Purpose:To contribute to the study of networks and graphs.Design/methodology/approach:We apply standard mathematical thinking.Findings:We show that the distance distribution in an undirected network Lorenz majorizes t...Purpose:To contribute to the study of networks and graphs.Design/methodology/approach:We apply standard mathematical thinking.Findings:We show that the distance distribution in an undirected network Lorenz majorizes the one of a chain.As a consequence,the average and median distances in any such network are smaller than or equal to those of a chain.Research limitations:We restricted our investigations to undirected,unweighted networks.Practical implications:We are convinced that these results are useful in the study of small worlds and the so-called six degrees of separation property.Originality/value:To the best of our knowledge our research contains new network results,especially those related to frequencies of distances.展开更多
目的探讨Lorenz散点图(LPs)矢量角的价值,及其联合B线斜率在提高心律失常诊断效能方面的作用。方法回顾性分析119例室性期前收缩(室早组)、97例室上性期前收缩(室上早组)、52例二度Ⅰ型房室传导阻滞(二度Ⅰ型组)和54例二度Ⅱ型房室/窦...目的探讨Lorenz散点图(LPs)矢量角的价值,及其联合B线斜率在提高心律失常诊断效能方面的作用。方法回顾性分析119例室性期前收缩(室早组)、97例室上性期前收缩(室上早组)、52例二度Ⅰ型房室传导阻滞(二度Ⅰ型组)和54例二度Ⅱ型房室/窦房传导阻滞(二度Ⅱ型组)患者的LPs,测量B线斜率及矢量角,比较各组间的差异。采用受试者工作特征曲线分析B线斜率、矢量角及两者联合在组间的诊断效能并使用MedCalc软件进行统计学比较。使用组内相关系数(ICC)、Bland-Altman图评估B线斜率、矢量角的观察者内和观察者间测量的一致性。结果室早组与室上早组、二度Ⅰ型组与二度Ⅱ型组间比较差异均有统计学意义(P<0.05)。B线斜率、矢量角以及两者联合鉴别室性与室上性期前收缩的曲线下面积(AUC)分别为0.81、0.84、0.87,鉴别二度Ⅰ型与二度Ⅱ型房室/窦房传导阻滞的AUC分别为0.76、0.78、0.80。矢量角的ICC优于B线斜率(观察者内0.99 vs 0.98、观察者间0.97 vs 0.96)。结论矢量角可用于鉴别心律失常类型,且具有较好的观察者内及观察者间一致性。其联合B线斜率诊断心律失常具有较高准确率,为临床诊疗提供了新的参考依据。展开更多
The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stab...The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stability under different parameter conditions are analyzed by applying Routh-Hurwitz criterion. The results indicate that the system may exist one, three and five equilibrium points for different system parameters. Based on the central manifold theorem and normal form theorem, the pitchfork bifurcation and Hopf bifurcation are studied respectively. By using the Hopf bifurcation theorem and calculating the first Lyapunov coefficient, the Hopf bifurcation of this system is obtained as supercritical for certain parameters. Finally, the results of theoretical parts are verified by some numerical simulations.展开更多
In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a...In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.展开更多
In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstr...In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.展开更多
In this article,we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative(CFFD).The requ...In this article,we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative(CFFD).The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii.Also,we enriched our work by establishing a stable result based on the Ulam-Hyers(U-H)concept.Also,the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method.We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders.The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier.Also,the suggested method results for the proposed system under the mentioned derivative are new.Furthermore,the adopted technique has some useful features,such as the lack of prior discrimination required by wavelet methods.our proposed method does not depend on auxiliary parameters like the homotopy method,which controls the method.Our proposed method is rapidly convergent and,in most cases,it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.展开更多
文摘Purpose:To contribute to the study of networks and graphs.Design/methodology/approach:We apply standard mathematical thinking.Findings:We show that the distance distribution in an undirected network Lorenz majorizes the one of a chain.As a consequence,the average and median distances in any such network are smaller than or equal to those of a chain.Research limitations:We restricted our investigations to undirected,unweighted networks.Practical implications:We are convinced that these results are useful in the study of small worlds and the so-called six degrees of separation property.Originality/value:To the best of our knowledge our research contains new network results,especially those related to frequencies of distances.
文摘目的探讨Lorenz散点图(LPs)矢量角的价值,及其联合B线斜率在提高心律失常诊断效能方面的作用。方法回顾性分析119例室性期前收缩(室早组)、97例室上性期前收缩(室上早组)、52例二度Ⅰ型房室传导阻滞(二度Ⅰ型组)和54例二度Ⅱ型房室/窦房传导阻滞(二度Ⅱ型组)患者的LPs,测量B线斜率及矢量角,比较各组间的差异。采用受试者工作特征曲线分析B线斜率、矢量角及两者联合在组间的诊断效能并使用MedCalc软件进行统计学比较。使用组内相关系数(ICC)、Bland-Altman图评估B线斜率、矢量角的观察者内和观察者间测量的一致性。结果室早组与室上早组、二度Ⅰ型组与二度Ⅱ型组间比较差异均有统计学意义(P<0.05)。B线斜率、矢量角以及两者联合鉴别室性与室上性期前收缩的曲线下面积(AUC)分别为0.81、0.84、0.87,鉴别二度Ⅰ型与二度Ⅱ型房室/窦房传导阻滞的AUC分别为0.76、0.78、0.80。矢量角的ICC优于B线斜率(观察者内0.99 vs 0.98、观察者间0.97 vs 0.96)。结论矢量角可用于鉴别心律失常类型,且具有较好的观察者内及观察者间一致性。其联合B线斜率诊断心律失常具有较高准确率,为临床诊疗提供了新的参考依据。
文摘The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stability under different parameter conditions are analyzed by applying Routh-Hurwitz criterion. The results indicate that the system may exist one, three and five equilibrium points for different system parameters. Based on the central manifold theorem and normal form theorem, the pitchfork bifurcation and Hopf bifurcation are studied respectively. By using the Hopf bifurcation theorem and calculating the first Lyapunov coefficient, the Hopf bifurcation of this system is obtained as supercritical for certain parameters. Finally, the results of theoretical parts are verified by some numerical simulations.
文摘In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.
文摘In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, the presence of the supercritical non-degenerate Hopf bifurcation is demonstrated. The technical tool used consists of the Central Manifold theorem, a well-known formula to calculate the Lyapunov coefficient and Hopf’s Theorem. For particular values of the parameters in the parameter space established in the main result of this work, a graph is presented that describes the evolution of the trajectories, obtained by means of numerical simulation.
基金support of Taif University Researchers Supporting Project No. (TURSP-2020/162),Taif University,Taif,Saudi Arabiafunding this work through research groups program under Grant No.R.G.P.1/195/42.
文摘In this article,we developed sufficient conditions for the existence and uniqueness of an approximate solution to a nonlinear system of Lorenz equations under Caputo-Fabrizio fractional order derivative(CFFD).The required results about the existence and uniqueness of a solution are derived via the fixed point approach due to Banach and Krassnoselskii.Also,we enriched our work by establishing a stable result based on the Ulam-Hyers(U-H)concept.Also,the approximate solution is computed by using a hybrid method due to the Laplace transform and the Adomian decomposition method.We computed a few terms of the required solution through the mentioned method and presented some graphical presentation of the considered problem corresponding to various fractional orders.The results of the existence and uniqueness tests for the Lorenz system under CFFD have not been studied earlier.Also,the suggested method results for the proposed system under the mentioned derivative are new.Furthermore,the adopted technique has some useful features,such as the lack of prior discrimination required by wavelet methods.our proposed method does not depend on auxiliary parameters like the homotopy method,which controls the method.Our proposed method is rapidly convergent and,in most cases,it has been used as a powerful technique to compute approximate solutions for various nonlinear problems.
