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In the last paragraph I explain my own opinion.

Introduction

The article starts with the following sentence.
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations.
To emphasize complex dynamical systems is a pity, because what a physisist want to understand are physical systems.
This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere.
All these systems are physical processes.
Much of modern research is focused on the study of chaotic systems.
All processes are chaotic systems. Its all, what is in the name.

1. Overview

Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems.
This reflects a physical and mathematical approach.
The only corect approach is to observe the (dynamical) system under consideration.
A mathematical approach allow you to study systems in general, but most probably does not answer the system under consideration.
Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does the long-term behavior of the system depend on its initial condition?"
Defining the equations which describe a (dynamical) is the most important and most difficult, if not impossible.
That includes to find the solutions, which require tuning in order to match the observations.

4.2 Chaos theory

Chaos theory describes the behavior of certain dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears random. This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply chaos.
From a physical point of view: there are no chaotic system.
Mathematical simulations, which show chaotic behavior, never show a realistic physical system, based on realistic initial conditions.
The outcome of a roulette depents on the initial conditions i.e. the angle of throwing the ball and the speed of the ball. To call the trajectory of the ball chaotic does not make sense. Its all in the name. The purpose of the game is that each outcome has the same chance and that completely depends how the roulette is designed. In principle the croupier has the possiblity to influence the outcome.

Reflection 1 - Dynamical systems theory.

All processes in nature are dynamical systems. They are subject of change.
All processes in nature are also unstable. That does not mean you cannot study a stable process. In principle a stable process exists forever. For example a stable system consists of two identical objects which both have the same rotational velocity. And if they have the correct speed they will rotate both in a circle. And assuming the objects follow Newton's Law this rotation will exist for ever.
But that is a strictly mathematical approach.

The whole issue is this picture accordingly to the reality?
When you observe the universe, our galaxy, or individual stars, you will realize that many stars are binary systems and that trajectories are not in the shape of a cirlce but an ellipse. This means the masses of the two stars are different. In fact I should have written the calculted masses using Newton's Law.
But that is also not enough. A detailed description is required how this calcultion is performed, such that others can repeat this experiment. In this case the calculation depends on two type of parameters: The positions of the two stars at a sequence of equally time spaced observations and the time of this observations.
One strategy to measure the positions is to define a 3D grid, consisting of 3 sets of equally spaced rods in the x,y and z direction. How smaller the rods, how finer the grid, how more accurate the positions can be measured. At the cross points of the rods there is a clock. The same rule applies: How finer the grid, how more accurate the time can be mesured. Using these observations and Newton's Law the masses can be measured.
But does this calculation explain why the two stars rotate around each other? What the calculation demonstrates that Newton's Law can be used to predict the behavior of both stars in the future and because Newton's Law involves forces, that forces are the basics of our understanding of the behavior of stellar systems. What is important that these forces are the same and when that is the case the trajectories are stable.
In reality this is not true, because the masses of both stars can change and as a result the trajectories will deviate from the past and specific will deviate from the predicted positions. One reason of this change can be the merging in due time, of each star with smaller objects. To solve that also these smaller objects have to included in the observations and calculations.
This mismatch will be larger the more stars are involved and the longer the timeframe.

Reflection 2 - The speed of light.

In Reflection 1, the strategy #1 to measure the positions of an object, is the 3D grid. A different strategy #2 is to use light rays.
The primary reason of this reflection is to identify the problems involved.

The biggest problem, when we want is to understand the whole of the universe, most of that is complete invisible to us because of distance. For the rest, what we see is only the present state of the stars in our immediate surroundings. All the rest of the present state is at a far distance and presently invisible to us, because it takes time for these stars, that, light emitted by these stars, becomes visible to us. When we observe these stars they can be used as observations in the past.
When you use a grid and clocks at each grid point, as in #1, you 'don't have' this problem because assuming all clocks run synchronised, all the observers, near each clock, together can decide for each star which clock is the closest and calculate the clock's position.

The problem with strategy #2 is that this requires the speed of light. That means you need the speed of light for the whole of the universe. The calculation of the speed of light involves the same strategy as the calculation of the speed of a star. That means you must establish two observation points to monitor the flash and two clocks two measure the events when the flash passes each of these points. Simple mathematics can be used to calculate the speed of light. The major problem is that the two clocks should be synchronized.
 ``` . . *A1 . H1* . . . / . . . / . . * . / * . . . / . . . / . .* . / * . . . / . . . / .G *G1 .H / * .\ .\ /. / . \ . \ / . / . \ *. \ / . / * . \ . \ / . / . \ . \ / ./ . \ * . . / * . \ . / \ /. . \ . / \ / . . * . / F1/ . * . \ . / / \ . . \./ / \. . * F. / . * . /.\ / .\ . / . \ / . \ . * . . . . \ * . / . / \ . \ . / ./ \ . \ . * / \ . *E1 . / /. \ . / . / / . \ . / . / * / . \ . / * . / / . \ . / ./ / . \./ .D * / . E. * .\ / . /. . \ / . / . . *D1 . / . * . \ . / . . \ . / . . * \ . / . * . \ . / . . \ . / . .* . . / .* . \ . / . . \./ . ................................ A C B v=0 --->v>0 ``` The picture at the left shows two mirrors at the two points A and B C is the point at the middle between the two points. From that point two flashes of light are emitted: One towards mirror A. This signal is reflected at point D. This event is also used to start a clock at mirror A. One towards mirror B. This signal is reflected at point E, This event is also used to start a clock at mirror B. Both clocks at mirror A and mirror B are now synchronised. Both reflected signals meet each other at point F. Thereafter The signal C E F will meet mirror A at point G, which services as a stop event. The signal C D F will meet mirror B at point H, which also services as a stop event. Observations should reveal that the time on the clock at mirror A is the same as the time on the clock at mirror B. That means this picture is a correct description of the physical situation. This implies that the distance between the clocks is constant. This also implies that the speed of light for both signals is the same What the left side of the picture shows is the situation for two clocks at rest. The right side shows the situation when both clocks move towards the right. mirror A is now represent by red *'s. This is a line from A to A1 mirror B is now represent by red *'s. This is a line from B to H1 C is the point at the middle between the two points. From that point two flashes of light are emitted: One towards mirror A. This signal is reflected at point D1. This event is used to start a clock at mirror A. One towards mirror B. This signal is reflected at point E1. This event is used to start a clock at mirror B. Both clocks at mirror A and mirror B are now assumed to be synchronised. Both reflected signals meet each other at point F1. Thereafter The signal C E1 F1 will meet mirror A at point G1, and stop the clock at A The signal C D1 F1 will meet mirror B at point H1, and stop the clock at B What the drawing shows is that the line-segment D1H1 is larger than the line-segment E1G1. This means that the time (dt) of clock B is larger as time (dt) of clock A. This in turn means that the calculated speed of light is different. The important point is that left side of the picture is symmetric, while the right side is not. In the left part both flashes from A to B and from B to A have travelled the same distance. In the right side this is not. The distance from A to B is longer than the distance from B to A. When you take that into account the calculated speed of light is the same. The problem is this calculation requires the speed v of both clocks towards the right, which is not known. This makes it impossible to calculate the speed of light in one direction To declaire the speed of light a constant is a soft solution. It should be mentioned that the speed v of an observer has physical nothing to do with the speed of light. The observer can have any speed to the left or to the right. Also the speed of light has nothing to do with the speed of the source of the light flash. If at the moment of emission at a certain point, there are two sources: one at rest and one moving, both will generate the same sphere of light, with the same center.

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Created: 20 August 2021

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