The sensitivity of 1. depends on the initial conditions: this means that the initial state of the system does not need to deviate from a certain order of magnitude. The smaller the deviation, the exponential increase will occur with the passage of time, making it possible to have a completely unpredictable state. This is why the chaotic model is defined as "unpredictable".
2. Steady state and convergence: Although the nonlinear dynamics of chaotic models are difficult to predict, in some cases, they tend to be steady state or converge according to specific laws. For example, within a certain parameter range, some dynamic models will continue to evolve in a stable and predictable way.
3. Multi-periodicity: Chaotic systems are usually multi-periodic, that is, the behavior of such systems will become periodic and then chaotic. A simple example is a smooth slope, and the rolling of the ball is a multi-period motion.
4. Fractal dimension: Chaotic systems usually do not conform to the basic assumptions of Euclidean geometry. They have a strange fractal structure, that is, some parts are more complex than others, and this complexity can be described in a higher dimension. In chaos science, this dimension is called "fractional dimension", which can help us better understand and predict the behavior of nonlinear dynamic systems.
These conclusions highlight the complexity and unpredictability of chaotic model, but also reveal some possible laws and trends. In current science and engineering, chaos theory has a wide range of applications, including weather forecasting, financial markets, fluid mechanics and other research fields.
The brief introduction of butterfly theorem is as follows:
Butterfly theorem is one of the most wonderful results in ancient Euclidean plane geometry. This proposition first appeared in 18 15 and was proved by W.G. Horner.
The name "Butterfly Theorem" first appeared in the February issue of American Mathematical Monthly (1944), with the title like a butterfly. There are countless proofs of this theorem, math lovers are still studying it, and there are various deformations in the exam.