In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured ...In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation （ODE） systems characterized by the nonexistence of homoclinic and heteroclinic orbits.展开更多
This paper applied the theory and method of non linear dynamic to study the integrated environ economic system( EES ). The results of the numerical computational experiment and theoretical inductions showed that the...This paper applied the theory and method of non linear dynamic to study the integrated environ economic system( EES ). The results of the numerical computational experiment and theoretical inductions showed that the system behaviour pattern of the EES will be changed with the variation of the force power level. When the force DP become higher, the system loss its stability gradually, until the chaos occurs. Based on these results, this paper presented an explanation for the long wave economic fluctuation, and proposed in order to guarantee the sustainable development of the specific EES, the DP value of the system should be limited within a reasonable range.展开更多
An analytical method for predicting chaos in perturbed planar non Hamiltonian integrable systems with slowly varying parameters was developed. Based on the analysis of the geometric structure of unperturbed systems, ...An analytical method for predicting chaos in perturbed planar non Hamiltonian integrable systems with slowly varying parameters was developed. Based on the analysis of the geometric structure of unperturbed systems, the condition of transversely homoclinic intersection was given. The generalized Melnikov function of the perturbed system was found by applying the theorem on the differentiability of ordinary differential equation solutions with respect to parameters.展开更多
A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence...A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.展开更多
Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Sh...Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Shilnikov chaos in continuous-time systems.Applied Mathematics and Mechanics(English Edition),33(3),1-4(2012)).It makes the main conclusion of the paper incorrect,that is to say,the non-existence of Shilnikov chaos in the continuous-time systems considered cannot be ensured.Furthermore,a counter-example shows that Theorem 1 in the paper is incorrect.展开更多
Based on a generalized chaos synchronization system and a discrete Sinai map, a non-symmetric true color(RGB) digital image secur e communication scheme is proposed. The scheme first changes an ordinary RGB digital im...Based on a generalized chaos synchronization system and a discrete Sinai map, a non-symmetric true color(RGB) digital image secur e communication scheme is proposed. The scheme first changes an ordinary RGB digital image with 8 bits into unrecognizable disorder codes and then transforms the disorder codes into an RGB digital image with 16 bits for transmitting. A receiver uses a non-symmetric key to verify the authentication of the received data origin,and decrypts the ciphertext. The scheme can encrypt and decrypt most formatted digital RGB images recognized by computers, and recover the plaintext almost without any errors. The scheme is suitable to be applied in network image communications. The analysis of the key space, sensitivity of key parameters, and correlation of encrypted images imply that this scheme has sound security.展开更多
In this paper, chaos synchronization in the presence of parameter uncertainty, observer gain perturbation and exogenous input disturbance is considered. A nonlinear non-fragile proportional-integral （PI） adaptive ob...In this paper, chaos synchronization in the presence of parameter uncertainty, observer gain perturbation and exogenous input disturbance is considered. A nonlinear non-fragile proportional-integral （PI） adaptive observer is designed for the synchronization of chaotic systems; its stability conditions based on the Lyapunov technique are derived. The observer proportional and integral gains, by converting the conditions into linear matrix inequality （LMI）, are optimally selected from solutions that satisfy the observer stability conditions such that the effect of disturbance on the synchronization error becomes minimized. To show the effectiveness of the proposed method, simulation results for the synchronization of a Lorenz chaotic system with unknown parameters in the presence of an exogenous input disturbance and abrupt gain perturbation are reported.展开更多
In this paper, the dynamics of chaos and the entanglement in triatomic molecular vibrations are investigated. On the classical aspect, we study the chaotic trajectories in the phase space. We employ the linear entropy...In this paper, the dynamics of chaos and the entanglement in triatomic molecular vibrations are investigated. On the classical aspect, we study the chaotic trajectories in the phase space. We employ the linear entropy to examine the dynamical entanglement of the two bonds on the quantum aspect. The correspondence between the classical chaos and the quantum dynamical entanglement is also investigated. As an example, we apply our algebraic model to molecule H2O.展开更多
The dynamic characteristic in a spatially distributed nonlinear system, a subset of lasers in an array of coupled lasers, has been studied and analysed numerically. The evolution, with the increasing coupling strength...The dynamic characteristic in a spatially distributed nonlinear system, a subset of lasers in an array of coupled lasers, has been studied and analysed numerically. The evolution, with the increasing coupling strength,from stable quiescent state to chaotic state, to hyper-chaotic state and, back to quasi-steady state has been observed in this system.展开更多
Due to the interactions among coupled spatio-temporal subsystems and the constant bias term of affine chaos, it is difficult to achieve tracking control for the affine coupled spatiotemporal chaos. However, every subs...Due to the interactions among coupled spatio-temporal subsystems and the constant bias term of affine chaos, it is difficult to achieve tracking control for the affine coupled spatiotemporal chaos. However, every subsystem of the affine coupled spatio-temporal chaos can be approximated by a set of fuzzy models; every fuzzy model represents a linearized model of the subsystem corresponding to the operating point of the controlled system. Because the consequent parts of the fuzzy models have a constant bias term, it is very difficult to achieve tracking control for the affine system. Based on these fuzzy models, considering the affine constant bias term, an H∞ fuzzy tracking control scheme is proposed. A linear matrix inequality is employed to represent the feedback controller, and parameters of the controller are achieved by convex optimization techniques. The tracking control for the affine coupled spatio-temporal chaos is achieved, and the stability of the system is also guaranteed. The tracking performances are testified by simulation examples.展开更多
The centrifugal flywheel governor （CFG） is a mechanical device that automatically controls the speed of an engine and avoids the damage caused by sudden change of load torque. It has been shown that this system exhi...The centrifugal flywheel governor （CFG） is a mechanical device that automatically controls the speed of an engine and avoids the damage caused by sudden change of load torque. It has been shown that this system exhibits very rich and complex dynamics such as chaos. This paper investigates the problem of robust finite-time synchronization of non-autonomous chaotic CFGs. The effects of unknown parameters, model uncertainties and external disturbances are fully taken into account. First, it is assumed that the parameters of both master and slave CFGs have the same value and a suitable adaptive finite-time controller is designed. Second, two CFGs are synchronized with the parameters of different values via a robust adaptive finite-time control approach. Finally, some numerical simulations are used to demonstrate the effectiveness and robustness of the proposed finite-time controllers.展开更多
This paper employs a multi-parameter multi-step chaos control method, which is built up on the OGY method, to stabilize desirable UPOs of a gear system with elastomeric web as a high-dimensional and non-hyperbolic cha...This paper employs a multi-parameter multi-step chaos control method, which is built up on the OGY method, to stabilize desirable UPOs of a gear system with elastomeric web as a high-dimensional and non-hyperbolic chaotic system, and the analyses are carried out. Three types of relations between components of a certain control parameter combination are defined in a certain control process. Special emphasis is put on the comparison of control efficiencies of the multi-parameter multi-step method and single-parameter multi-step method. The numerical experiments show the ability to switch between different orbits and the method can be a good chaos control alternative since it provides a more effective UPOs stabilization of high-dimensional and non-hyperbolic chaotic systems than the single-parameter chaos control, and according to the relation between components of each parameter combination, the best combination for chaos control in a certain UPO stabilization process are obtained.展开更多
This paper proposes a new search strategy using mutative scale chaos optimization algorithm （MSCO） for model selection of support vector machine （SVM）. It searches the parameter space of SVM with a very high effic...This paper proposes a new search strategy using mutative scale chaos optimization algorithm （MSCO） for model selection of support vector machine （SVM）. It searches the parameter space of SVM with a very high efficiency and finds the optimum parameter setting for a practical classification problem with very low time cost. To demonstrate the performance of the proposed method it is applied to model selection of SVM in ultrasonic flaw classification and compared with grid search for model selection. Experimental results show that MSCO is a very powerful tool for model selection of SVM, and outperforms grid search in search speed and precision in ultrasonic flaw classification.展开更多
The stable operation of first and second order Zero Crossing Digital Phase Locked Loop (ZCDPLL) is extended by using a Fixed Point Iteration (FPI) method with relaxation. The non-linear components of ZCDPLL such as sa...The stable operation of first and second order Zero Crossing Digital Phase Locked Loop (ZCDPLL) is extended by using a Fixed Point Iteration (FPI) method with relaxation. The non-linear components of ZCDPLL such as sampler phase detector and Digital Controlled Oscillator (DCO) lead to unstable and chaotic operation when the filter gains are high. FPI will be used to stabilize the chaotic operation and consequently extend the lock range of the loop. The proposed stabilized loop can work in higher filter gains which are needed for faster signal acquisition.展开更多
文摘In this paper, a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional, continuous-time, and smooth systems is obtained, Based on this result and an elementary example, it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation （ODE） systems characterized by the nonexistence of homoclinic and heteroclinic orbits.
文摘This paper applied the theory and method of non linear dynamic to study the integrated environ economic system( EES ). The results of the numerical computational experiment and theoretical inductions showed that the system behaviour pattern of the EES will be changed with the variation of the force power level. When the force DP become higher, the system loss its stability gradually, until the chaos occurs. Based on these results, this paper presented an explanation for the long wave economic fluctuation, and proposed in order to guarantee the sustainable development of the specific EES, the DP value of the system should be limited within a reasonable range.
文摘An analytical method for predicting chaos in perturbed planar non Hamiltonian integrable systems with slowly varying parameters was developed. Based on the analysis of the geometric structure of unperturbed systems, the condition of transversely homoclinic intersection was given. The generalized Melnikov function of the perturbed system was found by applying the theorem on the differentiability of ordinary differential equation solutions with respect to parameters.
基金supported by the Ministerio de Educacion y Ciencia,Plan Nacional I+D+I co-financed with FEDER Funds(No.MTM2010-20907-C02)the Consejeria de Educacion y Ciencia de la Juntade Andalucia(Nos.FQM-276,TIC-0130,and P08-FQM-03770)
文摘A paper, ＂Non-existence of Shilnikov chaos in continuous-time systems＂ was published in the journal Applied Mathematics and Mechanics （English Edition）. The authors gave sufficient conditions for the non-existence of homoclinic and heteroclinic orbits in an nth-order autonomous system. Unfortunately, we show in this comment that the proof presented is erroneous and the result is invalid. We also provide two counterexamples of the wrong criterion stated by the authors.
基金Project supported by the National Natural Science Foundation of China(No.11102156)the Northwestern Polytechnical University Foundation for Fundamental Research
文摘Comments on 'Non-existence of Shilnikov chaos in continuous-time systems' are given.An error has been found in the proof of Theorem 1 in the paper by Elhadj and Sprott(Elhadj,Z.and Sprott,J.Non-existence of Shilnikov chaos in continuous-time systems.Applied Mathematics and Mechanics(English Edition),33(3),1-4(2012)).It makes the main conclusion of the paper incorrect,that is to say,the non-existence of Shilnikov chaos in the continuous-time systems considered cannot be ensured.Furthermore,a counter-example shows that Theorem 1 in the paper is incorrect.
基金the National Natural Science Foundation of China under，the Foundation for University Key Teachers，高等学校博士学科点专项科研项目，教育部科学技术研究项目
文摘Based on a generalized chaos synchronization system and a discrete Sinai map, a non-symmetric true color(RGB) digital image secur e communication scheme is proposed. The scheme first changes an ordinary RGB digital image with 8 bits into unrecognizable disorder codes and then transforms the disorder codes into an RGB digital image with 16 bits for transmitting. A receiver uses a non-symmetric key to verify the authentication of the received data origin,and decrypts the ciphertext. The scheme can encrypt and decrypt most formatted digital RGB images recognized by computers, and recover the plaintext almost without any errors. The scheme is suitable to be applied in network image communications. The analysis of the key space, sensitivity of key parameters, and correlation of encrypted images imply that this scheme has sound security.
文摘In this paper, chaos synchronization in the presence of parameter uncertainty, observer gain perturbation and exogenous input disturbance is considered. A nonlinear non-fragile proportional-integral （PI） adaptive observer is designed for the synchronization of chaotic systems; its stability conditions based on the Lyapunov technique are derived. The observer proportional and integral gains, by converting the conditions into linear matrix inequality （LMI）, are optimally selected from solutions that satisfy the observer stability conditions such that the effect of disturbance on the synchronization error becomes minimized. To show the effectiveness of the proposed method, simulation results for the synchronization of a Lorenz chaotic system with unknown parameters in the presence of an exogenous input disturbance and abrupt gain perturbation are reported.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 91021009 and 10874102)the National Basic Research Program of China (Grant No. 2009CB929404)
文摘In this paper, the dynamics of chaos and the entanglement in triatomic molecular vibrations are investigated. On the classical aspect, we study the chaotic trajectories in the phase space. We employ the linear entropy to examine the dynamical entanglement of the two bonds on the quantum aspect. The correspondence between the classical chaos and the quantum dynamical entanglement is also investigated. As an example, we apply our algebraic model to molecule H2O.
文摘The dynamic characteristic in a spatially distributed nonlinear system, a subset of lasers in an array of coupled lasers, has been studied and analysed numerically. The evolution, with the increasing coupling strength,from stable quiescent state to chaotic state, to hyper-chaotic state and, back to quasi-steady state has been observed in this system.
文摘Due to the interactions among coupled spatio-temporal subsystems and the constant bias term of affine chaos, it is difficult to achieve tracking control for the affine coupled spatiotemporal chaos. However, every subsystem of the affine coupled spatio-temporal chaos can be approximated by a set of fuzzy models; every fuzzy model represents a linearized model of the subsystem corresponding to the operating point of the controlled system. Because the consequent parts of the fuzzy models have a constant bias term, it is very difficult to achieve tracking control for the affine system. Based on these fuzzy models, considering the affine constant bias term, an H∞ fuzzy tracking control scheme is proposed. A linear matrix inequality is employed to represent the feedback controller, and parameters of the controller are achieved by convex optimization techniques. The tracking control for the affine coupled spatio-temporal chaos is achieved, and the stability of the system is also guaranteed. The tracking performances are testified by simulation examples.
文摘The centrifugal flywheel governor （CFG） is a mechanical device that automatically controls the speed of an engine and avoids the damage caused by sudden change of load torque. It has been shown that this system exhibits very rich and complex dynamics such as chaos. This paper investigates the problem of robust finite-time synchronization of non-autonomous chaotic CFGs. The effects of unknown parameters, model uncertainties and external disturbances are fully taken into account. First, it is assumed that the parameters of both master and slave CFGs have the same value and a suitable adaptive finite-time controller is designed. Second, two CFGs are synchronized with the parameters of different values via a robust adaptive finite-time control approach. Finally, some numerical simulations are used to demonstrate the effectiveness and robustness of the proposed finite-time controllers.
基金Sponsored by the National High Technology Research and Development Program of China(Grant No.2009AA04Z404)
文摘This paper employs a multi-parameter multi-step chaos control method, which is built up on the OGY method, to stabilize desirable UPOs of a gear system with elastomeric web as a high-dimensional and non-hyperbolic chaotic system, and the analyses are carried out. Three types of relations between components of a certain control parameter combination are defined in a certain control process. Special emphasis is put on the comparison of control efficiencies of the multi-parameter multi-step method and single-parameter multi-step method. The numerical experiments show the ability to switch between different orbits and the method can be a good chaos control alternative since it provides a more effective UPOs stabilization of high-dimensional and non-hyperbolic chaotic systems than the single-parameter chaos control, and according to the relation between components of each parameter combination, the best combination for chaos control in a certain UPO stabilization process are obtained.
基金Project supported by National High-Technology Research and De-velopment Program of China (Grant No .863-2001AA602021)
文摘This paper proposes a new search strategy using mutative scale chaos optimization algorithm （MSCO） for model selection of support vector machine （SVM）. It searches the parameter space of SVM with a very high efficiency and finds the optimum parameter setting for a practical classification problem with very low time cost. To demonstrate the performance of the proposed method it is applied to model selection of SVM in ultrasonic flaw classification and compared with grid search for model selection. Experimental results show that MSCO is a very powerful tool for model selection of SVM, and outperforms grid search in search speed and precision in ultrasonic flaw classification.
文摘The stable operation of first and second order Zero Crossing Digital Phase Locked Loop (ZCDPLL) is extended by using a Fixed Point Iteration (FPI) method with relaxation. The non-linear components of ZCDPLL such as sampler phase detector and Digital Controlled Oscillator (DCO) lead to unstable and chaotic operation when the filter gains are high. FPI will be used to stabilize the chaotic operation and consequently extend the lock range of the loop. The proposed stabilized loop can work in higher filter gains which are needed for faster signal acquisition.
