I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, Reddit and other such web pages that more or less intuitively explain the connections with bifurcation diagrams and the Mandelbrot set being the parameter space for the logistic map, all of which I understand. However I am having difficulty finding reputable academic sources that research and explain these concepts in a formal mathematical style. I would be very grateful to anybody who could point me in the direction of academic papers or textbooks that cover this. Thank you all!
On the undergraduate level, I recommend the books by Robert Devaney. He wrote several books on this subject:
An introduction to chaotic dynamical systems. The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986 (there is a second and third editions, 1989, 2003).
A first course in chaotic dynamical systems. Theory and experiment. With a separately available computer disk. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1992.
There is also a video: Transition to chaos. The orbit diagram and the Mandelbrot set. Science Television, New York; distributed by the American Mathematical Society, Providence, RI, 1990. 1 videocassette (NTSC; 1/2 inch; VHS) (65 min.); sd., col. ISBN: 1-878310-08-9