Discuss the theme of homecoming in chapters 13–24. In particular, consider how both Odysseus and his country have changed in the twenty years he has been away.

It is well known that the theme of Homer's The Odyssey is return (in Greek, nostos). The second half of The Odyssey begins with the arrival of Odysseus back in his native land of Ithaca. Much has changed in the twenty years since Odysseus left for the Trojan War. In fact, when Odysseus returns in book 13, he initially has a difficult time recognizing his native Ithaca:

Alas, what mortal country have I reached? Are they lawless, cruel, uncivilised people here, or are they kind to strangers, minds fearful of the gods? (Kline translation)

Because Odysseus was gone for so long, Odysseus was assumed to be dead, and now his wife, Penelope, is being pestered to remarry by more than 100 suitors from Ithaca and its neighboring islands. Penelope, however, is one of the few people or things that have remained unchanged during Odysseus's absence. Penelope remains faithful to her long-lost husband.
Odysseus's son, Telemachus, however, has grown up during his father's absence. Telemachus was a mere infant when Odysseus left for the war. Telemachus has now grown into a mature young man, and the first four books of the Odyssey deal with Telemachus's search for news of his father.
While Odysseus was off at war, his mother, Anticlea, passed away. Odysseus encounters her spirit during his encounter with other spirits from the underworld in book 11.
Also during Odysseus's absence, Odysseus's father, Laertes, has retired from the main palace and left to live on his farm. After defeating Penelope's suitors, Odysseus goes to the farm, proves his identity to his father, and enlists his father's aid in dealing with the suitors' relatives, who are angry at the deaths of the suitors.
Not only is Odysseus's home infested by suitors during his absence, in addition, some of Odysseus' servants are no longer faithful to him (e.g., Melanthius and Melantho).
Interestingly enough, after Odysseus kills the suitors and faces his wife Penelope, it is her claim that something has changed that allows her to recognize that she is actually standing face to face with her husband. In book 23, Penelope tests Odysseus by telling him that she has moved the location of their marriage bed. When Odysseus becomes angry upon hearing about this change, he reveals the secret of how he built this bed so that moving it would be nearly impossible. This revelation convinces Penelope that the man before her is in fact her husband. Odysseus's return is well on its way to becoming complete.