1∶2 INTERNAL RESONANCE OF COUPLED DYNAMIC SYSTEM WITH QUADRATIC AND CUBIC NONLINEARITIES

ＩｎｔｒｏｄｕｃｔｉｏｎＦｒｏｍｔｈｅｂｅｇｉｎｎｉｎｇｏｆ 1 990ｓ,ｒｅｓｅａｒｃｈｏｎｓｙｓｔｅｍｗｉｔｈｅｎｇｉｎｅｅｒｉｎｇｐｒａｃｔｉｃａｌｂａｃｋｇｒｏｕｎｄｂｅｃｏｍｅｓｍｏｒｅａｎｄｍｏｒｅａｔｔｒａｃｔｉｖｅｂｙｕｓｉｎｇｔｈｅｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃｓｙｓｔｅｍｔｈｅｏｒｙ .Ａｓａｍｅｔｈｏｄｏｆｓｉｍｐｌｉｆｙｉｎｇｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓ,ｔｈｅｔｈｅｏｒｙｏｆＮｏｒｍａｌＦｏｒｍｇｉｖｅｓｔｈｅｓｉｍｐｌｅｓｔｆｏｒｍｓｏｆｄｉｆｆｅｒｅｎｔｉａ…     

HOPF BIFURCATION IN A THREE-DIMENSIONAL SYSTEM

In this paper,Liapunor-Schmidt reduction and singularity theory are employed todiscuss Hopf and degenerate Hopf bifurcations in global parametric region in a three-dimensional system x=-βx y,y=-x-βy(1-kz),z=β[α(1-z)-ky~2].Theconditions on existence and stability are given.     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

ＴｈｅｄｅｃｉｓｉｖｅｅｆｆｅｃｔｓｏｆｓｉｎｇｕｌａｒｐｏｉｎｔｓａｎｄｂｏｕｎｄａｒｉｅｓｂｅｌｏｎｇｉｎｇｔｏｓｏｌｕｔｉｏｎｐｒｏｃｅｓｓｅｓｏｆＩｔｏｓｔｏｃｈａｓｔｉｃｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｏｎｔｈｅｓｉｔｕａｔｉｏｎｓｏｆｄｉｓｔｒｉｂｕｔｉｏｎｓｏｆｅｒｇｏｄｉｃｃｏｍｐｏｎｅｎｔｓ,ｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｔｈｅｆｏｒｍｓｏｆｉｎｖａｒｉａｂｌｅｍｅａｓｕｒｅｓ,ｔｈｅｓａｍｐｌｅｐｒｏｐｅｒｔｉｅｓ（ｉｎｃｌｕｄｉｎｇｓａｍｐｌｅｓｔａｂｉｌｉｔｙ）…     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅰ)

For a real noise parametrically excited co-dimension two bifurcation system on three-dimensional central manifold, a model of enhanced generality is developed in the present paper by assuming the real noise to be an output of a linear filter system, namely, a zero-mean stationary Gaussian diffusion process that satisfies the detailed balance condition. On such basis, asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are established by use of Arnold asymptotic analysis approach in parallel with the eigenvalue spectrum of Fokker-Planck operator. Foundation item: the National Natural Science Foundation of China (19602016)     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅱ)

For a co-dimension two bifurcation system on a three-dimensional central manifold, which is parametrically excited by a real noise, a rather general model is obtained by assuming that the real noise is an output of a linear filter system-a zeromean stationary Gaussian diffusion process which satisfies detailed balance condition. By means of the asymptotic analysis approach given by L. Arnold and the expression of the eigenvalue spectrum of Fokker-Planck operator, the asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are obtained. Foundation item: the National Natural Science Foundation of China (19602016)     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO_DIMENSION TWO BIFURCATION SYSTEM (Ⅰ)

Ｐｒｅｓｅｎｔｌｙ,ｔｈｅｃｏｍｐｕｔａｔｉｏｎａｌａｐｐｒｏａｃｈｔｏｍａｘｉｍａｌＬｙａｐｕｎｏｖｅｘｐｏｎｅｎｔｓｆｏｒｎｏｎｌｉｎｅａｒｓｔｏｃｈａｓｔｉｃｓｙｓｔｅｍｓｈａｓｅｍｅｒｇｅｄａｓｏｎｅｐｒｉｍａｒｙｆｏｃｉｏｆｒｅｓｅａｒｃｈｉｎｔｅｒｅｓｔｉｎｔｈｅｆｉｅｌｄｏｆｓｙｓｔｅｍｓｔａｂｉｌｉｔｙａｎｄｓｔｏｃｈａｓｔｉｃｂｉｆｕｒｃａｔｉｏｎ．ＴｈｉｓｉｓｍａｉｎｌｙａｔｔｒｉｂｕｔｅｄｔｏｔｈｅｆａｃｔｔｈａｔｍａｘｉｍａｌＬｙａｐｕｎｏｖｅｘｐｏｎｅｎｔｈａｓｂｅｅｎ…     

ON INTERNAL RESONANCE OF NONLINEAR VIBRATING SYSTEMS WITH MANY DEGREES OF FREEDOM

The problem of periodic solutions of nonlinear autonomous systems with many degrees of freedom is considered. This is made possible by the development ora modified version of the KBM method. The method can be used to generate limit cycle phase portrait, amplitude, period and to indicate stability of the limit cycle.     

GLOBAL BIFURCATION AND CHAOS IN A VAN DER POL-DUFFING-MATHIUE SYSTEM WITH A SINGLE-WELL POTENTIAL OSCILLATOR

The global bifurcation and chaos are investigated in this paper for a van der Pol-Duff-ing-Mathieu system with a single-well potential oscillator by means of nonlinear dynamics. The au-tonomous system corresponding to the system under discussion is analytically studied to draw all globalbifureation diagrams in every parameter space, These diagrams are called basic bifurcation ones. Thenfixing parameter in every space and taking the parametrically excited amplitude as a bifurcation param-eter, we can observe how to evolve from a basic bifurcation diagram to a chaos pattern in terms of nu-merical methods. The results are sufficient to show that the system has distinct dynamic behavior, Fi-nally, the properties of the basins of attraction are observed and the appearance of fractal basin bound-aries heralding the onset of a loss of structural integrity is noted in order to consider how to control theextent and the rate of the erosion in the next paper.     

SINGULAR ANALYSIS OF BIFURCATION OF NONLINEAR NORMAL MODES FOR A CLASS OF SYSTEMS WITH DUAL INTERNAL RESONANCES

ＩｎｔｒｏｄｕｃｔｉｏｎＡｎｉｎｔｅｒｅｓｔｉｎｇｆｅａｔｕｒｅｉｎｔｈｅｆｒｅｅｖｉｂｒａｔｉｏｎｏｆａｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｉｓｔｈｅｆａｃｔｔｈａｔｔｈｅｎｕｍｂｅｒｏｆｅｘｉｓｔｉｎｇｎｏｒｍａｌｍｏｄｅｓｍａｙｅｘｃｅｅｄｔｈｅｎｕｍｂｅｒｏｆｄｅｇｒｅｅｓｏｆｆｒｅｅｄｏｍ ,ａｐｈｅｎｏｍｅｎｏｎｎｏｔｅｎｃｏｕｎｔｅｒｅｄｉｎａｌｉｎｅａｒｓｙｓｔｅｍａｎｄｃａｕｓｅｄｂｙｍｏｄｅｂｉｆｕｒｃａｔｉｏｎ .Ｔｈｅｒｅｆｏｒｅｍｕｃｈｗｏｒｋｈａｓｂｅｅｎｄｏｎｅｏｎｔｈｅｓｔ…     

HIGHER-ORDER THEORY OF INTERNAL SOLITARY WAVES WITH A FREE SURFACE IN TWO-LAYER FLUID SYSTEM OF FINITE DEPTH

A higher-order approximation theory of internal solitary waves with a frre surface ispresented.Using the method of strained co-ordinates,the third-order approximationevolution equation of interface has been found.An analytic expression of the wave velocityis given.The evolution equation has been solved numerically.It is found that the effects offree surface on the shape and wave velocity of solitary wave are O(ε_2),and the third-order numerical solutions,are closer to experimental data than the first-and second-ordersolutions.     

THE STUDY ON A KIND OF CONTROL SYSTEM WITH NONLINEAR PARABOLIC DISTRIBUTED PARAMETERS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅａｃｔｉｖｉｔｉｅｓｓｕｃｈａｓｔｈｅａｅｒｏｓｐａｃｅｅｎｇｉｎｅｅｒｉｎｇｈｅａｔｐｒｏｔｅｃｔｉｏｎ[1,2 ],ｔｈｅｌａｒｇｅ＿ｓｃａｌｅｃｏｎｃｒｅｔｅｐｏｕｒｉｎｇ ,ｔｈｅｃｈｅｍｉｃａｌｐｒｏｄｕｃｔｉｏｎ ,ｔｈｅｐｏｗｄｅｒｍｅｔａｌｌｕｒｇｙａｎｄｔｈｅｆｉｒｅｓａｆｅｔｙｅｎｇｉｎｅｅｒｉｎｇ[3,4],ｔｈｅｒｅｉｓｏｎｅｋｉｎｄｏｆｎｏｎｌｉｎｅａｒｐａｒａｂｏｌｉｃｄｉｓｔｒｉｂｕｔｅｄｐａｒａｍｅｔｅｒｃｏｎｔｒｏｌｓｙｓｔｅｍｗｉｔｈｍｏｖｉ…     

CO-DIMENSION 2 BIFURCATIONS AND CHAOS IN CANTILEVERED PIPE CONVEYING TIME VARYING FLUID WITH THREE-TO-ONE INTERNAL RESONANCES

The nonlinear behavior of a cantilevered fluid conveying pipe subjected to principal parametric and internal resonances is investigated in this paper. The flow velocity is divided into constant and sinusoidai parts. The velocity value of the constant part is so adjusted such that the system exhibits 3:1 internal resonances for the first two modes. The method of multiple scales is employed to obtain the response of the system and a set of four first-order nonlinear ordinary-differential equations for governing the amplitude of the response. The eigenvalues of the Jacobian matrix are used to assess the stability of the equilibrium solutions with varying parameters. The codimension 2 derived from the double-zero eigenvaiues is analyzed in detail. The results show that the response amplitude may undergo saddle-node, pitchfork, Hopf, homoclinic loop and period-doubling bifurcations depending on the frequency and amplitude of the sinusoidal flow. When the frequency of the sinusoidal flow equals exactly half of the first-mode frequency, the system has a route to chaos by period-doubling bifurcation and then returns to a periodic motion as the amplitude of the sinusoidal flow increases.     

STUDY FOR THE BIFURCATION TOPOLOGICAL STRUCTURE AND THE GLOBAL COMPLICATED CHARACTER OF A KIND OF NONLINEAR FINANCE SYSTEM(Ⅰ)

ＩｎｔｒｏｄｕｃｔｉｏｎＣｈａｏｔｉｃｉｓｔｈｅｉｎｈｅｒｅｎｔｒａｎｄｏｍｎｅｓｓｉｎｔｈｅｄｅｆｉｎｉｔｅｓｙｓｔｅｍ .Ｔｈｅｄｅｆｉｎｉｔｅｎｅｓｓｉｓｃａｕｓｅｄｂｙｔｈｅｓｙｓｔｅｍｉｎｔｅｒｎａｌｓｂｕｔｎｏｔｂｙｔｈｅｅｘｔｅｒｎａｌｄｉｓｔｕｒｂａｎｃｅ[1- 4],ｗｈｉｌｅｔｈｅｉｎｈｅｒｅｎｔｒａｎｄｏｍｎｅｓｓｉｓｔｈｅｉｒｒｅｇｕｌａｒａｎｄｄｉｆｆｉｃｕｌｔ＿ｔｏ＿ｂｅ＿ｐｒｅｄｉｃｔｅｄｂｅｈａｖｉｏｒｏｆｔｈｅｓｙｓｔｅｍ[5 ,6 ].Ｗｈａｔｉｓｍｏｓｔａｔｔｒａｃｔｉｖｅ…     

STUDY FOR THE BIFURCATION TOPOLOGICAL STRUCTURE AND THE GLOBAL COMPLICATED CHARACTER OF A KIND OF NONLINEAR FINANCE SYSTEM(Ⅱ)

Based on the work discussed on the former study, this article first starts from the mathematical model of a kind of complicated financial system, and analyses all possible things that the model shows in the operation of our country’s macro-financial system: balance, stable periodic, fractal, Hopf-bifurcation, the relationship between parameters and Hopf-bifurcation, and chaotic motion etc. By the changes of parameters of all economic meanings, the conditions on which the complicated behaviors occur in such a financial system, and the influence of the adjustment of the macro-economic policies and adjustment of some parameter on the whole financial system behavior have been analyzed. This study will deepen people’s understanding of the lever function of all kinds of financial policies.     

STUDY FOR THE BIFURCATION TOPOLOGICAL STRUCTURE AND THE GLOBAL COMPLICATED CHARACTER OF A KIND OF NONLINEAR FINANCE SYSTEM(Ⅱ)

ＩｎｔｒｏｄｕｃｔｉｏｎＣｈａｏｔｉｃｉｓｔｈｅｉｎｈｅｒｅｎｔｒａｎｄｏｍｎｅｓｓｉｎｔｈｅｄｅｆｉｎｉｔｅｓｙｓｔｅｍ .Ｔｈｅｄｅｆｉｎｉｔｅｎｅｓｓｉｓｃａｕｓｅｄｂｙｔｈｅｓｙｓｔｅｍｉｎｔｅｒｎａｌｓｂｕｔｎｏｔｂｙｔｈｅｅｘｔｅｒｎａｌｄｉｓｔｕｒｂａｎｃｅ,ｗｈｉｌｅｔｈｅｉｎｈｅｒｅｎｔｒａｎｄｏｍｎｅｓｓｉｓｔｈｅｉｒｒｅｇｕｌａｒａｎｄｄｉｆｆｉｃｕｌｔ＿ｔｏ＿ｂｅ＿ｐｒｅｄｉｃｔｅｄｂｅｈａｖｉｏｒｏｆｔｈｅｓｙｓｔｅｍ .Ｗｈａｔｉｓｍｏｓｔａｔｔｒａｃｔｉｖｅｏｆｔｈｅｃ…     

STUDY FOR THE BIFURCATION TOPOLOGICAL STRUCTURE AND THE GLOBAL COMPLICATED CHARACTER OF A KIND OF NONLINEAR FINANCE SYSTEM (Ⅰ)

Based on the mathematical model of a kind of complicated financial system, all possible things that the model shows in the operation of our country’s macro-financial system were analyzed, such as balance, stable periodic, fractal, Hopf-bifurcation, the relationship between parameters and Hopf-bifurcations,and chaotic motion etc. By the changes of parameters of all economic meanings, the conditions on which the complicated behaviors occur in such a financial system, and the influence of the adjustment of the macro-economic policies and adjustment of some parameter on the whole financial system behavior were analyzed. This study will deepen people’s understanding of the lever function of all kinds of financial policies.     

WENTZEL–KRAMERS–BRILLOUIN SOLUTIONS TO AN EQUATION OF INTERNAL GRAVITY WAVES IN A STRATIFIED MEDIUM WITH SLOWLY VARYING SHEAR FLOWS

Journal of Applied Mechanics and Technical Physics - Model buoyancy frequency distribution and the Wentzel–Kramers–Brillouin method are used to obtain an asymptotic solution to a...     

FREE CONVECTION FLOW OF WATER AT 4℃ PASSING AN INFINITE POROUS PLATE WITH CONSTANT SUCTION IN A ROTATING SYSTEM

In this work an analysis of steady free convection flow of waterat 4℃ passing a vertical infinite porous plate in a rotating systemis considered.Approximate solutions for the coupled nonlinear equa-tions are obtained for the velocity and the temperature.The effectof E’(Eckman number) is discussed,when P (Prandtl number)=11.4,which corresponds to water at 4℃.     

FREE CONVECTION FLOW OF WATER AT 4℃ PASSING AN INFINITE POROUS PLATE WITH CONSTANT SUCTION IN A ROTATING SYSTEM

In this work an analysis of steady free convection flow of water at 4℃ passing a vertical infinite porous plate in a rotating system is considered. Approximate solutions for the coupled nonlinear equations are obtained for the velocity and the temperature. The effect of E’ (Eckman number) is discussed, when P (prandtl number)=11.4, which corresponds to water at 4℃.     

A STUDY OF THE CATASTROPHE AND THE CAVITATION FOR A SPHERICAL CAVITY IN HOOKE’S MATERIAL WITH 1/2 POISSON’S RATIO

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｒｅｃｅｎｔｙｅａｒｓ,ｔｈｅｒｅｓｅａｒｃｈｅｓｏｎｃａｖｉｔａｔｉｏｎａｎｄｃａｔａｓｔｒｏｐｈｅｏｆａｃａｖｉｔｙｈａｖｅｓｕｐｐｌｉｅｄａｎｅｗｍｅｔｈｏｄｆｏｒｉｎｖｅｓｔｉｇａｔｉｎｇｔｈｅｍｅｃｈａｎｉｃｓｏ...