Application of new dynamical spectra of orbits in Hamiltonian systems

In the present article, we investigate the properties of motion in Hamiltonian systems of two and three degrees of freedom, using the distribution of the values of two new dynamical parameters. The distribution functions of the new parameters define the S(g) and the S(w) dynamical spectra. The first spectrum definition that is the S(g) spectrum will be applied in a Hamiltonian system of two degrees of freedom (2D), while the S(w) dynamical spectrum will be deployed in a Hamiltonian system of three degrees of freedom (3D). Both Hamiltonian systems, describe a very interesting dynamical system which displays a large variety of resonant orbits, different chaotic components, and also several sticky regions. We test and prove the efficiency and the reliability of these new dynamical spectra, in detecting tiny ordered domains embedded in the chaotic sea, corresponding to complicated resonant orbits of higher multiplicity. The results of our extensive numerical calculations suggest that both dynamical spectra are fast and reliable discriminants between different types of orbits in Hamiltonian systems, while requiring very short computation time in order to provide solid and conclusive evidence regarding the nature of an orbit. Furthermore, we establish numerical criteria in order to quantify the results obtained from our new dynamical spectra. A?comparison to other previously used dynamical indicators, reveals the leading role of the new spectra.     

STUDY ON THE PREDICTION METHOD OF LOW_DIMENSION TIME SERIES THAT ARISE FROM THE INTRINSIC NONLINEAR DYNAMIC

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Study on the Prediction Method of Low-Dimension Time Series that Arise from the Intrinsic Nonlinear Dynamics

The prediction methods and its applications of the nonlinear dynamic systems determined from chaotic time series of low-dimension are discussed mainly. Based on the work of the foreign researchers, the chaotic time series in the phase space adopting one kind of nonlinear chaotic model were reconstructed. At first, the model parameters were estimated by using the improved least square method. Then as the precision was satisfied, the optimization method was used to estimate these parameters. At the end by using the obtained chaotic model, the future data of the chaotic time series in the phase space was predicted. Some representative experimental examples were analyzed to testify the models and the algorithms developed in this paper. The results show that if the algorithms developed here are adopted, the parameters of the corresponding chaotic model will be easily calculated well and true. Predictions of chaotic series in phase space make the traditional methods change from outer iteration to interpolations. And if the optimal model rank is chosen, the prediction precision will increase notably. Long term superior predictability of nonlinear chaotic models is proved to be irrational and unreasonable. Paper from Chen Yu-shu, Member of Editorial of Committee, AMM Foundation item: the National Natural Science Foundation of China (19990510); the National Key Basic Research Special Fund(G1998020316) Biography: Ma Jun-hai(1965-), Professor, Doctor     

Analysis and applied study of dynamic characteristics of chaotic repeller in complicated system

Introduction Inrecentyears,thestudyondynamicbehaviorofnonlinearsystemhasbecomeanactive subjectinnonlinearscience[1-12].Chaosisakindofcomplicatedandirregularbehaviorcseated bynonlinearsystem,suchirregularphenomenonexistsinnatureandsocietywidely.Itiswell_ known,timeserieswithcomplicatedphenomenonandbehaviorincludingchaosexistinvarious complicatedsystemsandinengineeringtechniques,suchsituationsareusuallytreatedeffectively withchaotictheoriesandmethods.Uptonowseveralmaturedstatisticindexesformeasu…     

考虑时滞效应时耦合化学反应系统的动力学行为

在耦合自催化反应系统中,采用数值分析方法研究了考虑时滞效应和流速扰动时子系统的动力学行为.与原系统相比,该系统呈现出更加丰富的动力学现象.反应过程中出现了结构复杂的混沌吸引子和由在周期解邻域内振荡而产生的概周期运动,并且存在混沌由倍周期分岔演变为新的混沌吸引子的过程.这些结果对于解释耦合化学反应系统中的复杂现象、揭示其反应机理具有一定的指导意义.     

Dynamics on certain sets of stochastic matrices

We study iteration of polynomials on symmetric stochastic matrices. In particular, we focus on a certain one-parameter family of quadratic maps which exhibits chaotic behavior for a wide range of the parameters. The well-known dynamical behavior of the quadratic family on the interval, and its dependence on the parameter, is reproduced on the spectrum of the stochastic matrices. For certain subclasses of stochastic matrices the referred dynamical behavior is also obtained in the matrix entries. Since a stochastic matrix characterizes a Markov chain, we obtain a discrete dynamical system on the space of reversible Markov chains. Therefore, depending on the parameter, there are initial conditions for which the corresponding reversible Markov chains will lead under iteration to a fixed point, to a periodic point, or to an aperiodic point. Moreover, there are sensitivity to initial conditions and the coexistence of infinite repulsive periodic orbits, both features of chaos.     

Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits

We study the nature of motion in a 3D potential composed of perturbed elliptic oscillators. Our technique is to use the results obtained from the 2D potential in order to find the initial conditions generating regular or chaotic orbits in the 3D potential. Both 2D and 3D potentials display exact periodic orbits together with extended chaotic regions. Numerical experiments suggest that the degree of chaos increases rapidly as the energy of the test particle increases. About 97?% of the phase plane of the 2D system is covered by chaotic orbits for large energies. The regular or chaotic character of the 2D orbits is checked using the S(c) dynamical spectrum, while for the 3D potential we use the S(c) spectrum, along with the P(f) spectral method. Comparison with other dynamical indicators shows that the S(c) spectrum gives fast and reliable information about the character of motion.     

Chaotic oscillations in a piecewise linear spring-mass system

Chaotic oscillations are useful in assessing the health of a structure. Hence, simple chaotic systems which can easily be realized mechanically or electro-mechanically are highly desired. We study a new piecewise linear spring-mass system. The chaotic behaviour in this system is characterized using bifurcation diagrams and the invariant parameters of the dynamics. We also show that there exists a stochastic analogue of this system, which mimics the dynamical features of its deterministic counterpart. This allows a greater flexibility in practical designs as the chaotic oscillations are obtained either deterministically or stochastically. Also, the oscillations are low dimensional, which reduces the computational resources needed for obtaining the invariant parameters of this system.     

Nonlinear dynamics of a fluid-conveying curved pipe subjected to motion-limiting constraints and a harmonic excitation

This paper investigates the dynamical behaviour of a fluid-conveying curved pipe subjected to motion-limiting constraints and a harmonic excitation. Based on a Newtonian method, the in-plane equation of motion of this curved pipe is derived. Then a set of discrete equations in spatial space obtained by the differential quadrature method (DQM) is solved numerically. Emphasis is placed on the possible dynamical behaviour of the curved pipe conveying fluid. The numerical results show that the pipe without motion-limiting constraints but with a harmonic force behaves as an ordinary linear system. If, however, the pipe is subjected to cubic motion-limiting constraints, nonlinear dynamic phenomena of the system will occur. Calculations of bifurcation diagrams, phase-plane portraits, time responses, power spectrum diagrams, and Poincaré maps of the oscillations clearly demonstrate the existence of chaotic and quasiperiodic motions. Moreover, it is shown that the route to chaos is via a sequence of period-doubling bifurcations.     

Chaotic motions of the L-mode to H-mode transition model in tokamak

The chaotic dynamics of the transport equation for the L-mode to H-mode near the plasma in a tokamak is studied in detail with the Melnikov method. The transport equations represent a system with external and parametric excitation. The critical curves separating the chaotic regions and nonchaotic regions are presented for the system with periodically external excitation and linear parametric excitation, or cubic parametric excitation, respectively. The results obtained here show that there exist uncontrollable regions in which chaos always take place via heteroclinic bifurcation for the system with linear or cubic parametric excitation. Especially, there exists a controllable frequency, excited at which chaos does not occur via homoclinic bifurcation no matter how large the excitation amplitude is for the system with cubic parametric excitation. Some complicated dynamical behaviors are obtained for this class of systems.     

单层悬索体系在风中的随机稳定性研究

本文建立了单层悬索体系在含有较低紊流成分的自然风中的随机稳定方程，应用Ｓｃｈｕｓｓ随机稳定理论分析，并建立可靠性评判准则，实例分析指出，由于微小紊流成分的随机摄动，致系统产生不稳定运动。     

Complexity of chaotic binary sequence and precision of its numerical simulation

Numerical simulation is one of primary methods in which people study the property of chaotic systems. However, there is the effect of finite precision in all processors which can cause chaos to degenerate into a periodic function or a fixed point. If it is neglected the precision of a computer processor for the binary numerical calculations, the numerical simulation results may not be accurate due to the chaotic nature of the system under study. New and more accurate methods must be found. A quantitative computable method of sequence complexity evaluation is introduced in this paper. The effect of finite precision is evaluated from the viewpoint of sequence complexity. The simulation results show that the correlation function based on information entropy can effectively reflect the complexity of pseudorandom sequences generated by a chaotic system, and it is superior to the other measure methods based on entropy. The finite calculation precision of the processor has significant effect on the complexity of chaotic binary sequences generated by the Lorenz equation. The pseudorandom binary sequences with high complexity can be generated by a chaotic system as long as the suitable computational precision and quantification algorithm are selected and behave correctly. The new methodology helps to gain insight into systems that may exist in various application domains such as secure communications and spectrum management.     

First passage of stochastic fractional derivative systems with power-form restoring force

In this paper, the first-passage failure of stochastic dynamical systems with fractional derivative and power-form restoring force subjected to Gaussian white-noise excitation is investigated. With application of the stochastic averaging method of quasi-Hamiltonian system, the system energy process will converge weakly to an Itô differential equation. After that, Backward Kolmogorov (BK) equation associated with conditional reliability function and Generalized Pontryagin (GP) equation associated with statistical moments of first-passage time are constructed and solved. Particularly, the influence on reliability caused by the order of fractional derivative and the power of restoring force are also examined, respectively. Numerical results show that reliability function is decreased with respect to the time. Lower power of restoring force may lead the system to more unstable evolution and lead first passage easy to happen. Besides, more viscous material described by fractional derivative may have higher reliability. Moreover, the analytical results are all in good agreement with Monte-Carlo data.     

A novel pseudo-random number generator from coupled map lattice with time-varying delay

Chaos has been widely combined with cryptography in the field of information security, especially, a considerable amount of studies of generating pseudo-random numbers based on chaotic systems have been proposed in recent decades. However, many of them are easy to be attacked via utilizing the nonlinear prediction method based on phase space reconstruction and other analysis. Furthermore, under the finite precision environment of computer simulation, there does not exist a random sequence which is truly non-periodic. Unfortunately, few researches had made a related analysis on the above two discussions. This paper is devoted to designing a pseudo-random number generator based on coupled map lattice with time-varying delay, analyzing the random properties of the generated pseudo-random numbers and discussing the dynamical degradation of the system under finite precision of computer simulation. The proposed scheme merely depends on the determining equation; thus, the algorithm itself is not complex, which does not impose high demand on computer hardware and its efficiency is excellent. In order to meet the requirements of using the proposed pseudo-random number generator in cryptography and other practical engineering applications, the proposed pseudo-random number generator is subjected to statistical tests utilizing the well-known test suites, such as NIST SP800-22 and TestU01. Moreover, other related properties, such as permutation entropy, invariant distribution, degradation of dynamical characteristics and parameter test, are also investigated. All results illustrate that the new pseudo-random number generator can generate a high percentage of available pseudo-random numbers for scientific computer simulation and practical applications in the field of information security.     

A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid

In this paper, the non-linear dynamics of simply supported pipes conveying pulsating fluid is further investigated, by considering the effect of motion constraints modeled as cubic springs. The partial differential equation, after transformed into a set of ordinary differential equations (ODEs) using the Galerkin method with N=2, is solved by a fourth order Runge-Kutta scheme. Attention is concentrated on the possible motions of the system with a higher mean flow velocity. Phase portraits, bifurcation diagrams and power spectrum diagrams are presented, showing some interesting and sometimes unexpected results. The analytical model is found to exhibit rich and variegated dynamical behaviors that include quasi-periodic and chaotic motions. The route to chaos is shown to be via period-doubling bifurcations. Finally, the cumulative effect of two non-linearities on the dynamics of the system is discussed.     

含三次耦合项的两自由度Duffing系统的共振及混沌行为

研究了一类含三次耦合项的两自由度Duffing系统的动力学行为。首先应用多尺度方法近似求解系统的一阶稳态响应。通过讨论系统的主共振和1∶1内共振,分析了三次耦合项对系统响应的影响。随后研究系统随外加周期力强度变化的分岔过程,发现除了常见的倍周期分岔通向混沌外,还存在一种直接由周期运动进入混沌的突发路径。结合对系统的最大Lyapunov指数,相轨图及Poincar啨映射的分析验证了上述结论。     

Random impacts of a complex damped system

We investigated the random impacts of a complex damped system. Firstly the interested deterministic complex damped system was revisited and the unstable periodic attractors could be found by means of Poincaré map, time evolution and phase plot since the top Lyapunov exponent could not be applied to decide the unstable states of the proposed system. Secondly the stochastic complex damped system was examined and random impacts would be discovered, namely, the initial deterministic system will be stabilized using the stochastic force properly. The top Lyapunov exponent versus the noise intensity will be observed and one can find the change of dynamical behaviors from instability to stability. Also we implemented Poincaré map analysis, time history and phase plot to confirm the obtained results of top Lyapunov exponent, and we can find excellent agreement between these results. Therefore random noise can be applied to control the dynamical behaviors.     

Analysis of a new three-dimensional system with multiple chaotic attractors

In this paper, a new three-dimensional autonomous system with complex dynamical behaviors is reported. This new system has three quadratic nonlinear terms and one constant term. One remarkable feature of the system is that it can generate multiple chaotic and multiple periodic attractors in a wide range of system parameters. The presence of coexisting chaotic and periodic attractors in the system is investigated. Moreover, it is easily found that the new system also can generate four-scroll chaotic attractor. Some basic dynamical behaviors of the system are investigated through theoretical analysis and numerical simulation.     

New class of chaotic systems with equilibrium points like a three-leaved clover

This paper presents a new class of chaotic systems with infinite number of equilibrium points like a three-leaved clover. They signify an exciting class of dynamical systems which represent many major characteristics of regular and chaotic motions. These chaotic systems belong to the general class of chaotic systems with hidden attractors. By using a systematic computer search, three chaotic systems with three-leaved-clover-shaped equilibria were found which are classified into dissipative systems. Dynamics of the chaotic system with the three-leaved-clover-equilibria has been investigated by using phase portraits, bifurcation diagram, Lyapunov exponents, Kaplan–Yorke dimension and Poincaré map. Moreover, an electronic circuit implementation of the theoretical system is designed to check its effectiveness. Random number generator design has been realized with newly developed chaotic systems. The obtained random bit sequences are used for image encryption. Security analysis of image encryption processes has been performed.