Strange attractors are by definition aperiodic and do not self-intersect, and I'm not sure I entirely understand your question(s). For example, what do you mean

Strange attractors are by definition aperiodic and do not self-intersect, and I'm not sure I entirely understand your question(s). For example, what do you mean by "where the chaos is being inserted"? Chaotic systems are completely deterministic and are sensitive to any change in initial conditions. The chaos is there from the start. But it doesn't mean that once initial conditions are given that the system behaves chaotically for the time evolution of the system since, again, the system is deterministic. I'm also not sure if the book answers your question(s) on this apparent aymmetry aspect, but Chaos and Fractals by David Feldman is a fantastic introduction to these topics with lots of good pictures. It is not a proof-based book, but it does include mathematics at manageable level. The book gives a great introduction to strange attractors, namely the Henon map, Rossler attractor, and the Lorenz attractor.