A method of identifying the existence of horseshoe for a two-dimension diffeomorphism is introduced and utilized to generalize the Birkhoff-Smale Theorem to the saddle-node case.
The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations. It characterizes the nonisotropic cha...The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations. It characterizes the nonisotropic chaotic vibration by means of the total variation theory. Some results are derived on the exponential growth of total variation of the snapshots on the spatial interval in the long-time horizon when the map and the initial condition satisfy some conditions.展开更多
In this paper,the Lauwerier map F a,b (x,y)=(bx(1-2y)+y,ay(1-y)) is considered for a=4 . This map possesses a nontrivial topologically transitive attractor Λ which is the closure of the unstable set of...In this paper,the Lauwerier map F a,b (x,y)=(bx(1-2y)+y,ay(1-y)) is considered for a=4 . This map possesses a nontrivial topologically transitive attractor Λ which is the closure of the unstable set of some hyperbolic fixed point. Periodic points are dense in Λ and all of them are hyperbolic with eigenvalues uniformly bounded away from 1 in norm. Moreover,any two periodic points are heteroclinically related (transversal intersection of their stable and unstable sets). The Sinai Bowen Ruelle measure supported on the attractor is constructed and its properties are studied.展开更多
In this paper,we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space,such that for each point of the set the corresponding diffeomorphism possesses strange attractor...In this paper,we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space,such that for each point of the set the corresponding diffeomorphism possesses strange attractor.For some families of one-dimensional mapping satisfying certain transversality condition,we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor.Furthermore,we study the dynamics of this type of strange attractor.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No. 19531070).
文摘A method of identifying the existence of horseshoe for a two-dimension diffeomorphism is introduced and utilized to generalize the Birkhoff-Smale Theorem to the saddle-node case.
基金It was supported in part by the National Natural Foundation of China (No. 10371136) and the Guangdong Natural Science Foundation of Guangdong Province (No.021765,031603)
文摘The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations. It characterizes the nonisotropic chaotic vibration by means of the total variation theory. Some results are derived on the exponential growth of total variation of the snapshots on the spatial interval in the long-time horizon when the map and the initial condition satisfy some conditions.
基金Supported by the Special Funds for Major State Basic Research Projects and NSF(1 0 0 71 0 55)
文摘In this paper,the Lauwerier map F a,b (x,y)=(bx(1-2y)+y,ay(1-y)) is considered for a=4 . This map possesses a nontrivial topologically transitive attractor Λ which is the closure of the unstable set of some hyperbolic fixed point. Periodic points are dense in Λ and all of them are hyperbolic with eigenvalues uniformly bounded away from 1 in norm. Moreover,any two periodic points are heteroclinically related (transversal intersection of their stable and unstable sets). The Sinai Bowen Ruelle measure supported on the attractor is constructed and its properties are studied.
基金Project Supported by Fund of National Science of China
文摘In this paper,we consider the families of nearby singular diffeomorphism and the measure of a set in the parameter space,such that for each point of the set the corresponding diffeomorphism possesses strange attractor.For some families of one-dimensional mapping satisfying certain transversality condition,we prove that there is a positive measure set in the parameter space, such that the system in the corresponding families of nearly singular diffeomorphism has strange attractor.Furthermore,we study the dynamics of this type of strange attractor.
