[Music] the flap of a butterfly's Wings in Brazil can set off a tornado in Texas of course this may sound like a kind of joke or something but trust me it's way more serious than you think you see the world we live in operates in two fundamental ways which shape the fabric of reality the probabilistic mode and the deterministic mode in the probabilistic mode our universe unfolds with an element of chance small seemingly insignificant events like the butterflies Wings can Cascade into significant unpredictable consequences on the flip side we have the deterministic mode a world
governed by rules and predictability here cause and effect are tightly linked and outcomes can be traced back to specific identifiable factors it's a structured reality where theoretically if we knew all the initial conditions precisely we could predict the future with certainty so how do these modes coexist can the flutter of a butterfly fly truly influence something as powerful as a tornado how much of our world is governed by chance and how much by rules answering these questions might seem beyond the scope of science but things are changing this puzzle is Central to chaos theory a
tool that helps us see the surprising connection between things that appear disorderly and those that are highly organized here's How The Story Goes imagine playing a game called the chaos game all you need is a blank sheet of paper something to write with and a die here's how it works start by marking three dots on the paper like the vertices of a triangle let's call them A B and C also Mark another Point within the three marked dots as your starting point roll the die if you get a one or two Mark a DOT between
the starting point and a if it's a three or four Mark a DOT between the starting point and B for a five or six Mark a DOT between the starting point and C for example if you roll a two Mark a DOT between the starting point and a roll the die again let's say you get a two again now Mark a DOT between the last point you marked and a repeat this process keep rolling the die and marking dots accordingly for example if you roll a four draw a DOT between the last marked point and
B and continue this process after doing this for a while you'll end up with seemingly random dots on the paper but here's the interesting part what if these seemingly chaotic dots lead to something unexpected to see some results we can ask a computer to do this and when I tried one of them the result was very shocking in that I set the number of points to be3 click Start and set the speed to fast the starting point or referred to here as the trace Point started moving marking points after waiting for about a minute this
was what I noticed looks familiar right it is indeed the cinsky triangle random dots and Chaos lead us to such an orderly and symmetrical pattern this is a part of Chaos Theory which is usually defined as the branch of mathematics that focuses on the behavior of dynamical systems that are highly sensitive to initial conditions so the initial conditions in this case were the three dots from where we began and the rule by which we were marking the dots If instead of three you begin with five dots then you would get a different pattern in Chaos
Theory apparent Randomness leads to various patterns fractals Symmetry and many more you can head over to to the website to try it out yourself the link would be at the description of this video but don't go anywhere yet for another example we'll need to start with a random number like this number on your screen now in each step we multiply it by 10 and remove the digit in front of the decimal point multiply by 10 you'll get this remove the digit in front and you'll get this multiply by 10 again this is what you'll get
remove the digit in front again you'll get this this processing seems deterministic because it follows a fixed Rule now here's the catch can you predict the future of our number the answer is both yes and no initially You can predict the outcome accurately for a few steps but after a certain point the Precision becomes crucial computers for example store a limited number of digits after the decimal point after applying our formula around 15 times in the case of a 64-bit double the results become unpredictable this unpredictability arises because our formula discards information with each step
specifically when removing the digit in front of the decimal point even if you started with more accurate initial conditions like 100 or 1,000 digits after the decimal point the results become unpredictable after a certain number of steps due to the critical dependence on the initial condition this is a simple illustration of a deterministic system becoming chaotic and unpredictable over time chaos theory is about equations or systems that are deter terministic meaning that there are clear instructions on how to calculate the future from the current state without any random elements involved but that at the same
time depends so critically on the initial conditions that it is impossible to predict the long-term future but how does any of this affect us Chaos Theory provides a scientifically quantifiable but simultaneously poetically suggestive way of looking at how almost everything in life is connected with everything else anything that has definition and coherence such as a single- celled organism a war a novel or a flock of birds is a system anything that changes grows deepens or expands is dynamic anything that depends upon subtle chains of cause and effect such as a fender bender leading to a
conversation leading to a date leading to a marriage leading to a family is deterministic perhaps most importantly any system that is open to Rapid and unpredictable change not moving in a straight and unbroken trajectory is nonlinear the most interesting intriguing challenging and exciting things in life tend to be dynamic deterministic nonlinear systems Chaos Theory challenges the certainty of traditional science by acknowledging a fundamental truth that science has often hesitated to admit since the enlightenment we can't know all the answers while Chaos Theory doesn't replace the enlightenment's idea of scientific determinism it emphasizes that even with
an abundance of data and information we can never be entirely certain about the behavior of any system this uncertainty persists especially when systems interact with others and are susceptible to the influence of small changes even seemingly simple systems like a swinging pendulum or a child's House of Cards can quickly descend into disorder what's surprising however is how highly disordered and chaotic systems reveal unexpected patterns of Simplicity chaotic systems tend to follow specific stable Pathways over time as they navigate through their operation phase space these pathways are influenced by attractors with the strange attractor being a
unique type found in nonlinear systems it displays both long-term stability and short-term unpredictability when mapped out in its phase space the title strange attractor refers to a specific kind of attractor showing that Chaos Theory doesn't just lead to disorder but also unveils hidden patterns within the complexity when closely examined the simple lines of these attract factors reveal layers upon layers of intricate complexity patterns repeat themselves infinitely a concept known as self similarity across scale seen in natural systems such as snowflakes Fern leaves and mountain ranges all explained by fractal geometry fractal objects possess unique properties
and one of the key Concepts is a modern interpretation of what older scientific terminology referred to as microcosms and macrocosms the term fractal was introduced by mathematician benois mandelbrot and refers to the fraction that can exist between dimensions in natural systems like an Alpine mountain range the degree of jaggedness observed on a single Mountaintop will remain consistent with the degree of jaggedness across the entire range extending across the Horizon this concept of a consistent fractal Dimension illustrates how irregularity is a characteristic property that persists across different scales in natural phenomena this mathematical framework embraced The
Irregular the jagged and the ever repeating patterns that ukian geometry had deemed unruly with fractals mandelbrot could quantify the coastlines model the intricate branching of trees and even understand the turbulent flow of fluids a fractal is a never-ending pattern that is self-similar across different scales they are created by repeating a simple process over and over in an ongoing feedback loop driven by recursion fractals are images of dynamic systems the pictures of chaos fractals are the unique irregular patterns left behind by the unpredictable movements of the chaotic world at work fractals depict chaotic Behavior yet if
one looks closely enough it is always possible to spot glimpses of self-similarity within a fractal fractal order is a unique and surprising byproduct of chaos this is the difference between the chaotic and the simply random chaotic systems never repeat themselves while remaining an identifiable system whereas random systems can fall into repetitive Cycles because they are free to end back at a point where they once began for example five fish swimming in a tank is a chaotic system because all five will never be in exactly the same position with the same direction and momentum twice rolling
five dice is random because it is perfectly possible to roll the exact same combination several times each chaotic system exists within the borders of its own phase space so if the system never exactly repeats itself then a graph of the system's movement will necessarily have to have an infinite number of dots with no two ever taking up exactly the same space one would expect the dots to appear randomly on the paper and for a while they do sensitivity to the butterfly effect makes a dynamic system open to radical change usually as rapid as it is
transformative a favorite example of complexity theorists is the sand pile one grain of sand is unlikely to make a difference at any single Moment In Time but sooner or later one of those single grains of sand will inevitably trigger a collapsing Landslide these are called phase Transitions and Chaos theorists have identified unexpected stabilities in the ways one state of matter exponentially bifurcates into another systems that are most receptive to phase transitions are those that live on the edge of chaos at a fundamental level the universe is quantum mechanical in nature full of an inherent indeterminism
and uncertainty if you take a particle like an electron you might think to ask questions like where is this electron how fast and in what direction is this electron moving and if I look away right now and look back one second later where will the electron be they're all reasonable questions and we'd expect that they'd all have definitive answers what happens in the quantum world is quite strange and can be a bit unsettling even for physicists who've dedicated their lives to studying it imagine you want to know exactly where an electron is the odd thing
is the more precisely you try to figure out its location the less you know about how fast and in which direction it's moving its momentum on the flip side if you try to measure its momentum you become less certain about its exact position here's the tricky part to accurately predict where the electron will be in the future you need both its momentum and position however due to this uncertainty principle in quantum mechanics you can only predict a range range of possibilities for its future position so to find out exactly where it is at some point
in the future you'll need to make another measurement at that specific time it's a bit like trying to pin down both where and how fast a tiny particle is going and the more you know about One the less you know about the other quantum physics introduces fundamental uncertainty but when dealing with larger everyday objects Newtonian physics is usually more than enough unlike the unpredictable nature of quantum mechanics Newtonian in physics is entirely deterministic according to Newton's Laws of Motion especially the fundamental equation Force equals mass times acceleration if you know the starting conditions of an
object like its position and momentum you should theoretically be able to precisely predict its location and motion at any future time the equation describes what happens in the next moment and as time progresses it continues to guide you about what will occur in each subsequent moment as long as we're dealing with objects objects where Quantum effects can be ignored these rules of Newtonian physics work seamlessly providing a clear understanding of how an object will evolve over time even with completely deterministic equations there are limits to how accurately we can predict the behavior of a Newtonian
system this might be surprising and you're not alone in feeling that way many prominent physicists even those who extensively studied Newtonian systems initially believed there would be no such limit back in 184 14 mathematician Pierre Le expressed this perspective in his Treatise titled a philosophical essay on probabilities he envisioned a scenario where if we could gather enough information to determine the state of the entire universe at any given moment we could use the laws of physics to predict absolutely everything about the future without any uncertainty however as we've come to understand the actual limitations of
predictability in complex systems even in the realm of Newtonian physics have proven more nuanced than initially imagined the necessity of introducing probabilities into predictions about the future doesn't always arise due to our lack of knowledge about the universe or the peculiarities of quantum phenomena like Heisenberg's uncertainty principle instead it can be attributed to a classical phenomenon known as chaos even if we have a precise understanding of the initial conditions of a system deterministic equations such as Newton's Laws of Motion do not always result in a deterministic universe this Revelation dates back to the early 1960s
when Edward Loren a meteorology professor at MIT sought to improve weather forecasting using a powerful Mainframe computer despite having what he thought was a reliable weather model a comprehensive set of measurable data temperature pressure wind conditions Etc and a computer with considerable computational power Loren attempted to predict weather conditions far into the future he formulated a set of equations input the data and ran the program however when he re-entered the same data and ran the program for an extended period the results were not what he anticipated this unexpected outcome marked the recognition of Chaos in
deterministic systems remarkably during the second run of the program Edward Loren observed a surprising Divergence a minute difference at one point quickly led to the two systems behaving as if they were entirely unrelated evolving chaotically in relation to each other upon investigation Loren discovered the root cause when he re-entered the data for the second run he used the rounded off values from the computer's print out of the first run as input parameters though this discrepancy might have seemed insignificant possibly corresponding to the width of an atom or less it had a profound impact on the
outcomes especially when projecting the system far into the future this phenomenon where minuscule imperceptible differences in initial conditions result in drastically different outcomes is what we know as the butterfly fly effect it illustrates that even in entirely deterministic systems chaos can emerge from the tiniest variations in the starting conditions because we can never know all the initial conditions of a complex system in sufficient detail we cannot hope to predict its ultimate fate even slight errors in measuring the state of a system will be Amplified dramatically rendering any prediction useless since it is impossible to measure
the effects of all the butterflies in the world accurate long-range weather prediction will always remain impossible While most traditional science deals with supposedly predictable phenomena like gravity electricity or chemical reactions Chaos Theory deals with nonlinear things that are effectively impossible to predict or control like turbulence weather the stock market our brain States and so on these phenomena are often described by fractal mathematics which captures the infinite complexity of nature many natural objects exhibit fractal properties including Landscapes clouds trees organs rivers and many of the systems in which we live exhibit complex chaotic Behavior recognizing the
chaotic fractal nature of our world can give us new insight power and wisdom for example by understanding the complex chaotic dynamics of the atmosphere a balloon pilot can steer a balloon to a desired location by understanding that our ecos systems our social systems and our economic systems are interconnected we can hope to avoid actions that may end up being detrimental to our long-term well-being the universe is vast and unpredictable chaos is not just in the weather it's part of life reflecting our existence our choices have far-reaching impacts like the ripple effect of a butterfly's Wings
chaos often emerges in systems with something we call feedback to explain this you'll need to think of the stock market when a Stock's value goes up or down people react by buying or selling this action in turn influences the Stock's price creating a cycle of chaotic fluctuations The Continuous feedback loop in the stock market is a simple yet powerful example of how small actions can trigger larger unpredictable consequences in complex systems another example of this is a predator prey relationship in an ecosystem such as wolves and deer when the deer population is high there's an
abundant food supply for the Wolves causing the wolf population to increase as the wolf population grows they hunt more deer leading to a decline in the deer population now with fewer deer the wolves have less food causing their population to decrease this creates a feedback loop the abundance of prey influences the predator and the abundance of predators influences the prey this cycle continues with the populations of both wolves and deer fluctuating over time the feedback mechanism in this ecological system can result in Dynamic often un un predictable changes in population sizes any system where the
actions of its components influence each other can exhibit feedback leading to complex and sometimes chaotic behaviors feedback in chaos shows that perfect predictability is often an illusion accepting the inherent uncertainty can be liberating instead of trying to control everything we learn to adapt and navigate the emergent patterns that arise from the constant interplay of feedback loops observed in the world simple things chance encounters spoken words or quiet ideas are like the wings of Our Lives embracing the unpredictability Sparks creativity and turns us into improvisers shaping not only our lives but the world around us Chaos
Theory might be a spark for a new wave of thought not just for scientists but for everyone curious the ripple effect prompts us to rethink chance order and our role in the bigger picture the key takeaway of chaos is this even when your equations are perfectly deterministic you cannot know the initial conditions of arbitrary sensitivities when the present determines the future but the approximate present does not approximately determine the future is chaos if we ever find a final Theory chaos Theory's lessons will be a vital link it reminds us that beneath life's complexity there's Simplicity
waiting to be understood philosophers scientists and everyday people may then gather not as strangers to the universe but as explorers sharing the quest to understand why we are here in chaos we may not find all the answers but we might discover the wonder that connects us to the universe and each other thanks for watching And subscribe for more videos
