In this paper, we consider a two-parameter family of systems E<sub>ε</sub> in which E<sub>0</sub> has a contour consisting of a saddle point and two hyperbolic periodic orbits, i.e., the situa...In this paper, we consider a two-parameter family of systems E<sub>ε</sub> in which E<sub>0</sub> has a contour consisting of a saddle point and two hyperbolic periodic orbits, i.e., the situation is similar to that described by the Lorenz equations for parameters b=8/3, σ=10, r=r<sub>1</sub>≈24.06. For the generic unfolding E<sub>ε</sub>of E<sub>0</sub>, we find three kinds of infinitely many bifurcation curves and establish the correspondence of the trajectories which stay forever in a sufficiently small neighborhood of the contour with symbolic systems of finite or countably infinite symbols; these results can be used to explain the turbulence behaviors appearing at the critical value r=r<sub>1</sub>≈24.06 observed on computer for Lorenz equations in a precise mathematical way.展开更多
文摘In this paper, we consider a two-parameter family of systems E<sub>ε</sub> in which E<sub>0</sub> has a contour consisting of a saddle point and two hyperbolic periodic orbits, i.e., the situation is similar to that described by the Lorenz equations for parameters b=8/3, σ=10, r=r<sub>1</sub>≈24.06. For the generic unfolding E<sub>ε</sub>of E<sub>0</sub>, we find three kinds of infinitely many bifurcation curves and establish the correspondence of the trajectories which stay forever in a sufficiently small neighborhood of the contour with symbolic systems of finite or countably infinite symbols; these results can be used to explain the turbulence behaviors appearing at the critical value r=r<sub>1</sub>≈24.06 observed on computer for Lorenz equations in a precise mathematical way.
