The computation of stable or unstable manifold of two-dimensional is developed, which is an efficient method in studying stable structure analysis of system character geometrically. The Lorentz stable manifold is comp...The computation of stable or unstable manifold of two-dimensional is developed, which is an efficient method in studying stable structure analysis of system character geometrically. The Lorentz stable manifold is computed by the fixed arclength method and the hyperbolic equilibrium is a saddle. The two-dimensional stable structure of Lorentz manifold is significant in people’s usual view. We also introduce the V-function to compute the V-manifold correspondingly. The defined V-function is smooth in the unstable direction of the manifold. Especially, the routh to period-doubling attractor on manifold surface is discussed too.展开更多
OGY method is the most important method of controlling chaos. It stabilizes a hyperbolic periodic orbit by making small perturbations for a system parameter. This paper improves the method of choosing parameter, and g...OGY method is the most important method of controlling chaos. It stabilizes a hyperbolic periodic orbit by making small perturbations for a system parameter. This paper improves the method of choosing parameter, and gives a mathematics proof of it.展开更多
Let ■ be the linear space of all C<sup>1</sup> vector fields X on a compact n-dimensionalC<sup>∞</sup> Riemann manifold（n≥2）,endowed with the C<sup>1</sup> norm ‖X‖<sub>...Let ■ be the linear space of all C<sup>1</sup> vector fields X on a compact n-dimensionalC<sup>∞</sup> Riemann manifold（n≥2）,endowed with the C<sup>1</sup> norm ‖X‖<sub>1</sub>.Write θ（X）for the numberof contractible periodic orbits of X∈（?）,which may be finite or infinite.Let （?）<sup>*</sup> be the set ofall X∈（?） possessing the property that X has a neighbourhood （?） such that every Y∈（?） hasonly a finite number of singularities and at most a countable number of periodic orbits.Inthis paper,it is shown that any given S∈（?） has a neighbourhood （?） in （?） together with anumber λ=λ（?）】0 such that θ（X）≤λfor all X∈（?）.展开更多
In this paper, Henon map is considered. For a positive measure set of parameters (a, b), we construct a trapping region G of topologically transitive strange attractor Aa,b for Ta,b, and prove that Aa,b= ∩n≥0Ta,bnG,...In this paper, Henon map is considered. For a positive measure set of parameters (a, b), we construct a trapping region G of topologically transitive strange attractor Aa,b for Ta,b, and prove that Aa,b= ∩n≥0Ta,bnG, and the basin B(Aa,b) of Aa,b is exactly the union of domain whose boundary is contained in w5(p) ∪wu(p) and ws(p). Therefore, that the conjecture posed by Benedicks and Carleson about the basin of strange attactor is true is proved. Furthermore, B(Aa,b) is simply connected and path-connected, w4(p2) is contained in the attainable boundary set of B(Aa,b) (where p2 is another hyperbolic fixed point of Ta,b).展开更多
文摘The computation of stable or unstable manifold of two-dimensional is developed, which is an efficient method in studying stable structure analysis of system character geometrically. The Lorentz stable manifold is computed by the fixed arclength method and the hyperbolic equilibrium is a saddle. The two-dimensional stable structure of Lorentz manifold is significant in people’s usual view. We also introduce the V-function to compute the V-manifold correspondingly. The defined V-function is smooth in the unstable direction of the manifold. Especially, the routh to period-doubling attractor on manifold surface is discussed too.
文摘OGY method is the most important method of controlling chaos. It stabilizes a hyperbolic periodic orbit by making small perturbations for a system parameter. This paper improves the method of choosing parameter, and gives a mathematics proof of it.
基金Supported by National Natural Science Foundation of China
文摘Let ■ be the linear space of all C<sup>1</sup> vector fields X on a compact n-dimensionalC<sup>∞</sup> Riemann manifold（n≥2）,endowed with the C<sup>1</sup> norm ‖X‖<sub>1</sub>.Write θ（X）for the numberof contractible periodic orbits of X∈（?）,which may be finite or infinite.Let （?）<sup>*</sup> be the set ofall X∈（?） possessing the property that X has a neighbourhood （?） such that every Y∈（?） hasonly a finite number of singularities and at most a countable number of periodic orbits.Inthis paper,it is shown that any given S∈（?） has a neighbourhood （?） in （?） together with anumber λ=λ（?）】0 such that θ（X）≤λfor all X∈（?）.
基金Project supported by Fund of National Science of China.
文摘In this paper, Henon map is considered. For a positive measure set of parameters (a, b), we construct a trapping region G of topologically transitive strange attractor Aa,b for Ta,b, and prove that Aa,b= ∩n≥0Ta,bnG, and the basin B(Aa,b) of Aa,b is exactly the union of domain whose boundary is contained in w5(p) ∪wu(p) and ws(p). Therefore, that the conjecture posed by Benedicks and Carleson about the basin of strange attactor is true is proved. Furthermore, B(Aa,b) is simply connected and path-connected, w4(p2) is contained in the attainable boundary set of B(Aa,b) (where p2 is another hyperbolic fixed point of Ta,b).
