Nonsymmetric Lorenz Attractors from a Homoclinic Bifurcation

We consider a bifurcation of a flow in three dimensions from a double homoclinic connection to a fixed point satisfying a resonance condition between the eigenvalues. For correctly chosen parameters in the unfolding, we prove that there is a transitive attractor of Lorenz type. In particular we show the existence of a bifurcation to an attractor of Lorenz type which is semiorientable, i.e., orientable on one half and nonorientable on the other half. We do not assume any symmetry condition, so we need to discuss nonsymmetric one dimensional Poincaré maps with one discontinuity and absolute value of the derivative always greater than one. We also apply these results to a specific set of degree four polynomial differential equations. The results do not apply to the actual Lorenz equations because they do not have enough parameters to adjust to make them satisfy the hypothesis.