An open-plus-closed-loop （OPCL） control problem for the chaotic motion of a 3D rigid pendulum subjected to a constant gravitationM force is studied. The 3D rigid pendulum is assumed to be consist of a rigid body sup...An open-plus-closed-loop （OPCL） control problem for the chaotic motion of a 3D rigid pendulum subjected to a constant gravitationM force is studied. The 3D rigid pendulum is assumed to be consist of a rigid body supported by a fixed and frictionless pivot with three rotational degrees. In order to avoid the singular phenomenon of Euler＇s angular velocity equation, the quaternion kinematic equation is used to describe the motion of the 3D rigid pendulum. An OPCL controller for chaotic motion of a 3D rigid pendulum at equilibrium position is designed. This OPCL controller contains two parts： the open-loop part to construct an ideal trajectory and the closed-loop part to stabilize the 3D rigid pendulum. Simulation results show that the controller is effective and efficient.展开更多
This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an...This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.展开更多
The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical non...The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.展开更多
In this paper a nonlinear model of convective and inertial motion of atmosphere with damping is studied with Melnikov's function method. The results show that in the atmospheric motion under periodic forcing chaos...In this paper a nonlinear model of convective and inertial motion of atmosphere with damping is studied with Melnikov's function method. The results show that in the atmospheric motion under periodic forcing chaos can occur independently whether the atmospheric motion is stable or not; and it is the key to produce chaos that the distribution of relevant physical quantity is nonlinear and periodically drives exists. The formulae calculating the parameter ranger in which chaos occur are given as well.展开更多
Different types of models for describing the motion of a kicked ion were suggested and studied. It is shown that certain kinds of jumping behavior of the exerting electromagnetic field can lead to a type of noninverti...Different types of models for describing the motion of a kicked ion were suggested and studied. It is shown that certain kinds of jumping behavior of the exerting electromagnetic field can lead to a type of noninvertible property, which changes this conservative system into a 'quasi-dissipative' one. The quasi-dissipative behaviors allow the particle to move along a confined chaotic 'quasi-attractor' in many regions of the parameter space. If the exerting electromagnetic field is discontinuous but the system is still invertible, it will take an unbounded chaotic diffusion with similar parameter values. We hope that this discovery could provide a helpful idea for confining the plasma.展开更多
Microcavity photon dynamics in curved space is an emerging interesting area at the crossing point of nanophotonics,chaotic science,and non-Euclidean geometry.We report the sharp difference between the regular and chao...Microcavity photon dynamics in curved space is an emerging interesting area at the crossing point of nanophotonics,chaotic science,and non-Euclidean geometry.We report the sharp difference between the regular and chaotic motions of cavity photons subjected to the varying space curvature.While the island modes of regular motion rise in the phase diagram in the curved space,the chaotic modes show special mechanisms to adapt to the space curvature,including the fast diffusion of ray dynamics,and the localization and hybridization of the Husimi wavepackets among different periodic orbits.These observations are unique effects enabled by the combination of the chaotic trajectory,the wave nature of light,and the non-Euclidean orbital motion,and therefore make the system a versatile optical simulator for chaotic science under quantum mechanics in curved space-time.展开更多
Large deflection theory of thin anisotropic circular plates was used to analyze the bifurcation behavior and chaotic phenomena of a corrugated thin circular plate with combined transverse periodic excitation and an in...Large deflection theory of thin anisotropic circular plates was used to analyze the bifurcation behavior and chaotic phenomena of a corrugated thin circular plate with combined transverse periodic excitation and an in-plane static boundary load. The nonlinear dynamic equation for the corrugated plate was derived by employing Galerkin＇s technique. The critical conditions for occurrence of the homoclinic and subharmonic bifurcations as well as chaos were studied theoretically using the Melnikov function method. The chaotic motion was also simulated numerically using Maple, with the Poincar~ map and phase curve used to evaluate when chaotic motion appears. The results indicate some chaotic motion in the corrugated plate. The method is directly applicable to chaotic analysis of an isotropic circular plate.展开更多
Based on reconstructing the phase space and calculating the largest Lyapunov exponent, an improved method of detecting chaotic motion is presented for rotor-bearing systems. The method is an improvement to the Wolf me...Based on reconstructing the phase space and calculating the largest Lyapunov exponent, an improved method of detecting chaotic motion is presented for rotor-bearing systems. The method is an improvement to the Wolf method and the Rosenstein algorithm. The improved method introduces the correlation integral function method to estimate the embedding dimension and the reconstruction delay simultaneously, and it makes tracks for the evolutions of every pair of the nearest neighbors to improve the utilization of the reconstructed phase space. Numerical calculation and experimental verification show that the improved method can estimate the proper reconstruction parameters and detect chaotic motion of rotor-bearing systems accurately. In addition, the analytical results show that the current approach is robust to variations of the embedding dimension and the reconstruction delay, and it is applicable to small data sets.展开更多
At present, many chaos-based image encryption algorithms have proved to be unsafe, few encryption schemes permute the plain images as three-dimensional（3D） bit matrices, and thus bits cannot move to any position, th...At present, many chaos-based image encryption algorithms have proved to be unsafe, few encryption schemes permute the plain images as three-dimensional（3D） bit matrices, and thus bits cannot move to any position, the movement range of bits are limited, and based on them, in this paper we present a novel image encryption algorithm based on 3D Brownian motion and chaotic systems. The architecture of confusion and diffusion is adopted. Firstly, the plain image is converted into a 3D bit matrix and split into sub blocks. Secondly, block confusion based on 3D Brownian motion（BCB3DBM）is proposed to permute the position of the bits within the sub blocks, and the direction of particle movement is generated by logistic-tent system（LTS）. Furthermore, block confusion based on position sequence group（BCBPSG） is introduced, a four-order memristive chaotic system is utilized to give random chaotic sequences, and the chaotic sequences are sorted and a position sequence group is chosen based on the plain image, then the sub blocks are confused. The proposed confusion strategy can change the positions of the bits and modify their weights, and effectively improve the statistical performance of the algorithm. Finally, a pixel level confusion is employed to enhance the encryption effect. The initial values and parameters of chaotic systems are produced by the SHA 256 hash function of the plain image. Simulation results and security analyses illustrate that our algorithm has excellent encryption performance in terms of security and speed.展开更多
This paper presents the results from a numerical study on the nonlinear dynamic behaviors including bifurcation and chaos of a truss spar platform. In view of the mutual influences between the heave and the pitch mode...This paper presents the results from a numerical study on the nonlinear dynamic behaviors including bifurcation and chaos of a truss spar platform. In view of the mutual influences between the heave and the pitch modes, the coupled heave and pitch motion equations of the spar platform hull were established in the regular waves. In order to analyze the nonlinear motions of the platform, three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs were constructed, the Poincare maps and the power spectrums of the platform response were calculated. It was found that the platform motions are sensitive to wave fre- quency. With changing wave frequency, the platform undergoes complicated nonlinear motions, including 1/2 sub-harmonic motion, quasi-periodic motion and chaotic motion. When the wave frequency approaches the natural frequency of the heave mode of the platform, the platform moves with quasi-periodic motion and chaotic motional temately. For a certain range of wave frequencies, the platform moves with totally chaotic motion. The range of wave frequencies which leads to chaotic motion of the platform increases with increasing wave height. The three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs reveal the nonlinear motions of the spar platform under different wave conditions.展开更多
In this paper, the relationship between external current stimulus and chaotic behaviors of a Hindmarsh–Rose(HR)neuron is considered. In order to find out the range of external current stimulus which will produce chao...In this paper, the relationship between external current stimulus and chaotic behaviors of a Hindmarsh–Rose(HR)neuron is considered. In order to find out the range of external current stimulus which will produce chaotic behaviors of an HR neuron, the Shil’nikov technique is employed. The Cardano formula is taken to obtain the threshold of the chaotic motion, and series solution to a differential equation is utilized to obtain the homoclinic orbit of HR neurons. This analysis establishes mathematically the value of external current input in generating chaotic motion of HR neurons by the Shil’nikov method. The numerical simulations are performed to support the theoretical results.展开更多
In this paper based on [1]we go further into the study of chaotic behaviour of theforced oscillator containing a square nonlinear term by the methods of multiple scalesand numerical simulation. Relation between the c...In this paper based on [1]we go further into the study of chaotic behaviour of theforced oscillator containing a square nonlinear term by the methods of multiple scalesand numerical simulation. Relation between the chaotic domain and principal resonance curve is discussed. By analyzing the stability of principal resonance curve weinfer that chaotic motion would occur near the frequency at which the principalresonance curve has vertical tangent.Results of numerical simulation confirm thisinference.Thus we offer an effective way to seek the chaotic motion of the systems which are hard to he investigated by Melnikoy method.展开更多
基金supported by the National Natural Science Foundation of China(No.11072038)the Municipal Key Programs of Natural Science Foundation of Beijing(No.KZ201110772039)
文摘An open-plus-closed-loop （OPCL） control problem for the chaotic motion of a 3D rigid pendulum subjected to a constant gravitationM force is studied. The 3D rigid pendulum is assumed to be consist of a rigid body supported by a fixed and frictionless pivot with three rotational degrees. In order to avoid the singular phenomenon of Euler＇s angular velocity equation, the quaternion kinematic equation is used to describe the motion of the 3D rigid pendulum. An OPCL controller for chaotic motion of a 3D rigid pendulum at equilibrium position is designed. This OPCL controller contains two parts： the open-loop part to construct an ideal trajectory and the closed-loop part to stabilize the 3D rigid pendulum. Simulation results show that the controller is effective and efficient.
文摘This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.
基金Project supported by the Natural Science Foundation of Gansu Province of China (No.ZS021-A25-007-Z)
文摘The nonlinear dynamical equations of axle symmetry are established by the method of quasi-shells for three-dimensional shallow conical single-layer lattice shells. The compatible equations are given in geometrical nonlinear range. A nonlinear differential equation containing the second and the third order nonlinear items is derived under the boundary conditions of fixed and clamped edges by the method of Galerkin. The problem of bifurcation is discussed by solving the Floquet exponent. In order to study chaotic motion, the equations of free oscillation of a kind of nonlinear dynamics system are solved. Then an exact solution to nonlinear free oscillation of the shallow conical single-layer lattice shell is found as well. The critical conditions of chaotic motion are obtained by solving Melnikov functions, some phase planes are drawn by using digital simulation proving the existence of chaotic motion.
文摘In this paper a nonlinear model of convective and inertial motion of atmosphere with damping is studied with Melnikov's function method. The results show that in the atmospheric motion under periodic forcing chaos can occur independently whether the atmospheric motion is stable or not; and it is the key to produce chaos that the distribution of relevant physical quantity is nonlinear and periodically drives exists. The formulae calculating the parameter ranger in which chaos occur are given as well.
基金the National Natural Foundation of China under Grant No. 19975039.
文摘Different types of models for describing the motion of a kicked ion were suggested and studied. It is shown that certain kinds of jumping behavior of the exerting electromagnetic field can lead to a type of noninvertible property, which changes this conservative system into a 'quasi-dissipative' one. The quasi-dissipative behaviors allow the particle to move along a confined chaotic 'quasi-attractor' in many regions of the parameter space. If the exerting electromagnetic field is discontinuous but the system is still invertible, it will take an unbounded chaotic diffusion with similar parameter values. We hope that this discovery could provide a helpful idea for confining the plasma.
基金supported by the National Key R&D Program of China(Grant No.2023YFA1407100)the National Natural Science Foundation of China(Grant Nos.12074303,and 11804267)+2 种基金the Shaanxi Key Science and Technology Innovation Team Project(Grant No.2021TD-56)the Sichuan Science and Technology Program(Grant No.2022NSFSC1811)the Xiaomi Young Scholar Program。
文摘Microcavity photon dynamics in curved space is an emerging interesting area at the crossing point of nanophotonics,chaotic science,and non-Euclidean geometry.We report the sharp difference between the regular and chaotic motions of cavity photons subjected to the varying space curvature.While the island modes of regular motion rise in the phase diagram in the curved space,the chaotic modes show special mechanisms to adapt to the space curvature,including the fast diffusion of ray dynamics,and the localization and hybridization of the Husimi wavepackets among different periodic orbits.These observations are unique effects enabled by the combination of the chaotic trajectory,the wave nature of light,and the non-Euclidean orbital motion,and therefore make the system a versatile optical simulator for chaotic science under quantum mechanics in curved space-time.
文摘Large deflection theory of thin anisotropic circular plates was used to analyze the bifurcation behavior and chaotic phenomena of a corrugated thin circular plate with combined transverse periodic excitation and an in-plane static boundary load. The nonlinear dynamic equation for the corrugated plate was derived by employing Galerkin＇s technique. The critical conditions for occurrence of the homoclinic and subharmonic bifurcations as well as chaos were studied theoretically using the Melnikov function method. The chaotic motion was also simulated numerically using Maple, with the Poincar~ map and phase curve used to evaluate when chaotic motion appears. The results indicate some chaotic motion in the corrugated plate. The method is directly applicable to chaotic analysis of an isotropic circular plate.
基金the National Basic Research Program (973) of China (No. 2009CB724302)
文摘Based on reconstructing the phase space and calculating the largest Lyapunov exponent, an improved method of detecting chaotic motion is presented for rotor-bearing systems. The method is an improvement to the Wolf method and the Rosenstein algorithm. The improved method introduces the correlation integral function method to estimate the embedding dimension and the reconstruction delay simultaneously, and it makes tracks for the evolutions of every pair of the nearest neighbors to improve the utilization of the reconstructed phase space. Numerical calculation and experimental verification show that the improved method can estimate the proper reconstruction parameters and detect chaotic motion of rotor-bearing systems accurately. In addition, the analytical results show that the current approach is robust to variations of the embedding dimension and the reconstruction delay, and it is applicable to small data sets.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.41571417 and 61305042)the National Science Foundation of the United States(Grant Nos.CNS-1253424 and ECCS-1202225)+4 种基金the Science and Technology Foundation of Henan Province,China(Grant No.152102210048)the Foundation and Frontier Project of Henan Province,China(Grant No.162300410196)China Postdoctoral Science Foundation(Grant No.2016M602235)the Natural Science Foundation of Educational Committee of Henan Province,China(Grant No.14A413015)the Research Foundation of Henan University,China(Grant No.xxjc20140006)
文摘At present, many chaos-based image encryption algorithms have proved to be unsafe, few encryption schemes permute the plain images as three-dimensional（3D） bit matrices, and thus bits cannot move to any position, the movement range of bits are limited, and based on them, in this paper we present a novel image encryption algorithm based on 3D Brownian motion and chaotic systems. The architecture of confusion and diffusion is adopted. Firstly, the plain image is converted into a 3D bit matrix and split into sub blocks. Secondly, block confusion based on 3D Brownian motion（BCB3DBM）is proposed to permute the position of the bits within the sub blocks, and the direction of particle movement is generated by logistic-tent system（LTS）. Furthermore, block confusion based on position sequence group（BCBPSG） is introduced, a four-order memristive chaotic system is utilized to give random chaotic sequences, and the chaotic sequences are sorted and a position sequence group is chosen based on the plain image, then the sub blocks are confused. The proposed confusion strategy can change the positions of the bits and modify their weights, and effectively improve the statistical performance of the algorithm. Finally, a pixel level confusion is employed to enhance the encryption effect. The initial values and parameters of chaotic systems are produced by the SHA 256 hash function of the plain image. Simulation results and security analyses illustrate that our algorithm has excellent encryption performance in terms of security and speed.
基金supported by the National Natural Science Foundation of China under Grant No.51179125the Innovation Foundation of Tianjin University under Approving No.1301
文摘This paper presents the results from a numerical study on the nonlinear dynamic behaviors including bifurcation and chaos of a truss spar platform. In view of the mutual influences between the heave and the pitch modes, the coupled heave and pitch motion equations of the spar platform hull were established in the regular waves. In order to analyze the nonlinear motions of the platform, three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs were constructed, the Poincare maps and the power spectrums of the platform response were calculated. It was found that the platform motions are sensitive to wave fre- quency. With changing wave frequency, the platform undergoes complicated nonlinear motions, including 1/2 sub-harmonic motion, quasi-periodic motion and chaotic motion. When the wave frequency approaches the natural frequency of the heave mode of the platform, the platform moves with quasi-periodic motion and chaotic motional temately. For a certain range of wave frequencies, the platform moves with totally chaotic motion. The range of wave frequencies which leads to chaotic motion of the platform increases with increasing wave height. The three-dimensional maximum Lyapunov exponent graphs and the bifurcation graphs reveal the nonlinear motions of the spar platform under different wave conditions.
基金supported by the Beijing Natural Science Foundation,China(Grant No.4132005)the National Natural Science Foundation of China(Grant No.61403006)+1 种基金the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions,China(Grant No.YETP1449)the Project of Scientific and Technological Innovation Platform,China(Grant No.PXM2015 014213 000063)
文摘In this paper, the relationship between external current stimulus and chaotic behaviors of a Hindmarsh–Rose(HR)neuron is considered. In order to find out the range of external current stimulus which will produce chaotic behaviors of an HR neuron, the Shil’nikov technique is employed. The Cardano formula is taken to obtain the threshold of the chaotic motion, and series solution to a differential equation is utilized to obtain the homoclinic orbit of HR neurons. This analysis establishes mathematically the value of external current input in generating chaotic motion of HR neurons by the Shil’nikov method. The numerical simulations are performed to support the theoretical results.
文摘In this paper based on [1]we go further into the study of chaotic behaviour of theforced oscillator containing a square nonlinear term by the methods of multiple scalesand numerical simulation. Relation between the chaotic domain and principal resonance curve is discussed. By analyzing the stability of principal resonance curve weinfer that chaotic motion would occur near the frequency at which the principalresonance curve has vertical tangent.Results of numerical simulation confirm thisinference.Thus we offer an effective way to seek the chaotic motion of the systems which are hard to he investigated by Melnikoy method.
