and two-dimensional probability distribution of fatigue limit

ＩｎｔｒｏｄｕｃｔｉｏｎＷｈｅｎａｃｏｍｐｏｎｅｎｔｆｏｒｉｎｓｔａｎｃｅｔｈｅｔｒａｎｓｍｉｓｓｉｏｎｓｈａｆｔａｎｄｔｈｅｖｉｂｒａｔｉｏｎｃｏｍｐｏｎｅｎｔ,ｅｔｃ.ｅｎｄｕｒｅｒａｔｈｅｒｌｏｗｓｔｒｅｓｓｌｅｖｅｌａｎｄｌｏｎｇｓｔｒｅｓｓｃｙｃｌｅｓ,ｉｔｓｓａｆｅｔｙｉｓｃｏｎｔｒｏｌｌｅｄｂｙｆａｔｉｇｕｅｓｔｒｅｓｓａｎｄｉｔｃａｎｂｅｄｅｓｉｇｎｅｄａｃｃｏｒｄｉｎｇｔｏｔｈｅｉｎｆｉｎｉｔｅｌｉｆｅｄｅｓｉｇｎｍｅｔｈｏｄｏｆｈｉｇｈｃｙｃｌｅｆａｔｉｇｕｅ .Ｗｈｅｎａｃｏｍ…     

Constrained multiobjective games in locally convex<Emphasis Type="Italic">H</Emphasis>-spaces

A new class of constrained multiobjective games with infinite players in noncompact locally convex H-spaces without linear structure are introduced and studied. By applying a Fan-Glicksberg type fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values and a maximum theorem, several existence theorems of weighted Nath-equilibria and Pareto equilibria for the constrained multiobjective games are proved in noncompact locally convex H-spaces. These theorems improve, unify and generalize the corresponding results of the multiobjective games in recent literatures. Contributed by Ding Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province ([2000]25) Biography: Ding Xie-ping (1938-)     

On probabilistic norm of a linear operators and space of operators

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ［１］,ＸｉａｏｉｎｔｒｏｄｕｃｅｄｔｈｅｎｏｔｉｏｎｏｆｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｏｆａｌｉｎｅａｒｏｐｅｒａｔｏｒｏｎＭｅｎｇｅｒｐｒｏｂａｂｉｌｉｓｔｉｃｎｏｒｍｅｄｓｐａｃｅ（ｂｒｉｅｆｙ,ＭｅｎｇｅｒＰＮ＿ｓｐａｃｅ）．Ａｎｄｂｙｕｓｉｎｇｔｈｅｎｏｔｉｏｎ,ｕｎｄｅｒｔｈｅｃｏｎｄｉｔｉｏｎｏｆｔ＿ｎｏｒｍΔｓａｔｉｓｆｙｉｎｇΔ（ｔ,ｔ）≥ｔ（ｔ∈（０,１］）,ｈｅｇａｖｅａｄｅｓｃｒｉｐｔｉｏｎｏｆｂｏｕｎｄｅｄｎｅｓｓｏｆｌｉｎｅａｒｏｐｅ…     

THE HAMILTONIAN SYSTEM AND COMPLETENESS OF SYMPLECTIC ORTHOGONAL SYSTEM

I.IntroductionThemethodofseparationofvariablesisimportanttosolvethesoluti0n0fprobIem0fmathematicalphysics,butmanyproblen1sofmathematicalphysicscannotseparatet'ariab1es,thereforeitrestrictstheranget0appIicatemethodofseparationofvariable.Inthepaperlll,Zhong…     

Adaptive control for autonomous rendezvous of spacecraft on elliptical orbit

A strategy for spacecraft autonomous rendezvous on an elliptical orbit in situation of no orbit information is developed. Lawden equation is used to describe relative motion of two spacecraft. Then an adaptive gain factor is introduced, and an adaptive control law for auton- omous rendezvous on the elliptical orbit is designed using Lyapunov approach. The relative motion is proved to be ultimately bounded under this control law, and the final relative position error can achieve the expected magnitude. Simulation results indicate that the adaptive control law can realize autonomous rendezvous on the elliptical orbit with relative state information only.     

Mathematical foundation of a new complexity measure

For many continuous bio-medieal signals with both strong nonlinearity and non-stationarity, two criterions were proposed for their complexity estimation ： （1） Only a short data set is enough for robust estimation; （2） No over-coarse graining preproeessing, such as transferring the original signal into a binary time series, is needed. Co complexity measure proposed by us previously is one of such measures. However, it lacks the solid mathematical foundation and thus its use is limited. A modified version of this measure is proposed, and some important properties are proved rigorously. According to these properties, this measure can be considered as an index of randomness of time series in some senses, and thus also a quantitative index of complexity under the meaning of randomness finding complexity. Compared with other similar measures, this measure seems more suitable for estimating a large quantity of complexity measures for a given task, such as studying the dynamic variation of such measures in sliding windows of a long process, owing to its fast speed for estimation.     

Singular Characteristics of Nonlinear Normal Modes in a Two Degrees of Freedom Asymmetric Systems with Cubic Nonlinearities

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｍｏｄｅｒｎａｎａｌｙｓｉｓａｎｄｍｅｔｈｏｄｓｆｏｒｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓｈａｖｅｇｒｅａｔｌｙｐｒｏｍｏｔｅｄｔｈｅｄｅｖｅｌｏｐｍｅｎｔｉｎｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅ.ＴｈｅｓｅｉｎｃｌｕｄｅＬ＿Ｓｒｅｄｕｃｅ[1],ｓｉｎｇｕｌａｒｉｔｙｔｈｅｏｒｙ[2 ],ｐｅｒｔｕｒｂａｔｉｏｎｔｅｃｈｎｉｑｕｅ[3 ],Ｍｅｌｎｉｋｏｖｆｕｎｃｔｉｏｎ[4 ],Ｃ＿Ｌｍｅｔｈｏｄ[5 ]ａｎｄｃｅｎｔｅｒｍａｎｉｆｏｌｄ[6],ｅｔｃ .Ｈｏｗｅｖｅｒ,ｔｈｅｃｏｕｐｌｉｎｇｂｅ…     

Stability of plane oscillations of a satellite in a circular orbit

We study motions of a rigid body (a satellite) about the center of mass in a central Newtonian gravitational field in a circular orbit. There is a known particular motion of the satellite in which one of its principal central axes of inertia is perpendicular to the orbit plane and the satellite itself exhibits plane pendulum-like oscillations about this axis. Under the assumption that the satellite principal central moments of inertia A, B, and C satisfy the relation B = A + C corresponding to the case of a thin plate, we perform rigorous nonlinear analysis of the orbital stability of this motion.In the plane of the problem parameters, namely, the oscillation amplitude ε and the inertial parameter, there exist countably many domains of orbital stability of the satellite oscillations in the linear approximation. Nonlinear orbital stability analysis was carried out in thirteen of these domains. Isoenergetic reduction of the system of equations of the perturbed motion is performed at the energy level corresponding to the unperturbed periodic motion. Further, using the algorithm developed in [1], we construct the symplectic mapping generated by the equations of the reduced system, normalize it, and analyze the stability. We consider resonance and nonresonance cases. For small values of the oscillation amplitude, we perform analytic investigations; for arbitrary values of ε, numerical analysis is used.Earlier, numerical analysis of stability of plane pendulum-like motions of a satellite in a circular orbit was performed in several special cases in [1–4].     

Some problems in the Z-C-X space

The new concepts of the Z-C-X space and excellent cone are introduced. Some problems of random semiclosed 1-set-contractive operator are investigated in the Z-C-X space. At first, an important inequality is proved. Secondly, several new conclusions are proved by means of random fixed point index in the theory of random topological degree. A random solution of a class of random operator equations under conditions of imitating the parallelogram law is obtained, famous Altman’s theorem is generalized in partially ordered Z-C-X space. Therefore, some new results are obtained.     

Analysis of nonlinear dynamic stability of liquid-converying pipes

Nonlinearly dynamic stability of flexible liquid-conveying pipe in fluid structure interaction was analyzed by using modal disassembling technique. The effects of Poisson, Junction and Friction couplings in the wave-flowing-vibration system on the pipe dynamic stability were included in the analytical model constituted by four nonlinear differential equations. An analyzing example of cantilevered pipe was done to illustrate the dynamic stability characteristics of the pipe in the full coupling mechanisms, and the phase curves related to the first four modal motions were drawn. The results show that the dynamic stable characteristics of the pipe are very complicated in the complete coupling mechanisms, and the kinds of the singularity points corresponding to the various modal motions are different. Foundation items: the National Natural Science Foundation of China (50079007); the Hydraulic Science Foundation of China Hydraulic Ministry (SZ9830); the Natural Science Foundation of Yunnan Provicce (98E003G) Biographies: ZHANG Li-xiang (1959−) HUANG Wen-hu (1926−)     

The study on the chaotic motion of a nonlinear dynamic system

ｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓ,ｃｈａｏｓｉｎｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓｙｓｔｅｍｓｈａｓｂｅｎａｒｏｕｓｉｎｇｍｏｒｅａｎｄｍｏｒｅｉｎｔｅｒｅｓｔ［１～３］．Ｔｈｅｃｈａｏｔｉｃｍｏｔｉｏｎｉｓｒｅｇａｒｄｅｄａｓａｎａ...