These are three related schemes that bear comparison since they often coincide. Anadiplosis is a repetition of the same word or phrase at the end of one clause and the beginning of the next. The essential feature of climax is the increase of magnitude, intensity or amplification in a series of related terms arranged in sequence. Gradatio combines anadiplosis and climax to create a "staircase parallel" of increasing intensity.
Basic Definitions
Anadiplosis
Corbett and Connors 392-393: Anadiplosis: Repetition of the last word of one clause at the beginning of the following clause.
“Once you change your philosophy, you change your thought pattern. Once you change your thought pattern, you change your attitude. Once you change your attitude, it changes your behavior pattern and then you go on into some action.”—Malcolm X
Climax
Gradatio and climax are often treated as synonyms. Notice, for instance, that Silva Rhetoricae subsumes gradatio into its definition of climax. This, however, leaves no way to distinguish simple climax, any series of words in increasing order of magnitude, from the kind that also involves anadiplosis. For this special case of climax we reserve use of the term gradatio.
Illustrations
Anadiplosis
Climax
Simple Climax: Veni vidi vici ("I came, I saw, I conquered")
Gradatio
"Who controls Berlin, controls Germany; who controls Germany controls Europe; who controls Europe controls the world" (qtd. in Burke 57-58). See "Burke Roadmap" for more discussion.
References
Burke, Kenneth. A Rhetoric of Motives. Berkeley: U of California P, 1969.
Corbett, Edward P. J. and Robert J. Connors. Classical Rhetoric for the Modern Student. 4th ed. New York: Oxford UP, 1999.
Walker, William. "Anadiplosis in Shakespearean Drama." Rhetorica: A Journal of the History of Rhetoric, Vol. 35 No. 4, Autumn 2017; (pp. 399-424) DOI: 10.1525/rh.2017.35.4.399