(1) The chaos we are discussing here generally evolves from an ordered state to a chaotic state, so it is called non-equilibrium chaos.
(2) Chaos is the inherent randomness of a deterministic system, which is very different from the random phenomena such as dice toss and coin toss that we used to know. The short-term behavior of chaotic systems can be known, and the results are uncertain only after long-term evolution.
(3) The sensitive dependence of chaos on initial values. In linear systems, small disturbances will only produce small deviations in the results, but for chaotic systems, it is "a drop in the bucket, a thousand miles a day."
(4) Chaos is not a simple disorder, nor is it an order in the usual sense. First of all, chaotic motion is a typical aperiodic motion, which breaks the symmetry of periodic motion, and the breaking of symmetry essentially means the improvement of order, so chaotic motion is another type of order; The system behavior in the chaotic region is not really a mess. The chaotic spectrum itself has an infinite internal structure, in which various periodic windows are nested, and aperiodic and periodic are inextricably intertwined, indicating that chaotic behavior is a non-mediocre order; Infinite nested structure in chaos has the invariance of scale transformation, and the structure is similar to the whole after local amplification, and this self-similarity is also symmetrical in a sense. Therefore, chaos can be regarded as an ordered state with higher symmetry characteristics. Secondly, non-equilibrium chaos follows some laws: strange attractor behavior. Attractor is a group of state points that describe the state of mechanical system in phase space, and these points or point sets have an attractive effect on the motion trajectory of the system phase space; And some points, which the state can't reach, are called repulsion. The trajectory starting from any point in the phase space is always closer to an attractor and far away from the repulsive attractor. Chaotic attractors are different from those of general systems. When the phase trajectory of a chaotic system enters the attractor, two tracks that are very close to each other will be exponentially separated. On the one hand, the evolution of the state will eventually enter the attractor, on the other hand, the sensitive dependence of the initial value makes the system show random characteristics and form a contradictory unity.
Chaos is by no means an interesting mathematical phenomenon. Chaos is a more common phenomenon than order. It gives us a deeper understanding of the material world, opens a way for us to study the complexity of nature, and also triggers some philosophical thinking on the epistemology of the material world.
6.2 Philosophical thinking
The chaos theory of 1 provides people with more things than order and stability. In the words of Hamlet, there are more things in the world than your philosophy imagined. Chaos makes people understand that a complete description of nature must include complex behaviors.
Chaos theory forces us to face up to our limitations. Usually, our perceptual knowledge of the world is limited by our knowledge of nature. The concept of chaos will change our view of the world and liberate us from the bell-shaped universe, especially in the contradictory relationship between decision and randomness, necessity and contingency, order and disorder, stability and instability, simplicity and complexity, part and whole, and the conditions and mechanisms of dialectical transformation.
(1) Determinism and Indeterminism
In physics, there are two recognized views on understanding nature, one is the causal determinism established by Newton's classical mechanics, and the other is the probability theory of the development of statistical mechanics and quantum mechanics. These two laws are tested on different objects.
Chaos is unique in that it skillfully combines the disorder of expression with the internal decision mechanism, and chaos is synonymous with internal randomness. "Decisive chaos" shows that there is a bridge between decisiveness and randomness, which greatly enriches our understanding of the basic categories of dialectics: contingency and inevitability. First of all, chaos follows the uncertainty principle of quantum mechanics, which once again implies that contingency is not insignificant in science. Secondly, chaos means that for some decisive equations, our ability to predict the future is fundamentally limited, and the uncertainty of initial measurement will extend to the whole attractor. Chaos combines determinism and randomness, which is both accidental and inevitable. It is proved that there is a strange disorder hidden behind the superficial order, and a more strange order hidden in the depths of disorder.
(2) Stability and instability
No matter how chaotic chaos is, it can be described by attractors, and the size of attractors is limited, so random and disorderly movements can only occupy a limited measure space. The two orbits of chaotic attractors should be exponentially separated, mutually exclusive and antagonistic, and kept in a limited measure space, that is, restricted by attractors, thus forming a perfect unity of opposites of attraction and repulsion. All the states at the attractor in the system are close to the attractor, which reflects the "stable" side of the system movement. Once they reach the attractor, their movements repel each other, corresponding to the "unstable" side, and "stability" and instability form a contradictory unity.
Chaos theory brings us closer to reality.
Nature is a unified whole. In natural science, there are two sets of description systems: determinism and probability theory. The scientific tradition since Newton respects the determinism system, while statistical mechanics focuses on probability description. However, complete determinism and pure probability theory are abstract limit cases, and the real nature is somewhere in between. The study of chaos will help us to understand the world from a more practical perspective, free us from the deep-rooted artificial opposition between determinism and probability theory, and people's understanding of these philosophical categories of contingency and inevitability will also deepen.