Chaotic attractors in one dimension generated by a singular Sil\u27nikov orbit

Chaotic attractors containing [special characters omitted]il\u27nikov\u27s saddle-focus homoclinic orbit have been observed numerically in the physical sciences, such as neural dynamics, ecological systems, chaotic circuits, nonlinear laser systems, and fluid dynamics. Past and current research of this type of structure has focused on the orbit and nearby structure only. Here we will look at the role the orbit plays in the attractor, using symbolic dynamics, kneading theory, and measure theory on the one-dimensional return maps generated by a singular [special characters omitted]il\u27nikov orbit