Understanding Chaos Theory, Fractal Math, and Nature

[shunyadragon]

There seems to be a lot of misunderstanding concerning the nature of the outcomes of cause-and-effect events and the nature of our physical existence including mutations, evolution, Quantum Mechanics,, and the question of randomness in how the variation of the outcomes of cause-and-effect events.

Chaos Theory is basically based on non-linear math with more than one variable that we learned in high school and demonstrates the fractal relationship between the variables involved. Scientists developed the Chaos Theory to make predictions with complex natural and applied science situations and problems with many variables The lesson from Chaos Theory is that everything with fractal properties will always be unpredictable. The outcome of cause-and-effect events will always be predictable only within a possible range of outcomes.

An example in everyday life is weather predictions. There are many complex variables in predicting the weather, and computer models use Chaos Theory to make weather predictions. Weather predictions can never be perfect, but . . . They run different models many times with a range of factors to come up with a range of possible outcomes for weather within the possible range of prediction. Because the variable factors change over time the ability to accurately predict the weather diminishes with time. Nonetheless, even long-term weather predictions are becoming more accurate as the sophistication of the programs increases. The scientist also programs in the history of weather to help predict the weather.

The conclusion is the nature of our physical existence is not random except for the timing of individual cause-and-effect events. What is described as sensitivity problems in Chaos models such as weather is the extreme number of variables that only allow for predictions within a possible range of outcomes.

As previously referenced the best primer is Chaos: Making a New Science by James Gleick.

What we will explore is the deterministic predictability of the nature of our physical existence versus the problem of the unpredictability in the outcome of cause-and-effect events within a range of outcomes.

Chaos theory explained: A deep dive into an unpredictable universe

Chaos theory is why we will never be able to perfectly predict the weather.

www.space.com

Chaos theory explained: A deep dive into an unpredictable universe​

By Paul Sutter
published March 18, 2022
Chaos theory is why we will never be able to perfectly predict the weather.

Chaos theory explains the behavior of dynamic systems like weather, which are extremely sensitive to initial conditions.

It would be really nice to know the weather forecast not just a week in advance but a month or even a year into the future. But predicting the weather presents several tricky problems that we will never be able to entirely solve.

The reason why isn't just complexity — scientists regularly tackle complex problems with ease — it's something much more fundamental. It's something discovered in the mid-20th century: the truth that we live in a chaotic universe that, in many ways, is completely unpredictable. But hidden deep within that chaos are surprising patterns, patterns that, if we are ever able to fully understand them, might lead to some deeper revelations.

One of the beautiful things about physics is that it's deterministic. If you know all the properties of a system (where "system" can mean anything from a single particle in a box to weather patterns on the Earth or even the evolution of the universe itself) and you know the laws of physics, then you can perfectly predict the future. You know how the system will evolve from state to state as time marches forward. This is determinism. This is what allows physicists to make predictions about how particles and the weather and the entire universe will evolve.

It turns out, though, that nature can be both deterministic and unpredictable. We first got hints of this way back in the 1800s, when the king of Sweden offered a prize to anyone who could solve the so-called three-body problem. This problem deals with predicting motion according to Isaac Newton's Laws. If two objects in the solar system are interacting only through gravity, then Newton's laws tell you exactly how those two objects will behave well into the future. But if you add a third body and let that play the gravitational game, too, then there is no solution and you won't be able to predict the future of that system.

rench mathematician Henri Poincaré (arguably a supergenius) won the prize without actually solving the problem. Instead of solving it, he wrote about the problem, describing all the reasons why it couldn't be solved. One of the most important reasons he highlighted was how small differences at the beginning of the system would lead to big differences at the end.


This idea was largely put to rest, and physicists continued, assuming that the universe was deterministic. That is, they did until the mid-20th century when mathematician Edward Lorenz was studying a simple model of the Earth's weather on an early computer. When he stopped and restarted his simulation, he ended up with wildly different results, which shouldn't be a thing. He was putting in the same inputs, and he was solving the problem on a computer, and computers are good at doing the same thing over and over again.


What he found was a surprising sensitivity to the initial conditions. One tiny rounding error, no more than 1 part in a million, would lead to a completely different behavior of the weather in his model.

 

[viole]

This idea was largely put to rest, and physicists continued, assuming that the universe was deterministic.

Chaos theory, does not defeat determinism. It defeats predictability, to a certain extent. Not the same things.

Ciao

- viole

 

[shunyadragon]

Chaos theory, does not defeat determinism. It defeats predictability, to a certain extent. Not the same things.

Ciao

- viole

Read the article more carefully. Yes, Chaos Theory does not defeat determinism and science depends on our physical existence being deterministic. Before modern Physics, Theory of Relativity, and Quantum Mechanics the rule of science was Newtonian determinism. The article stated that both determinism and unpredictability are now fundamental principles of science.

When I cited the article I did say I had a few problems in the wording. The reference to unpredictability is incomplete, and the proper wording is that the unpredictability of the outcome of cause-and-effect events is within a deterministic range of outcomes.

Note" The article stated: 'It turns out, though, that nature can be both deterministic and unpredictable. '

 

[bobhikes]

Read the article more carefully. Yes, Chaos Theory does not defeat determinism and science depends on our physical existence being deterministic. Before modern Physics, Theory of Relativity, and Quantum Mechanics the rule of science was Newtonian determinism. The article stated that both determinism and unpredictability are now fundamental principles of science.

When I cited the article I did say I had a few problems in the wording. The reference to unpredictability is incomplete, and the proper wording is that the unpredictability of the outcome of cause-and-effect events is within a deterministic range of outcomes.

Note" The article stated: 'It turns out, though, that nature can be both deterministic and unpredictable. '

I would like to comment on this quote "It turns out, though, that nature can be both deterministic and unpredictable" It is only unpredictable however intelligence patternizes nature so that we can easily deal with it. We know that the quantum realm is random. We know that all materials are created from the quantum realm. We know that in every bit of time those materials change at random. Our mind builds a pattern of the material ignoring changes we can't see and discounting small changes we can see. Examples Bob's Car gets into an accident and is damaged. The damage is random and radically change bob's car but as long as it is fixable it is still Bobs Car. Bob himself was born in 1965, and no longer is physically the same or mentally the same and none of the changes were predictable yet he is still Bob to those that have known him throughout this time. If you place an ICE Cube on a table in the summer can you determine it will melt but when is it not the ice cube you placed there. The mind will latch on to it being that same Ice Cube until it becomes water even though its Shape, Size and quantity of Ice have all been changed unpredictably. In my opinion Determinism is the mind simplifying reality for survival purposes and it works well but reality is always and will always be quantumly random and you will never be able to determine an outcome or form 100%.

 

[shunyadragon]

I would like to comment on this quote "It turns out, though, that nature can be both deterministic and unpredictable" It is only unpredictable however intelligence patternizes nature so that we can easily deal with it. We know that the quantum realm is random. We know that all materials are created from the quantum realm. We know that in every bit of time those materials change at random. Our mind builds a pattern of the material ignoring changes we can't see and discounting small changes we can see. Examples Bob's Car gets into an accident and is damaged. The damage is random and radically change bob's car but as long as it is fixable it is still Bobs Car. Bob himself was born in 1965, and no longer is physically the same or mentally the same and none of the changes were predictable yet he is still Bob to those that have known him throughout this time. If you place an ICE Cube on a table in the summer can you determine it will melt but when is it not the ice cube you placed there. The mind will latch on to it being that same Ice Cube until it becomes water even though its Shape, Size and quantity of Ice have all been changed unpredictably. In my opinion Determinism is the mind simplifying reality for survival purposes and it works well but reality is always and will always be quantumly random and you will never be able to determine an outcome or form 100%.

You need to think this over and come back with references. Quantum Mechanics is not random. Just as the macroscale world is unpredictable and deterministic. but the outcome of individual events is predictable within a range that is limited by Natural Laws. If this were not the case we would not have laws and predictable principles that explain behavior on the smallest scale in terms of Quantum Mechanics. Just as on the macro scale, the timing of individual events is random, but the outcome is predictable in terms of a pattern within a range of possible outcomes.


Note the bold.

Start with the basics: Quantum chaos - Wikipedia


Quantum chaos is a branch of physics that studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?[1][2]

In seeking to address the basic question of quantum chaos, several approaches have been employed:

1. Development of methods for solving quantum problems where the perturbation cannot be considered small in perturbation theory and where quantum numbers are large.
2. Correlating statistical descriptions of eigenvalues (energy levels) with the classical behavior of the same Hamiltonian (system).
3. Study of probability distribution of individual eigenstates (see scars and Quantum ergodicity).
4. Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the dynamical system with quantum features.
5. Direct application of the correspondence principle.

History​

Experimental recurrence spectra of lithium in an electric field showing birth of quantum recurrences corresponding to bifurcations of classical orbits.[3]
During the first half of the twentieth century, chaotic behavior in mechanics was recognized (as in the three-body problem in celestial mechanics), but not well understood. The foundations of modern quantum mechanics were laid in that period, essentially leaving aside the issue of the quantum-classical correspondence in systems whose classical limit exhibit chaos.

Approaches​

Comparison of experimental and theoretical recurrence spectra of lithium in an electric field at a scaled energy of �=−3.0

.[4]
Questions related to the correspondence principle arise in many different branches of physics, ranging from nuclear to atomic, molecular and solid-state physics, and even to acoustics, microwaves and optics. However, classical-quantum correspondence in chaos theory is not always possible. Thus, some versions of the classical butterfly effect do not have counterparts in quantum mechanics.[5]

Important observations often associated with classically chaotic quantum systems are spectral level repulsion, dynamical localization in time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (as in scattering). In the semiclassical approach of quantum chaos, phenomena are identified in spectroscopy by analyzing the statistical distribution of spectral lines and by connecting spectral periodicities with classical orbits. Other phenomena show up in the time evolution of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular amplitude distributions.

Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes (see e.g. Dyson series). Simple and exact solutions are precluded by the fact that the system's constituents either influence each other in a complex way or depend on temporally varying external forces.

More to follow . . .

 

[shunyadragon]

Signatures of Quantum Mechanics in Chaotic Systems

We examine the quantum-classical correspondence from a classical perspective by discussing the potential for chaotic systems to support behaviors normally associated with quantum mechanical systems. Our main analytical tool is a chaotic system’s ...

www.ncbi.nlm.nih.gov

Signatures of Quantum Mechanics in Chaotic Systems​

Kevin M. Short1,* and Matthew A. Morena2
Author information Article notes Copyright and License information Disclaimer

Go to:

Abstract​

We examine the quantum-classical correspondence from a classical perspective by discussing the potential for chaotic systems to support behaviors normally associated with quantum mechanical systems. Our main analytical tool is a chaotic system’s set of cupolets, which are highly-accurate stabilizations of its unstable periodic orbits. Our discussion is motivated by the bound or entangled states that we have recently detected between interacting chaotic systems, wherein pairs of cupolets are induced into a state of mutually-sustaining stabilization that can be maintained without external controls. This state is known as chaotic entanglement as it has been shown to exhibit several properties consistent with quantum entanglement. For instance, should the interaction be disturbed, the chaotic entanglement would then be broken. In this paper, we further describe chaotic entanglement and go on to address the capacity for chaotic systems to exhibit other characteristics that are conventionally associated with quantum mechanics, namely analogs to wave function collapse, various entropy definitions, the superposition of states, and the measurement problem. In doing so, we argue that these characteristics need not be regarded exclusively as quantum mechanical. We also discuss several characteristics of quantum systems that are not fully compatible with chaotic entanglement and that make quantum entanglement unique.

1. Introduction​

Chaotic behavior is generally attributed to a sensitive dependence on initial conditions and is characterized by a positive maximal Lyapunov exponent that causes nearby trajectories to diverge from each other exponentially fast. Despite its ubiquity in classical physics, chaos is yet to be rigorously established within quantum settings. One explanation for this disparity is that, unlike chaotic or classical systems, whose states may be completely described by a set of dynamical variables, in quantum mechanics, conjugate observables such as position and momentum cannot take on well-defined values at the same time. Particle dynamics are instead determined in part by the uncertainity principle and by the linearity of the Schrödinger equation, which preserves the overlap between quantum states. In other words, the nonlinearity required for chaotic dynamics and the exponential divergence of neighboring trajectories are fundamentally incompatible with quantum mechanics in its present formulation.

In addition, however, much effort has recently been devoted to detecting signatures of chaos in quantum systems [1,2,3,4,5]. One such signature is the sensitivity of some quantum systems to perturbation. This has been experimentally observed in the decay in the overlap between quantum states that are evolving under slightly different Hamiltonians and is attributed to the positivity of a classically-derived Lyapunov exponent [1,4]. In fact, the rate of overlap decay is known to transpire at different rates depending on whether the evolution begins from initial conditions that correspond classically to chaotic versus regular regimes [5]. A second signature is quantum scarring, which refers to the scenario in which a quantum system’s associated wave function concentrates on paths that represent periodic orbits in the classical limit [6,7]. This phenomenon has been experimentally observed in several recent studies [8,9].
Entanglement in the purely quantum sense has also been observed to be a reliable indicator of classical chaos [10,11,12,13]. In Chaudhury et al.’s recent kicked top experiments of laser-cooled Cesium (133Cs) atoms, each atom’s initial state is followed for several periods of the “kicked” Hamiltonian, and the corresponding classical phase space reveals islands of regular motion surrounded by a sea of chaos [5]. When entropy is used to measure the entanglement, stronger entanglement is detected between entangled atoms whose states are initially prepared from chaotic regimes, whereas weaker entanglement is measured between atoms that evolve from regular regions. It is as if the quantum regime respects an underlying classical presence [13].

One feature of chaotic systems typically encountered in investigations is the infinite set of unstable periodic orbits (UPOs) that are found densely embedded in associated attractors. These orbits collectively provide a rich source of qualitative information about the parent chaotic system and are the focus of numerous theoretical and practical applications [14,15,16]. As a result, several control schemes have been designed to detect and stabilize these orbits [17,18,19]. In Section 2 of this paper, we discuss an adaptation of one particular control method that very efficiently stabilizes the cupolets of chaotic systems (Chaotic, Unstable, Periodic, Orbit-LETS) [20,21,22,23].

Cupolets are controlled and stabilized periodic orbits of a chaotic system that would normally be unstable without the presence of the control mechanism. These orbits represent approximations to the UPOs but are distinguished because their stabilization supports a one-to-one correspondence between a given sequence of controls and a specific cupolet, with each cupolet able to be generated independently of initial condition. In Section 4.2, we derive the functional form of a cupolet, which further establishes the suitability of cupolets for analyzing chaotic systems.

 

[bobhikes]

You need to think this over and come back with references. Quantum Mechanics is not random. Just as the macroscale world is unpredictable and deterministic. but the outcome of individual events is predictable within a range that is limited by Natural Laws. If this were not the case we would not have laws and predictable principles that explain behavior on the smallest scale in terms of Quantum Mechanics. Just as on the macro scale, the timing of individual events is random, but the outcome is predictable in terms of a pattern within a range of possible outcomes.

Yes the cat is Neither dead nor alive but in a quantum state and only through observation can we determine the state. The spin is either left or right, The electron state is either here or there. Until observation it is in a quantum state. All this goes along with what I said our intelligence frames the pattern as long as 99.994 percent of the time is correct then the outcome is deterministic. It works for us, allows us to create things, to live simpler lives and to still be surprised by unexpected events yet believe that we still have control. If it is only predictable in a range of possible outcomes, even if it is just 2 possible outcomes, that is still random. Scientists and Skeptics don't like that term. I can rewrite it as It is unpredictable for a range of only 2 possible outcomes and it means the same thing random between 2 number's or events.

I get it is hard to agree and I don't see any need to agree, we need patterns and determined events to survive in this world and everything being random at the basic level helps no-one, but if you evaluate science as a whole without bias it is the only conclusion you can make. Also 99.994 % accuracy is really good. For our life span over 80 years you will only be surprised 93 days.

 

[shunyadragon]

Yes the cat is Neither dead nor alive but in a quantum state and only through observation can we determine the state. The spin is either left or right, The electron state is either here or there. Until observation it is in a quantum state. All this goes along with what I said our intelligence frames the pattern as long as 99.994 percent of the time is correct then the outcome is deterministic. It works for us, allows us to create things, to live simpler lives and to still be surprised by unexpected events yet believe that we still have control. If it is only predictable in a range of possible outcomes, even if it is just 2 possible outcomes, that is still random. Scientists and Skeptics don't like that term. I can rewrite it as It is unpredictable for a range of only 2 possible outcomes and it means the same thing random between 2 number's or events.

I get it is hard to agree and I don't see any need to agree, we need patterns and determined events to survive in this world and everything being random at the basic level helps no-one, but if you evaluate science as a whole without bias it is the only conclusion you can make. Also 99.994 % accuracy is really good. For our life span over 80 years you will only be surprised 93 days.

You still apparently do not understand randomness, and have not read the references concerning the fractal nature of Quantum Mechanics.

No, two possible outcomes do not make anything random. With randomness, you cannot have a limited number of possibilities limited by Natural Laws and processes when the outcome shows a fractal pattern. It is too predictable for true randomness.

A common example is the through of dice whether one dice or more possible outcomes of each throw. Each throw of the dice can have only outcomes within the possible range of outcomes. Analysis of a series of dice throws has been determined to have a pattern of outcome that is fractal and predictable. Gambling casinos know the odds and patterns of dice throws and have an advantage over the customers, and fleece their customers for billions of dollars.

The evidence as referred to in the references is overwhelming that there are no 'unexpected outcomes' of cause and effect events out side the expected range of possible outcomes.

This is not just my opinion. You are arguing against actual scientific references.

 

[bobhikes]

You still apparently do not understand randomness, and have not read the references concerning the fractal nature of Quantum Mechanics.

No, two possible outcomes do not make anything random. With randomness, you cannot have a limited number of possibilities limited by Natural Laws and processes when the outcome shows a fractal pattern. It is too predictable for true randomness.

A common example is the through of dice whether one dice or more possible outcomes of each throw. Each throw of the dice can have only outcomes within the possible range of outcomes. Analysis of a series of dice throws has been determined to have a pattern of outcome that is fractal and predictable. Gambling casinos know the odds and patterns of dice throws and have an advantage over the customers, and fleece their customers for billions of dollars.

The evidence as referred to in the references is overwhelming that there are no 'unexpected outcomes' of cause and effect events out side the expected range of possible outcomes.

This is not just my opinion. You are arguing against actual scientific references.

There are many human definitions for everything but there are always only two states Something or Nothing, Light or No light, Spin or No spin. If you can't predetermine the state change then the state change is random. The randomness you are talking about is through infinite moments and then probability gets involved. Though you can predict your dice throw you cannot determine your dice throw. Each throw has its own elements and each of these elements are affected randomly. Strength of throw, Air quality, point of impact,...etc. Point of impact is the easiest to explain, the table itself is changing moment to moment based on temperature, moisture content of the air, lighting, sound ...etc. You can break these changes all the way down to the molecules, then the atoms that make up the molecules then the electrons that make up the atoms but you will never be able to determine the actual dice roll. When looking at the simple environment, The roller generally throws at the same strength, the dice are minimally changed the table is also minimally changed we determine a probability. This is not a factual representation but a guess based on data collected over a period of time. If you flip a coin long enough the data should get close to 50 % but in that data there will be long periods of time where it will be heads 100% and there is no guarantee in the next 10 flips it will be 50%.

Being able to predict things less than 100% is not determining them. Not being able to predict things because there are to many variables is an excuse. It is not an opinion it is a fact everything is randomly changing every moment. Predictability is only possible because of how much of everything there is. The larger the form the more possible predictability is. Its good for us that we can get by dealing with the macro world for the most part.

 

[shunyadragon]

There are many human definitions for everything but there are always only two states Something or Nothing, Light or No light, Spin or No spin. If you can't predetermine the state change then the state change is random. The randomness you are talking about is through infinite moments and then probability gets involved. Though you can predict your dice throw you cannot determine your dice throw. Each throw has its own elements and each of these elements are affected randomly. Strength of throw, Air quality, point of impact,...etc. Point of impact is the easiest to explain, the table itself is changing moment to moment based on temperature, moisture content of the air, lighting, sound ...etc. You can break these changes all the way down to the molecules, then the atoms that make up the molecules then the electrons that make up the atoms but you will never be able to determine the actual dice roll. When looking at the simple environment, The roller generally throws at the same strength, the dice are minimally changed the table is also minimally changed we determine a probability. This is not a factual representation but a guess based on data collected over a period of time. If you flip a coin long enough the data should get close to 50 % but in that data there will be long periods of time where it will be heads 100% and there is no guarantee in the next 10 flips it will be 50%.

Being able to predict things less than 100% is not determining them. Not being able to predict things because there are to many variables is an excuse. It is not an opinion it is a fact everything is randomly changing every moment. Predictability is only possible because of how much of everything there is. The larger the form the more possible predictability is. Its good for us that we can get by dealing with the macro world for the most part.

You still apparently do not understand randomness, and have not read the references concerning the fractal nature of Quantum Mechanics.

No, two possible outcomes do not make anything random. With randomness, you cannot have a limited number of possibilities limited by Natural Laws and processes when the outcome shows a fractal pattern. It is too predictable for true randomness.

The evidence as referred to in the references is overwhelming that there are no 'unexpected outcomes' of cause and effect events outside the expected range of possible outcomes.

This is not just my opinion. You are arguing against actual scientific references.

 

[bobhikes]

You still apparently do not understand randomness, and have not read the references concerning the fractal nature of Quantum Mechanics.

No, two possible outcomes do not make anything random. With randomness, you cannot have a limited number of possibilities limited by Natural Laws and processes when the outcome shows a fractal pattern. It is too predictable for true randomness.

The evidence as referred to in the references is overwhelming that there are no 'unexpected outcomes' of cause and effect events outside the expected range of possible outcomes.

This is not just my opinion. You are arguing against actual scientific references.

Have a nice day.

 

[shunyadragon]

Have a nice day.

Take a walk in the rain without your hat and experience reality as it is.

 

[bobhikes]

Take a walk in the rain without your hat and experience reality as it is.

I find that enjoying sometimes. Singing in the rain is a favorite movie of mine. So thanks for the happy thoughts.

 

[syo]

Modern science is scary.

 

[viole]

Note" The article stated: 'It turns out, though, that nature can be both deterministic and unpredictable. '

No objections there. almost

ciao

- viole

 

[shunyadragon]

No objections there. almost

ciao

- viole

I hope I qualified their use of unpredictability. It is not referring to 'randomness' except in the timing of events. The limited randomness in outcomes of cause-and-effect events is in the timing of each individual event, which would be unpredictable. The outcomes are predictable within a limited range of possible outcomes The pattern of outcomes over time is fractal and explained by Chaos Theory.