Bifurcation and chaos analysis for aeroelastic airfoil with freeplay structural nonlinearity in pitch

The dynamics character of a two degree-of-freedom aeroelastic airfoil with combined freeplay and cubic stiffness nonlinearities in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincare mapping method and Floquet theory are adopted to analyse the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the Poincare section also approves the previous conclusion.

Bifurcations and Chaos in Mira 2 Map

In this paper, Mira 2 map is investigated. The conditions of the existence for fold bifurcation, flip bifurcation and Naimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. And the conditions of the existence for chaos in the sense of Marroto are obtained. Numerical simulation results not only show the consistence with the theoretical analysis but also display complex dynamical behaviors, including period-n orbits, crisis, some chaotic attractors, period-doubling bifurcation to chaos, quasi-period behaviors to chaos, chaos to quasi-period behaviors, bubble and onset of chaos.

Bifurcations and Chaos of a Discrete Mathematical Model for Respiratory Process in Bacterial Culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.

Hopf-flip bifurcation of high dimensional maps and application to vibro-impact systems

This paper addresses the problem of Hopf-flip bifurcation of high dimensional maps. Using the center manifold theorem, we obtain a three dimensional reduced map through the projection technique. The reduced map is further transformed into its normal form whose coefficients are determined by that of the original system. The dynamics of the map near the Hopf-flip bifurcation point is approximated by a so called “time-2τ^2 map” of a planar autonomous differential equation. It is shown that high dimensional maps may result in cycles of period two, tori T^1 （Hopf invariant circles）, tori 2T^1 and tori 2T^2 depending both on how the critical eigenvalues pass the unit circle and on the signs of resonant terms＇ coefficients. A two-degree-of-freedom vibro-impact system is given as an example to show how the procedure of this paper works. It reveals that through Hopf-flip bifurcations, periodic motions may lead directly to different types of motion, such as subharmonic motions, quasi-periodic motions, motions on high dimensional tori and even to chaotic motions depending both on change in direction of the parameter vector and on the nonlinear terms of the first three orders.

Discrete Chaos in Fractional Henon Map

In this study, a discrete fractional Henon map is proposed in the Caputo discrete delta’s sense. The results show that the discrete fractional calculus is an efficient tool and the maps derived in this way have simpler forms but hold rich dynamical behaviors.

TWISTED BIFURCATIONS AND STABILITY OF HOMOCLINIC LOOP WITH HIGHER DIMENSIONS

By using the linear independent solutions of the linear variational equation along the homoclinic loop as the demanded local coordinates to construct the Poincaré map,the bifurcations of twisted homoclinic loop for higher dimensional systems are studied.Under the nonresonant and resonant conditions,the existence,number and existence regions of the 1-homoclinic loop,1-periodic orbit,2-homoclinic loop,2-periodic orbit and 2-fold 2-periodic orbit were obtained.Particularly,the asymptotic repressions of related bifurcation surfaces were also given.Moreover, the stability of homoclinic loop for higher dimensional systems and nontwisted homoclinic loop for planar systems were studied.

On the Regularity and Chaos of the Hydrogen Atom Subjected to External Fields

In this paper, the integrable classical case of the Hydrogen atom subjected to three static external fields is investigated. The structuring and evolution of the real phase space are explored. The bifurcation diagram is found and the bifurcations of solutions are discussed. The periodic solutions and their associated periods for singular common-level sets of the first integrals of motion are explicitly described. Numerical investigations are performed for the integrable case by means of Poincaré surfaces of section and comparing them with nearby living nonintegrable solutions, all generic bifurcations that change the structure of the phase space are illustrated;the problem can exhibit regularity-chaos transition over a range of control parameters of system.

A kind of noise-induced transition to noisy chaos in stochastically perturbed dynamical system

We investigate a kind of noise-induced transition to noisy chaos in dynamical systems. Due to similar phenomenological structures of stable hyperbolic attractors excited by various physical realizations from a given stationary random process, a specific Poincar6 map is established for stochastically perturbed quasi-Hamiltonian system. Based on this kind of map, various point sets in the Poincar6＇s cross-section and dynamical transitions can be analyzed. Results from the customary Duffing oscillator show that, the point sets in the Poincare＇s global cross-section will be highly compressed in one direction, and extend slowly along the deterministic period-doubling bifurcation trail in another direction when the strength of the harmonic excitation is fixed while the strength of the stochastic excitation is slowly increased. This kind of transition is called the noise-induced point-overspreading route to noisy chaos.

Chaos Existence in Surface Discharge of Tracking Test

Tracking tests for different polymer materials were carried out to investigate the chaotic behavior of surface discharge. The discharge sequences measured during the discharge process were analyzed for finding the evidence of chaos existence. Four kinds of nonlinear analysis methods were adopted： estimating the largest Lyapunov exponent, calculating the fractal dimension with increasing the embedding dimension, drawing the recurrence plots, and plotting the Poincare maps. It is found that the largest Lyapunov exponent of the discharge is positive, and the plot of fractal dimension, as a function of embedding dimension, will saturate at a value. The recur- rence plots show the chaotic frame-work patterns, and the Poincar6 maps also have the chaotic characteristics. The results indicate that the chaotic behavior does exist in the discharge currents of the tracking test.

Bifurcations of rough heteroclinic loop with two saddle points

The bifurcation problems of rough 2-point-loop are studied for the caseρ 1 1 >λ 1 1 ,ρ 2 1 <λ 2 1 ,ρ 1 1 ρ 2 1 <λ 1 1 λ 2 1 , where ?ρ i 1 <0 and λ i 1 >0 are the pair of principal eigenvalues of unperturbed system at saddle point pi, i = 1,2. Under the transversal and nontwisted conditions, the authors obtain some results of the existence of one 1-periodic orbit, one 1-periodic and one 1-homoclinic loop, two 1-periodic orbits and one 2-fold 1-periodic orbit. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graph is drawn.

The Bifurcation of the Equatorial Periodic Orbit of the Planar Polynomial Vector Fields

In this paper,we have obtained the necessary and sufficient condition for the set of points at infinity on the plane R^2 to be a periodic orbit which is called an equatorial periodic orbit of a planar vector field X(x),and the formulae about the multiplicity of the equatorial periodic orbit of X(x).We have also proved that the main result of [9] is erroneous with regard to the formlae.

Spatial chaos-based image encryption design

In recent years, the chaos based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques, but the drawbacks of small key space and weak security in one-dimensional chaotic cryptosystems are obvious. In this paper, permutation and sub- stitution methods are incorporated to present a stronger image encryption algorithm. Spatial chaotic maps are used to realize the position permutation, and to confuse the relationship between the ci- pher-image and the plain-image. The experimental results demonstrate that the suggested encryption scheme of image has the advantages of large key space and high security; moreover, the distribution of grey values of the encrypted image has a random-like behavior.

Analysis of a Shil’nikov Type Homoclinic Bifurcation

The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinie bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinie connections to the periodic orbit is proved.

Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincare map near the homoclinic orbit, the existence and uniqueness of 1-homoclinic orbit and 1-periodic orbit are given. Also considered is the existence of 2-homoclinic orbit and 2-periodic orbit. In additon, the corresponding bifurcation surfaces are given.

THE CRITERION OF HIGHER PERIOD ON CONTROLLING CHAOS IN A VIBRO-IMPACT SYSTEM

针对工程实际中一类碰撞振动系统的混沌行为，设计了一种动态间歇反馈控制策略，仅对不碰撞时刻进行动态反馈，该方法易于操作，选择参数范围较广，能有效节约控制能量，理论分析表明混沌吸引子中的不稳定周期轨道可以稳定化，并以此控制方法为例，采用Poincare映射和功率谱相结合技术能有效判别倍化出的各种形式的时·稳定高周期轨道和nH概周期轨道。该方法能准确识别混沌控制后高倍周期数和概周期吸引不变圈数，它对于碰撞振动系统的优化设计、振动控制及安全运行提供了理论参考。

BIFURCATIONS OF ROUGH 3-POINT-LOOP WITH HIGHER DIMENSIONS

The authors study the bifurcation problems of rough heteroclinic loop connecting threesaddle points for the case β1 ＞ 1, β2 ＞ 1, β3 ＜ 1 and β1β2β3 ＜ 1. The existence, number, co-existence and incoexistence of 2-point-loop, 1-homoclinic orbit and 1-periodic orbit are studied.Meanwhile, the bifurcation surfaces and existence regions are given.

Bifurcations of Fine 3-point-loop in Higher Dimensional Space

By using the linear independent fundamental solutions of the linearvariational equation along the heteroclinic loop to establish a suitable local coordinate system insome small tubular neighborhood of the heteroclinic loop, the Poincaré map is constructed to studythe bifurcation problems of a fine 3–point loop in higher dimensional space. Under some transversalconditions and the non–twisted condition, the existence, coexistence and incoexistence of2–point–loop, 1–homoclinic orbit, simple 1–periodic orbit and 2–fold 1–periodic orbit, and thenumber of 1–periodic orbits are studied. Moreover, the bifurcation surfaces and existence regionsare given. Lastly, the above bifurcation results are applied to a planar system and an insidestability criterion is obtained.

Nonlinear dynamics analysis of a new autonomous chaotic system

In this paper,a new nonlinear autonomous system introduced by Chlouverakis and Sprott is studied further,to present very rich and complex nonlinear dynamical behaviors. Some basic dynamical properties are studied either analytically or nu-merically,such as Poincaré map,Lyapunov exponents and Lyapunov dimension. Based on this flow,a new almost-Hamilton chaotic system with very high Lyapunov dimensions is constructed and investigated. Two new nonlinear autonomous systems can be changed into one another by adding or omitting some constant coefficients.

THE STUDY ON THE CHAOTIC MOTION OF A NONLINEAR DYNAMIC SYSTEM

In this paper, the system of the forced vibration -λ 1T+λ 2T 2+λ 3T 3=ε(g cos ωt-ε′) is discussed, which contains square and cubic items. The critical condition that the system enters chaotic states is given by the Melnikov method. By Poincaré map, phase portrait and time_displacement history diagram, whether the chaos occurs is determined.

CHAOTIC MOTION OF AN ELASTIC CIRCULAR PLATE

The chaotic motion of a harmonically forced circular plate is studied in the paper. The virtual displacement principle is used to derive the dynamic equation of motion, with the effect of large deflection of plate taken into account. By means of Garlerkin approach and Melnikov function method, the critical condition for chaotic motion is obtained. A demonstrative example is discussed through the Poincare mapping, phase portrait and time history.