In this paper, global dynamics of the Maxwell-Bloch system is discussed. Firstly, the complete description of its dynamic behavior on the sphere at infinity is presented by using the Poincar\'{e} compactification in $R^{3}$. Secondly, the existence of singularly degenerate heteroclinic cycles is investigated. It is proved that for a suitable choice of the parameters, there is an infinite set of singularly degenerate heteroclinic cycles in Maxwell-Bloch system. Specially, it is emerged that for a special parameter value, nearby singularly degenerate heteroclinic cycles Maxwell-Bloch system has the chaotic attractors by combining theoretical and numerical analysis. It is hoped that these theoretical and numerical value results are given a contribution in an understanding of the physical essence for chaos in the Maxwell-Bloch system.
This paper devotes to the qualitative geometric analysis of the traveling wave solutions of MEW-Burgers wave equation. Firstly, MEW-Burgers equation is transformed into an equivalent planar dynamical system by using traveling wave transformation. Then the global structure of the planar system is presented, and solitary waves, kink waves (anti-kink waves), and periodic waves are found. Secondly, Jacobi stability for the planar system is studied based on KCC theory, and Jacobi stability of any point on the trajectory of system is dicussed. The dynamical behavior of the deviation vector near the equilibrium points is analyzed, and the numerical simulation is consistent with the theoretical analysis. Lyapunov stability and Jacobi stability of the equilibrium points are also compared and analyzed. The obtaining results show that Lyapunov stability of the equilibrium points of the system is not exactly consistent with Jacobi stability. Finally, the planar system with periodic disturbance is transformed into a six dimensional nonlinear system, and the periodic, quasi-periodic, and chaotic dynamical behaviors of the system are numerically simulated.
In this paper, chaos and bifurcation are explored for the controlled chaotic system, which is put forward based on the hybrid strategy in an unusual chaotic system. Behavior of the controlled system with variable parameter is researched in detain. Moreover, the normal form theory is used to analyze the direction and stability of bifurcating periodic solution.
With the development of quality education, Ordinary Differential Equation, as a basic course of mathematics major in colleges and universities, has been reformed, innovated and developed. Especially under the support of education informatization, micro class and flipped classroom are widely used in the teaching of Ordinary Differential Equation, which has a positive impact on improving its teaching quality and efficiency. This paper starts with the necessity of the application of flipped classroom teaching mode supported by micro course in the teaching of Ordinary Differential Equation, and explores the specific application strategy research in the course practice, aiming to provide reference for other professional courses teaching reform.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.