Interesting username, butterfly! It reminded me right away the technical notion of sensitive dependence on initial conditions in Chaos Theory I read. Small variations of the initial condition of a non-linear dynamical system that may produce large variations in the long term behavior of the system. So this is sometimes presented as esoteric behavior, but can be exhibited by very simple systems: for example, a ball placed at the crest of a hill might roll into any of several valleys depending on slight differences in initial position. The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that ultimately cause a tornado to appear (or prevent a tornado from appearing). The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.
These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, the other in yellow) for the same period of time in the Lorenz attractor starting at two initial points that differ only by 10-5 in the x-coordinate. Initially, the two trajectories seem coincident, as indicated by the small difference between the z coordinate of the blue and yellow trajectories, but for t > 23 the difference is as large as the value of the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at t=30.
Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather).
In the movie "Run Lola Run" (Lola rennt in German-1998), the butterfly effect is represented clearly. There, minor and almost sub-conscious actions in everyday life can be seen to have gross and wide spread effects upon the future. For example, the fact that Lola bumps into someone instead of passing by may lead to a painful death after suffering paralysis. As such, seemingly inconsequential actions can be seen to have drastic long-term results. Lola's boyfriend Manni is trying to prove his loyalty to a gang boss. Manni's final task in a particular job is to deliver 100,000 Deutsche Mark (about 51,000 Euro at the final exchange rate) to his boss Ronnie. Everything goes wrong. Lola's moped is stolen and she is unable to transport Manni to the meeting place. After waiting for her Manni decides to use the metro. He accidentally leaves the bag, with its 100,000 Mark, in the underground after an encounter with a bum and two ticket-controllers. The money is then found by the homeless man. Manni realizes what he's done and soon makes a desperate phone call to Lola, asking her to think of something, to help him. If he does not have the money by the meeting at 12 noon, he will certainly be killed. Lola promises to get him the 100,000 Mark. Manni warns her that he will rob a supermarket on the street corner if Lola has not come in 20 minutes. Can Lola get him the money and save his life? It is at this point that the three sequential alternative realities begin.
First reality: Manni and Lola rob a supermarket.
As Lola flees from her apartment, a punk with a dog is shown on the staircase. The dog growls at her, causing her to scream and sprint faster. With little time and no vehicle, Lola runs through the streets of Berlin to get to her father's bank, with the intention of asking him for the money. Lola's father refuses, and says that he feels unappreciated at home, and that he is leaving Lola and her mother for his mistress. He also announces that he is not Lola's real father. Lola runs to where Manni is anyway, passing an ambulance that stops in front of a crew of workers carrying a window pane. She arrives at the street corner a few moments too late; Manni's robbery is already in progress. Lola decides to help Manni rob the store. The two flee on foot afterwards but find themselves surrounded by police, and a nervous police officer accidentally shoots Lola in the chest after Manni throws the bag with its stolen money into the air. While Lola is dying, a sequence of her memory (or, possibly, her consciousness) is shown. In it, Lola and Manni are together talking in bed. Lola questions Manni about his love for her and remains unconvinced that it is genuine. The scene fades in a sea of red.
As she dies, the film suddenly starts again. It jumps back to the end of Lola's first phone call from Manni and again she tries to get the money from her father. This time, the punk with a dog in the stairway trips her. Lola arrives at the bank a few moments later because of her limp, which leaves enough time for her father's mistress to explain that she is pregnant by someone else. Lola hears more of the argument this time, and becomes infuriated. She then robs her father's bank with a gun grabbed from the security officer, and takes off with the money to meet Manni, and tries to hitch a ride from the red ambulance. But her distracting the driver makes the ambulance crash into the window pane, stopping it for a few seconds. When Lola reaches Manni he is run down by the same ambulance as he crosses the street to meet her. After Manni is killed by the ambulance another memory sequence ensues in which Lola and Manni's roles are reversed: Manni questions Lola about her love for him.
The story starts a third time. Lola is a split second faster, as she leaps over the punk on the steps and stops on Mr. Meyer's (her father's co-worker, as it now turns out) car long enough to prevent his accident in the first two realities. This allows Mr. Meyer to get to work and pick up Lola's father. As a result, Lola misses her father completely. Not knowing what to do, she decides to simply keep running. However her father, along with Mr. Meyer now end up in an apparently fatal car crash as the tramp with the money distracts the driver. Lola enters a casino, buys a single 100-mark chip, and finds a roulette table. She wins two consecutive bets on the number "20" (an echo of the 20 minutes of her journey), which gives Lola 127,000 Marks. More than sufficient money to help Manni, but she still must catch him in time. She hitches a ride in the same ambulance, unnoticed by the driver, as it stops in front of the crew with the window pane. The ambulance is carrying Schuster, the security guard from her father's bank who has apparently suffered a heart attack, as foreshadowed by his clutching his chest and his loud heartbeats in the Second Run earlier in the film. Although some English subtitles here have Lola saying "I'll stay with him," the actual German line is "Ich gehöre zu ihm", which translates as "I'm with him" or "I belong with him." She holds Schuster's hand, and moments later, his heart rate begins to return to normal. Meanwhile, Manni has borrowed a phone card from a blind woman (portrayed for the third time) to make a phone call as he futilely seeks a loan. As in the other sequences, he returns the phone card to the woman, but this time the woman gestures with her head, and Manni looks up to notice the tramp with his money riding by on a bicycle. Manni chases him down, recovers his money, gives him his pistol in exchange for the bag of cash and then delivers it to Ronnie. Lola arrives to find Manni stepping out of Ronnie's car and shaking his boss's hand. The movie ends with Manni asking Lola what is in the bag she is carrying.
Connections between the runs
Throughout the film, Lola bumps into people, talks to them, or simply passes them by. Details of that person's future are subsequently shown in a series of still frames. The futures are widely divergent from encounter to encounter. In one scenario, a woman whom Lola accidentally bumps into remains poor and kidnaps an unattended baby after her child was taken away by social workers. In another scenario the woman wins the lottery and becomes rich. In the third scenario, the woman experiences a religious conversion. Several moments in the film allude to a supernatural awareness of the characters. For example, in the first reality, a nervous Lola is shown by Manni how to use a gun by removing the safety, whereas she does this as if remembered from a previous experience in the second reality. Lola's encounters with Schuster also contain an air of the supernatural. The movie itself begins by posing questions pertaining to the unpredictability of the world and the unknowable nature of its meaning. It suggests that drastically disparate consequences can alter the fates of different people from a one second change in the time of one person's running.