How does pattern emerge and are they as natural as randomness?
From Kepler’s laws of planetary motion to genes propagating through generations, within vast amount of data about nature we usually find order. All these processes could have been absolutely random, but in reality they aren’t. On the other had share market (not exactly natural) and radioactive decays are examples of random events.
Given a set of initial conditions of a system, can we say which system is going to generate order and which one will operate randomly? Is there any field of science that studies how and under what universal abstract conditions- patteren emerge?
In: 5
Chaos is essentially the idea that tiny discrepancies in initial conditions can result in wildly varying behavior at some time in the future.
An example of a non-chaotic system is a pendulum, or a ball rolling down a hill, or a trajectory of a thrown object. In all of these examples, if you know the *initial state* of the object, you can predict all future behavior to some degree of precision. If I roll two balls down a hill, starting one of them *slightly* higher will mean they reach the bottom at *slightly* different times, regardless of how big the hill is.
Chaotic systems are different. If I take two double pendulums, and I raise one slightly higher than the other before I release it, then very soon that difference will explode into completely different behaviour for each system. [The wikipedia page](https://en.wikipedia.org/wiki/Double_pendulum) has a great illustration of this.
Importantly: a classical chaotic system *isn’t random*. The double pendulum is fully deterministic, and precisely modeled by a very simple set of equations. The same is true of a three-body orbit. The chaotic nature arises directly from those equations as a mathematical consequence. Chaos theory applies the theories of stochastic analysis, probability, and differential equations to study this kind of chaotic behavior.
The “chaos” (in a simplified way) comes from the fact that when multiple variables interact with each other, they can form “positive feedback loops”. For instance, a ball at the top of a hemispherical mound is stationary, but any push, no matter how small, will set if off and it will keep accelerating. Contrast this with a ball on a flat surface, where a small push sets it rolling at a small speed, or a ball at the bottom a crater, where a small push leads to small oscillations that returns the ball to equilibrium. Error terms in chaotic systems behave like the first ball, while error terms in nonchaotic systems behave like the other two.
It’s an insanely rich and beautiful theory, and there are certainly many complex patterns. You can ask, for instance, what these feedback loops look like, and how you can classify them into different patterns of behaviour. You can ask if there are equilibrium points, and whether they are stable. You can draw connections to fractal geometry. The theory of differential equations is the gateway to all this.
A differential equation is a way of modeling variables whose behavior depends on their own rate of change. For instance, because of air resistance, the acceleration of a falling object depends on its current speed. If the rate at which speed changes depends on its speed, how can we work out the behaviour of the object? This is the domain of differential equations, so it should be clear why differential equations are the language of chaos theory.
Chaos is essentially the idea that tiny discrepancies in initial conditions can result in wildly varying behavior at some time in the future.
An example of a non-chaotic system is a pendulum, or a ball rolling down a hill, or a trajectory of a thrown object. In all of these examples, if you know the *initial state* of the object, you can predict all future behavior to some degree of precision. If I roll two balls down a hill, starting one of them *slightly* higher will mean they reach the bottom at *slightly* different times, regardless of how big the hill is.
Chaotic systems are different. If I take two double pendulums, and I raise one slightly higher than the other before I release it, then very soon that difference will explode into completely different behaviour for each system. [The wikipedia page](https://en.wikipedia.org/wiki/Double_pendulum) has a great illustration of this.
Importantly: a classical chaotic system *isn’t random*. The double pendulum is fully deterministic, and precisely modeled by a very simple set of equations. The same is true of a three-body orbit. The chaotic nature arises directly from those equations as a mathematical consequence. Chaos theory applies the theories of stochastic analysis, probability, and differential equations to study this kind of chaotic behavior.
The “chaos” (in a simplified way) comes from the fact that when multiple variables interact with each other, they can form “positive feedback loops”. For instance, a ball at the top of a hemispherical mound is stationary, but any push, no matter how small, will set if off and it will keep accelerating. Contrast this with a ball on a flat surface, where a small push sets it rolling at a small speed, or a ball at the bottom a crater, where a small push leads to small oscillations that returns the ball to equilibrium. Error terms in chaotic systems behave like the first ball, while error terms in nonchaotic systems behave like the other two.
It’s an insanely rich and beautiful theory, and there are certainly many complex patterns. You can ask, for instance, what these feedback loops look like, and how you can classify them into different patterns of behaviour. You can ask if there are equilibrium points, and whether they are stable. You can draw connections to fractal geometry. The theory of differential equations is the gateway to all this.
A differential equation is a way of modeling variables whose behavior depends on their own rate of change. For instance, because of air resistance, the acceleration of a falling object depends on its current speed. If the rate at which speed changes depends on its speed, how can we work out the behaviour of the object? This is the domain of differential equations, so it should be clear why differential equations are the language of chaos theory.
Chaos is essentially the idea that tiny discrepancies in initial conditions can result in wildly varying behavior at some time in the future.
An example of a non-chaotic system is a pendulum, or a ball rolling down a hill, or a trajectory of a thrown object. In all of these examples, if you know the *initial state* of the object, you can predict all future behavior to some degree of precision. If I roll two balls down a hill, starting one of them *slightly* higher will mean they reach the bottom at *slightly* different times, regardless of how big the hill is.
Chaotic systems are different. If I take two double pendulums, and I raise one slightly higher than the other before I release it, then very soon that difference will explode into completely different behaviour for each system. [The wikipedia page](https://en.wikipedia.org/wiki/Double_pendulum) has a great illustration of this.
Importantly: a classical chaotic system *isn’t random*. The double pendulum is fully deterministic, and precisely modeled by a very simple set of equations. The same is true of a three-body orbit. The chaotic nature arises directly from those equations as a mathematical consequence. Chaos theory applies the theories of stochastic analysis, probability, and differential equations to study this kind of chaotic behavior.
The “chaos” (in a simplified way) comes from the fact that when multiple variables interact with each other, they can form “positive feedback loops”. For instance, a ball at the top of a hemispherical mound is stationary, but any push, no matter how small, will set if off and it will keep accelerating. Contrast this with a ball on a flat surface, where a small push sets it rolling at a small speed, or a ball at the bottom a crater, where a small push leads to small oscillations that returns the ball to equilibrium. Error terms in chaotic systems behave like the first ball, while error terms in nonchaotic systems behave like the other two.
It’s an insanely rich and beautiful theory, and there are certainly many complex patterns. You can ask, for instance, what these feedback loops look like, and how you can classify them into different patterns of behaviour. You can ask if there are equilibrium points, and whether they are stable. You can draw connections to fractal geometry. The theory of differential equations is the gateway to all this.
A differential equation is a way of modeling variables whose behavior depends on their own rate of change. For instance, because of air resistance, the acceleration of a falling object depends on its current speed. If the rate at which speed changes depends on its speed, how can we work out the behaviour of the object? This is the domain of differential equations, so it should be clear why differential equations are the language of chaos theory.
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