In this paper,a simplest fractional-order hyperchaotic(SFOH)system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system,which possesses seven terms without any quadratic or h...In this paper,a simplest fractional-order hyperchaotic(SFOH)system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system,which possesses seven terms without any quadratic or higher-order polynomials.The numerical solution of the SFOH system is investigated based on the Adomian decomposition method(ADM).The methods of segmentation and replacement function are proposed to solve this system and analyze the dynamics.Dynamics of this system are demonstrated by means of phase portraits,bifurcation diagrams,Lyapunov exponent spectrum(LEs)and Poincarésection.The results show that the system has a wide chaotic range with order change,and large Lyapunov exponent when the order is very small,which indicates that the system has a good application prospect.Besides,the parameter a is a partial amplitude controller for the SFOH system.Finally,the system is successfully implemented by digital signal processor(DSP).It lays a foundation for the application of the SFOH system.展开更多
The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling an...The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.展开更多
The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation ar...The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.展开更多
In the present paper the authors prove that all the generalised entropy,solutions of the CJ-model, which for the given Riemann initial data are a oneparameter family u(eta) characterised by weak detonation discontinui...In the present paper the authors prove that all the generalised entropy,solutions of the CJ-model, which for the given Riemann initial data are a oneparameter family u(eta) characterised by weak detonation discontinuity point eta, are just the limits of the admissible solutions u(ek) Of the selfsimilar combustion model. In fact, the authors prove that for any possible eta, there exists a constant B>0 s.t.展开更多
We successfully constructed wide classes of exact solutions for the Burgers equation by using the generalized simplest equation method. This method yielded a Bäcklund transformation between the Burgers equatio...We successfully constructed wide classes of exact solutions for the Burgers equation by using the generalized simplest equation method. This method yielded a Bäcklund transformation between the Burgers equation and a related constraint equation. By dealing with the constraint equation, we obtained the traveling wave solutions and non-traveling wave solutions of the Burgers equation.展开更多
In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov–Kuznetsov(Kd V-ZK) equation and complex coupled Kd V system by using extended simplest equa...In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov–Kuznetsov(Kd V-ZK) equation and complex coupled Kd V system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations(NLEEs) in mathematical physics.展开更多
Teicoplanin (TCP) is a multiple drug-resistant lipoglycopeptide antibiotic produced by fermenting Actinoplanes teichomyceticus. In this study, a mixture of TCP with the Tris-HCl buffer (TCP-Tris- HCl) was used to simu...Teicoplanin (TCP) is a multiple drug-resistant lipoglycopeptide antibiotic produced by fermenting Actinoplanes teichomyceticus. In this study, a mixture of TCP with the Tris-HCl buffer (TCP-Tris- HCl) was used to simulate TCP fermentation broth. The reagent-free, rapid and simultaneous quantitative analysis models for TCP and Tris in the TCP-Tris-HCl mixtures were established by near-infrared (NIR) spectroscopy. The equidistant combination partial least squares (EC-PLS) method and the equivalent model sets were proposed, the simplest equivalent model with the smallest number of wavelengths were further selected. The initial wavelength, number of wavelengths, number of wavelength gaps, number of PLS factors were 1520 nm, 28, 5, 5 for TCP and 1084 nm, 13, 6, 4 for Tris, respectively. Compared with the optimal EC-PLS models, the simplest equivalent models adopted fewer wavelengths. Thus, the redundant wavelengths were removed, the models were further simplified. The root-mean-square errors (SEP) and correlation coefficients (R<sub>P</sub>) for prediction were 0.043 mg·mL<sup>-1</sup> and 0.9998 for TCP, and 0.222 mg·mL<sup>-1</sup> and 0.9989 for Tris, respectively. The results indicate that NIR method can be applied to highly accurate quantitative analysis for TCP and provide valuable references for further application to TCP fermentation broth.展开更多
The total π-bond energy shows a quantum-chemical characteristic of the conjugated molecules. Based on HMO methods, a lot of people try to search for the simple general formulas of an approximated calculation for a wi...The total π-bond energy shows a quantum-chemical characteristic of the conjugated molecules. Based on HMO methods, a lot of people try to search for the simple general formulas of an approximated calculation for a wide use. So far, a large number of calculation methods have been suggested, but not the best. On展开更多
基金supported by the National Natural Science Foundation of China (61161006 and 61573383)supported by the Research and Innovation Project of Graduate Students of Central South University (2018ZZTS348)
文摘In this paper,a simplest fractional-order hyperchaotic(SFOH)system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system,which possesses seven terms without any quadratic or higher-order polynomials.The numerical solution of the SFOH system is investigated based on the Adomian decomposition method(ADM).The methods of segmentation and replacement function are proposed to solve this system and analyze the dynamics.Dynamics of this system are demonstrated by means of phase portraits,bifurcation diagrams,Lyapunov exponent spectrum(LEs)and Poincarésection.The results show that the system has a wide chaotic range with order change,and large Lyapunov exponent when the order is very small,which indicates that the system has a good application prospect.Besides,the parameter a is a partial amplitude controller for the SFOH system.Finally,the system is successfully implemented by digital signal processor(DSP).It lays a foundation for the application of the SFOH system.
基金National Natural Science Foundation of China (No 10372068)
文摘The coefficients of the simplest normal forms of both high-dimensional generalized Hopf and high-dimensional Hopf bifurcation systems were discussed using the adjoint operator method. A particular nonlinear scaling and an inner product were introduced in the space of homogeneous polynomials. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of both the conventional normal forms of Hopf and generalized Hopf bifurcation systems. A symbolic manipulation was designed to perform the calculation of the coefficients of the simplest normal forms using Mathematica. The original ordinary differential equation was required in the input and the simplest normal form could be obtained as the output. Finally, the simplest normal forms of 6-dimensional generalized Hopf singularity of type 2 and 5-dimensional Hopf bifurcation system were discussed by executing the program. The output showed that the 5th- and 9th-order terms remained in 6-dimensional generalized Hopf singularity of type 2 and the 3rd- and 5th-order terms remained in 5-dimensional Hopf bifurcation system.
基金Supported by National Natural Science Foundation of China (No10872141)Doctoral Foundation of Ministry of Education of China (No20060056005)Natural Science Foundation of Tianjin University of Science and Technology (No20070210)
文摘The normal forms of generalized Neimark-Sacker bifurcation are extensively studied using normal form theory of dynamic system. It is well known that if the normal forms of the generalized Neimark-Sacker bifurcation are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal forms. In this paper, five theorems are presented to show that the conventional Neimark-Sacker bifurcation can be further simplified. The simplest normal forms of generalized Neimark-Sacker bifurcation are calculated. Based on the conventional normal form, using appropriate nonlinear transformations, it is found that the generalized Neimark-Sacker bifurcation has at most two nonlinear terms remaining in the amplitude equations of the simplest normal forms up to any order. There are two kinds of simplest normal forms. Their algebraic expression formulas of the simplest normal forms in terms of the coefficients of the generalized Neimark-Sacker bifurcation systems are given.
文摘In the present paper the authors prove that all the generalised entropy,solutions of the CJ-model, which for the given Riemann initial data are a oneparameter family u(eta) characterised by weak detonation discontinuity point eta, are just the limits of the admissible solutions u(ek) Of the selfsimilar combustion model. In fact, the authors prove that for any possible eta, there exists a constant B>0 s.t.
文摘We successfully constructed wide classes of exact solutions for the Burgers equation by using the generalized simplest equation method. This method yielded a Bäcklund transformation between the Burgers equation and a related constraint equation. By dealing with the constraint equation, we obtained the traveling wave solutions and non-traveling wave solutions of the Burgers equation.
文摘In this paper, we investigate the traveling wave solutions for the nonlinear dispersive equation, Korteweg-de Vries Zakharov–Kuznetsov(Kd V-ZK) equation and complex coupled Kd V system by using extended simplest equation method, and then derive the hyperbolic function solutions include soliton solutions, trigonometric function solutions include periodic solutions with special values for double parameters and rational solutions. The properties of such solutions are shown by figures. The results show that this method is an effective and a powerful tool for handling the solutions of nonlinear partial differential equations(NLEEs) in mathematical physics.
文摘Teicoplanin (TCP) is a multiple drug-resistant lipoglycopeptide antibiotic produced by fermenting Actinoplanes teichomyceticus. In this study, a mixture of TCP with the Tris-HCl buffer (TCP-Tris- HCl) was used to simulate TCP fermentation broth. The reagent-free, rapid and simultaneous quantitative analysis models for TCP and Tris in the TCP-Tris-HCl mixtures were established by near-infrared (NIR) spectroscopy. The equidistant combination partial least squares (EC-PLS) method and the equivalent model sets were proposed, the simplest equivalent model with the smallest number of wavelengths were further selected. The initial wavelength, number of wavelengths, number of wavelength gaps, number of PLS factors were 1520 nm, 28, 5, 5 for TCP and 1084 nm, 13, 6, 4 for Tris, respectively. Compared with the optimal EC-PLS models, the simplest equivalent models adopted fewer wavelengths. Thus, the redundant wavelengths were removed, the models were further simplified. The root-mean-square errors (SEP) and correlation coefficients (R<sub>P</sub>) for prediction were 0.043 mg·mL<sup>-1</sup> and 0.9998 for TCP, and 0.222 mg·mL<sup>-1</sup> and 0.9989 for Tris, respectively. The results indicate that NIR method can be applied to highly accurate quantitative analysis for TCP and provide valuable references for further application to TCP fermentation broth.
文摘The total π-bond energy shows a quantum-chemical characteristic of the conjugated molecules. Based on HMO methods, a lot of people try to search for the simple general formulas of an approximated calculation for a wide use. So far, a large number of calculation methods have been suggested, but not the best. On
