Chaos theory is a method that combines qualitative thinking and quantitative analysis. It is used to explore behaviors that cannot be explained and predicted by a single data relationship in a dynamic system, but must be explained and predicted by a whole and continuous data relationship.
“The original state of everything is a bunch of seemingly unrelated fragments, but after this chaotic state ends, these inorganic fragments will organically merge into a whole.”
The term chaos originally referred to the state of chaos before the universe was formed. Ancient Greek philosophers held the theory of chaos on the origin of the universe, claiming that the universe gradually formed the orderly world from the beginning of chaos. In the orderly universe, Western natural scientists have discovered many laws in nature one by one after long-term research, such as the well-known gravity, the principle of leverage, and the theory of relativity. These natural laws can be described by a single mathematical formula, and the behavior of objects can be accurately predicted based on this formula.
For nearly half a century, scientists have discovered that many natural phenomena can be transformed into simple mathematical formulas, but their behavior cannot be predicted. For example, meteorologist Edward Lorenz discovered that simple thermal convection phenomena can actually cause unimaginable meteorological changes, resulting in the so-called “butterfly effect.” In the 1960s, American mathematician Stephen Smale discovered that the behavior of certain objects undergoes a certain regularity. After the change, the subsequent development has no certain trajectory to follow, showing a chaotic state of disorder.
What is chaos theory, the definition and characteristics of chaos theory-Yuhui International
Chaos Characteristics of Chaos Theory
(1) Randomness: The chaotic state of a system is an irregular behavior caused by the dynamic randomness of the system, which is often called internal randomness. For example, in one-dimensional nonlinear mapping, even if the mathematical model describing the evolution of the system does not contain any additional random terms, even if the control parameters and initial values are determined, the behavior of the system in the chaotic zone is still random. . This kind of randomness is spontaneously generated inside the system, and has a completely different source and mechanism from external randomness. It is obviously an internal randomness and mechanism function within the deterministic system. The local instability in the system is the characteristic of internal randomness and the reason for the sensitivity to the initial value.
(2) Sensitivity: The chaotic motion of the system, whether it is discrete or continuous, low-dimensional or high-dimensional, conservative or dissipative. Both time evolution and spatial distribution have a basic feature, that is, the system’s motion orbit is extremely sensitive to the initial value. This sensitivity, on the one hand, reflects the strong influence of the stochastic system movement trend in the nonlinear dynamic system; on the other hand, it will also lead to the unpredictability of the long-term time behavior of the system. The so-called “butterfly effect” proposed by meteorologist Lorentz is a prominent and vivid explanation of this sensitivity.
(3) Fractal dimension: Chaos has the nature of fractal dimension, which means that the geometric form of the system motion orbit in the phase space can be described by fractal dimension. For example, the fractal dimension of the Koch snowflake curve is 1.26; the fractal dimension of the Lorentz model describing atmospheric chaos is 2.06. The chaotic motion of the system is infinitely entangled, folded and kinked in the phase space, forming a self-similar structure with infinite levels.
(4) Universality: When the system tends to chaos, the displayed characteristics have universal meaning. Its characteristics do not change due to differences in specific systems and system motion equations. Such systems are all related to Feigenbaum’s constant.
(5) Scaling law: Chaos is a non-periodic ordered state, with infinite levels of self-similar structure, and there is a scale-free area. As long as the accuracy of the numerical calculation or the resolution of the experiment is high enough, small-scale chaotic orderly movement patterns can be found from it, so it has the nature of the scaling law. For example, in the process of period-doubling bifurcation, the infinitely nested similar structure of chaotic attractors, from the perspective of hierarchical relationship, has structural self-similarity and structural invariance under scale transformation, thus showing order.