Here is the famous dynamic equation x=kx(1-x) drawn for values of k=[0,4]. The fixed point x=(k-1)/k by simple algebra is drawn in red. There is another fixed point at x = 0 which accounts for the starting part of the graph.
One way to visualise what is happening here is to see the logistic curve with the y=x line. As the dynamic system progresses the starting x value finds a y value through the function, which is then returned to the x axis which is the same as reflecting in the y=x. Using k=2 the start of the path is drawn in green. It will spiral down to the fixed point where the graph meets the y=x line.
A quick analysis of the differentials should be done as this will probably reveal the reason why certain values never find the fixed point. TODO.
Above is an analysis of the first iteration or Order 1. The Logistic equation for the second iteration is :
x = -k^2 (-1 + x) x (1 + k (-1 + x) x)
And this has 4 fixed points. The original 2 from the 1st Order and two new fixed points:
x = 0
x = (-1 + k)/k,
x = (k + k^2 - k Sqrt[-3 - 2 k + k^2])/(2 k^2)
x = (k + k^2 + k Sqrt[-3 - 2 k + k^2])/(2 k^2)
Drawn in red here:
Recent note on the fractal nature of the brain. Its an interesting analogy that the self is very much the fixed point around which our chaotic lives orbit.
Starting into our own eyes in a mirror and trying to fathom that "what we are seeing is doing the seeing" or more abstract in Descartes "what is doing the thinking is the thinking" these kinds of mental operations are just dynamic systems where the output maps at least to the input. I say "maps" because obviously you cannot see "the self" anymore than a camera can take a picture of the image within the camera, or that you cannot see light until it falls on your retina. The fact that there is "only one way" to be "in the picture" is the key to self. No one can share your experience is the key to the dynamic system. There is no leakage. Consider mirrors exactly facing one another. It is possible that a photon of light gets trapped and bounces back and forward. Being trapped between the mirrors is the experience of a human staring directly into a mirror and seeing "them self." But this "self" is a mental process deduced from the situation. A camera taking a photo of itself in a mirror does not get this experience, just the human looking at the picture so taken. Its an isomorphism that is set up in the thinking mind. What I am seeing, maps to "me." Now that thought is a dynamic system with a fixed point that is "self." Interestingly you never get to "self" it is just the fascinating possibility around which you orbit. And soon like Descartes the orbit ends, and we escape the meditation. But like Descartes we realise that we can get that same experience looking at anything as long as we set up the isomorphism. He ended up realising that he didn't need to just meditate on thoughts of doubt, but any thought would do and eventually any perception was proof of existence. He was back where he started after spinning down almost to the fixed point. I wonder whether Tat Tvam Asi is similar. The realisation that the Atman and the Brahman are the same. What you see in the mirror (the world) is the same as what is looking (the self). Certainly there is only one process really, its not like you are jumping between outside and inside when looking in a mirror there is just the way things are. But there is conserably spinning to begin with.
Now all this crazy speculation here is not seeking to discover anything!! The point is to show that all possible mental processes like this go no where. What we are really trying to do here is to let go and just see mental processes and what they are. Not seeking to come to an end point and discovery. Mankind has been thinking and discovering for millennia. There is the belief that it is all going somewhere and one day we will come face to face with God; the library will be complete and mankind will know everything. We will be complete. Existence will have laid bare all its secrets. But surly we sense the problem here, that whatever we find will just raise yet more questions. Why for instance was there a limit to knowledge at all? Thinking never ends until you end it yourself! Not that we become a corpse but we let go and let the world just be. But thinking still exists, its useful, its what our brains do, so we don't become a stone either. But we realise that the "self" at the heart of our enquiries into the world is a phantom. And the Hindu's would add that the idea of a fixed soul to the Universe is a phantom too.
Theological detail. The Hindus wouldn't say this actually, its the Buddhists. But by the time the difference is important the point has been made.
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