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The butterfly effect is a term used in chaos theory to describe how small changes to a seemingly unrelated thing or condition (also known as an initial condition) can affect large, complex systems. The term comes from the suggestion that the flapping of a butterfly's wings in South America could affect the weather in Texas, meaning that the tiniest influence on one part of a system can have a huge effect on another part. Taken more broadly, the butterfly effect is a way of describing how, unless all factors can be accounted for, large systems like the weather remain impossible to predict with total accuracy because there are too many unknown variables to track.

Origins in Weather Prediction

Anarchy OK 3. Silence is Violence 4. The River Song 9. Doors (Now & Then My Life Feels Like It's Going Nowhere).

The concept of the butterfly effect is attributed to Edward Norton Lorenz, a mathematician and meteorologist, who was one of the first proponents of chaos theory. Lorenz was running global climate models on his computer one day and, hoping to save himself some time, ran one model from the middle rather than the beginning. The two weather predictions, one based on the entire process, including initial conditions, and another based on a portion of the data, starting with the process already part way completed, diverged drastically. Lorenz, along with most scientists of his time, had expected the computer models to be identical regardless of where they started. Instead, tiny, unpredictable variations caused the two models to differ.

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All of the prison scenes were filmed in a real prison (Washington State) with real prisoners.
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During one of Evan's 'flashback' scenes, he can be heard reading part of Ray Bradbury's short story 'A Sound of Thunder.' In this story, a group of people travel millions of years into the past to hunt dinosaurs. One of them accidentally steps on and kills a butterfly, which dramatically alters the future.
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Ashton Kutcher did extensive research on psychology, mental disorders, and chaos theory to prepare for his role in this film.
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In early versions of the script, the character of Evan was originally Chris Treborn. When the 'T' is moved over, it becomes 'Christ Reborn'. This was changed to Evan Treborn, which is a play on 'Event Reborn'.
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Was one of the most widely read unproduced scripts in the industry. It wasn't until Ashton Kutcher signed on as an executive producer of the movie, that it was greenlit.
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Elden Henson shot all of the later 'normal' scenes with his character first because he had to gain around twenty pounds in one month for the later timelines in which his character is crazy. He was to look bigger as crazy Lenny and thinner as normal Lenny. To the filmmaker's amazement he accomplished this.
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The lights flicker in the background of the psychiatrist's office when Evan is being hypnotized, after they blew up the mailbox. This was not a planned special effect - it was an actual short in the wiring on set. The directors thought it fit well with the scene and used that take in the final cut of the movie.
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They only had three of the postboxes to blow up and they only got it right on the third try.
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Logan Lerman (Evan at 7) wore dark contact lenses to match Ashton Kutcher's eyes.
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The picture over Evan's bed in his dorm room is 'Sleep' by Salvador Dalí.
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The title is a reference to a concept in Chaos Theory whereby small events may lead unpredictably to large events. This is explained in the opening quotation which explains that a butterfly flapping its wings may result in a typhoon. It may also be seen secondarily as a reference to the short story, 'A Sound of Thunder', by Ray Bradbury.
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When Evan wakes up in the sorority house, there is a 'Bradbury University' pennant on the wall - a reference to Ray Bradbury.
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The wig that Eric Stoltz wears in the film was made of Stoltz's real hair.
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The teenage versions of Evan, Kayleigh, and Tommy go see Se7en (1995) at a movie theater that's also showing Dumb and Dumber (1994). Both of those movies, as well as this one, were released by New Line Cinema. And Elden Henson who plays the grown-up Lenny can also be seen in Dumb and Dumberer: When Harry Met Lloyd (2003).
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Evan's cigarette burn scar looks very much like an outline of the Mandelbrot set, often a symbol of Chaos theory at the heart of the butterfly effect.
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Josh Hartnett, Seann William Scott, and Joshua Jackson were each offered the role of Evan.
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Evan's diaries have the same cover layout as the diaries of John Doe in Se7en (1995), also released by New Line Cinema. They are standard composition notebooks that are used by school children across the country every day.
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Ali Larter was offered the role of Kayleigh, but dropped out.
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The main character, Evan, wrote a total of ~4745 entries from age 6 to age 19 (basically 365 days x 13 years). Therefore 4745 entries divided by 192 pages(front/back) in a standard notebook would take up ~24 notebooks in this movie.
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The Hilltop Café, Kayleigh (Amy Smart)'s workplace, was also used in Bates Motel (2013). In episode 2.2, Bates Motel: Shadow of a Doubt (2014), Dylan (Max Thieriot) & Bradley (Nicola Peltz) wait for the bus there.
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When Evan is placing the phone call at 'State', the number he dials is 555-5785: a number often repeated by Frank Rizzo of The Jerky Boys.
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Tommy may suffer from antisocial personality disorder.
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Kevin G. Schmidt (13 years old Lenny) and Ashton Kutcher (adult Evan) have previously played in Cheaper by the Dozen (2003).
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Spoilers

The trivia items below may give away important plot points.

There are four alternate endings. In the first, Evan turns around and talks to Kayleigh at the same time. In the second, Evan turns around after Kayleigh has already turned around, and follows her. In the third ending, Kayleigh turns around first and then Evan turns around and Kayleigh walks away. Evan stands there for a minute then walks away and does not follow Kayleigh. In the fourth one, Evan watches a home movie of his birth, rather than him meeting Kayleigh. He travels into the movie, and strangles himself in the womb. (This can be seen at the end of the Director's cut.)
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The Butterfly Effect Explanation

Evan's mother makes mention of having previous late term miscarriages before Evan was finally born. This information does more than just set the stage for the alternate ending in which Evan watches his birth video and strangles himself in the womb. It also adds intrigue as to whether these were natural events, or if perhaps Evan had siblings who suffered the same fate, and decided to end their own lives just before being born.
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The scene where Evan has no arms was achieved by using two shots: one with an empty bed, and one with Ashton Kutcher lying in the bed with green gloves on his hands (which were erased later). Both shots had identical camera movements. For most films, this would be achieved by using machine-controlled cameras, which can replicate the exact same movement for multiple takes. However, since this film was fairly low-budget, the filmmakers were not able to afford this kind of equipment and the two identical shots were achieved by manually moving the camera while using a stopwatch for reference. Any small changes in the two shots were fixed digitally in post-production.

The Butterfly Effect Theory

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The movie's title screen is a scan of the brain (as shown from the front). The left and right lateral ventricles of the corpus callosum (the middle of the brain) in this picture are displaced. Such is the case with many schizophrenic patients. The displaced ventricles in the middle of the brain are sometimes referred to as looking like a butterfly. In the movie, Evan is not schizophrenic, but in the next to last universe he 'jumps' to (after he accidentally kills Kayleigh as a child), he is treated as if he is; the doctors tell him that everything that happened to him throughout the course of the movie was made-up.
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In the original ending, Evan travels back through time to the day of his birth and he commits suicide in his mother's womb as an unborn baby. That ending was used in the director's cut. The film was given a different ending in the theatrical version and the conclusion of the film was changed to Evan traveling back to the day he met Kayleigh when he was child, and he upsets her when he tells her that he hates her and her family, and threatens to kill them all if she comes near him again. Evan did this because he felt the only way for he and his family and close friends to be happy with their lives was to avoid interacting with Kayleigh or her troubled, protective brother Tommy. Doing this made it possible for Kayleigh and Tommy to live with their mother, instead of getting the abusive upbringing from their father that led to the original chain of events, because as Kayleigh explains in an earlier scene, being near Evan was the only reason she did not go live with her mother in the first place.
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In the scene where Evan and his mother are standing at Jason's grave, Kayleigh comes up and takes Evan's hand. We don't know what she says to him because of the distance of the shot and the score playing in the background. She actually says: 'you're probably better off now.'
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There is a scene where the alternate reality is created where Evan finds himself at College as a fraternity brother. When his memory is being 'rebuilt', Kayleigh hitches a ride to visit Evan at age 13 after he moved away. People don't realize (and it's not even mentioned in the DVD commentary) that the driver is Karl, the bald prisoner who would exist in that reality before going to the same prison as Evan.
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The film takes place in 1989, 1996, 2002 and 2010.
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Aside from appearing for a split second in a picture during the last-to-final flashback, co-director Eric Bress also appears as a patient in the mental hospital in the beginning of the movie. In the alternate ending on the director's cut, he also has a cameo as Andrea's new husband (in the alternate time line created by Evan).
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During Evan's penultimate flashback (the one after which he wakes up in the mental hospital), there is a quick succession of shots with people making funny faces. One of the people appearing on several pictures is co-director Eric Bress.
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