On a class of nonlocal bifurcation concerning the Lorenz attractor

In this paper, we consider a two-parameter family of systemsE in whichE ₀ 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 parametersb= 8/3,=10,r=r ₁ 24.06. For the generic unfoldingE ofE ₀, 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 valuer=r ₁ 24.06 observed on computer for Lorenz equations in a precise mathematical way.