This is an overview of the various factors in-game that are used in triggering randomly-occurring events across the dream world.
Per Dream Session
Every dream session, or when sleeping in the bed in Urotsuki's Dream Apartments, there are certain elements in the dream world that may appear differently, such the Nexus' background, or there may be some events that occur during certain dream sessions. This is determined by Variable #44, a fixed value that falls under the range of 0 and 255. A selection of commonly-known events involving this variable, including ways of identifying the approximate value of this variable can be found here.
Seasonal System
Seasons change every three dream sessions, where the seasonal system follows the four seasons: winter, spring, summer, and fall. One example of its application is the Zodiac constellations that you can encounter in the Constellation World based on the season in the Northern Hemisphere.
The easier way of checking the current season is with the drying clothes hanged in the balcony of Urotsuki's Room while sleeping, which will change depending on the current season as follows:
Winter or Spring
Summer
Fall
Aditionally, the Teru Teru Bozu effect makes snow instead of rain when the current season is winter, which can be used to identify the season when the first case shown in the previous pictures is presented.
Optionally, a visual indicator of the current season can be found in the Subterranean Research Center and a coffee cup in Coffee Cup World which changes its contents depending of the season.
Season Calculator
It is also possible to calculate the current season by using the amount of dream sessions that have been had since the game started. This value can be obtained when loading a game, saving a game or in the fourth page of the data section on the Computer.
Input the aforementioned value in the following field to calculate the season in your next dream.
Coin Acquisition
When killing creatures across the Dream World, there is a probability that you earn money upon doing so, and it is determined by the Coin Acquisition variable, Variable #15, where the range is assigned to different NPCs that can be killed. One example of a random event linked to this variable is in the random events on the TV of Urotsuki's Dream Room.