"It is impossible for nature herself to 'know' the state of the Universe to the required precision."
In John Gribbin's "Q is for Quantum, Particle Physics from A to Z", entry for 'Chaos'
Precision and predictability. Predicting the outcome a certain behaviour in a system would lead to, branding a behaviour as unpredictable or not. Leading to chaos or order, the eternal conundrum. Precise calculations required of the state in a system, the parametric descriptors of the state, the system is, that define the conditions prevailing in the system. Parameters as quantities to be measured, assigned to be completed by a number-crunching entity, a calculator, hand-held or not, which its operators employ to produce a value which would feed into their formulas describing functions defining future states of systems under observation.
Nature herself employs a similar, but attuned to its overwhelming size, cosmic calculator. Its role to achieve a similar feat, that the human individual counterparts attempt to elucidate observed systems states, discerning their lapse into chaos or not.
Cosmic calculator, emphasized entity to produce the numbers at a cosmic scale dealing with infinities that will be fed into the laws of nature to determine the fate of the universe in every time scale possible, envisaged, imagined.
A tool employed in understanding the workings of nature, the notion of position the meaning conferred and the devise of numbers to ascertain such meaning, the relationships existing between quantities expressed as parameters as if nature herself relies upon the notions human individuals invented to make meaningful nature to themselves.
Human individuals in their efforts to understand nature, their methods employed mirroring nature's own workings, out of the same blueprint according to the law of self-similarity. Nature herself is unable to confer the precision required, amenable to predictability, predictability the essence of evolution in a cosmic scale. Nature attempts to predict the outcome of the laws is endowed with, predictability determines the very structures favoured by the laws themselves. And nature runs into trouble as she is faced with a daunting task even for an entity as mighty as her. Precision evades her in the face of irrational numbers.
"To take just one simple example, known to the Ancient Greeks, the very concept of labelling the points on a straight line is an approximation to the truth. To label every point on a line (no matter how short the line is) you would need an infinite number of numbers. In the interval from 0 to 1, many of the points can be expressed in terms of the familiar fractions, such as 1/2 or 1/4 (or by less familiar fractions, such as 137/731). But some of the points can only be represented by 'irrational' numbers which cannot be expressed as the ratio of two integers in a fraction, and which require an infinite number of digits after the decimal point in order to be described fully."
An infinite number of digits, precision evaded, predictability un-achievable, unpredictable behaviour, chaos itself prevailing. In essence systems at that breakpoint separate into chaotic and non-chaotic behaviour. For what a position stabilises into a halt, a stop, a whole fraction, such a point provides an anchor for the state of the system to materialise. Its linear evolution becomes possible, a discernible pattern emerges. For any other position where its precision is impossible due to the appearance of irrational numbers in the presented 'solutions', confers unpredictability, system prevailing behaviour chaotic.
Fibonacci numbers, their implication in chaos, their involvement in the make-up of the irrational solutions might raise notions about their implied involvement.
Critical values of defining parameters in a system, the points where parameter relationships are defined by solutions running into the realm of irrational numbers. Precision affected, predictability compromised, chaotic behaviour ensues.
Chaotic behaviour ends, as the device of cosmic calculator, in running through the irrational number calculated provides a solution amenable to states involved, order emerges.