Odysseus waits for the cyclops to return in hope that the cyclops will be a good host and give him a present. It is really Odysseus' curiousity that compels him to stay.
1 year
Odysseus and his men steal food and other items from Polyphemus' cave (a cyclops). They planned to take the materials and leave but Odysseus is curious to see what the cyclops looks like. They stay to meet Polyphemus, getting trapped in his cave. To get free they sharpen a wooden pole, get the cyclops drunk, and blind him by stabbing his eye while he's asleep. Polyphemus moves the large rock blocking the cave and Odysseus and his men escape clinging to the bellies of sheep.
Odysseus chose to stay in the cave because he was curious about the Cyclops and wanted to see him, believing that he and his men could benefit from the giant's hospitality. He also hoped to gather information and assess the situation before deciding to leave. However, this decision ultimately put his crew in danger when the Cyclops returned and trapped them inside the cave.
because Polyphemus put a large stone in the door way so is his sheep would not get out
Calypso wants Odysseus to stay with her on the island forever.
Odysseus waits for the cyclops to return in hope that the cyclops will be a good host and give him a present. It is really Odysseus' curiousity that compels him to stay.
1 year
Odysseus spent seven years on the island of Ogygia with the nymph Calypso.
Odysseus insists on staying to confront the cyclops because he wants to find out if the cyclops can offer him any hospitality, believing it is the custom of the gods to offer hospitality to travelers. He also wants to test the cyclops's strength and cunning.
Odysseus and his men steal food and other items from Polyphemus' cave (a cyclops). They planned to take the materials and leave but Odysseus is curious to see what the cyclops looks like. They stay to meet Polyphemus, getting trapped in his cave. To get free they sharpen a wooden pole, get the cyclops drunk, and blind him by stabbing his eye while he's asleep. Polyphemus moves the large rock blocking the cave and Odysseus and his men escape clinging to the bellies of sheep.
The cave that Odysseus and his men found when they landed on the Island of the Cyclopes belongs to Polyphemus. Odysseus decided to stay in the cave until its owner returned. When Polyphemus returned and saw the intruders, he ate Odysseus' men.
Calypso
Calypso offers to make Odysseus immortal and forever young if he will stay with her on her island. Odysseus respectfully refuses her offer.
Odysseus stays on Circe's island for one year.In a movie version, he stays for what seems like 7 days, but is in reality 7 years.
Calypso's island for seven years
I believe he wanted to meet the owner of the cave because he didn't know who inhabited the island. When he first came in, he and his men just discovered a cave with food (and sheep I think).