1, aperiodic: The butterfly model is a nonlinear dynamic system, and its state changes aperiodically with time. This means that the trajectory of the butterfly model does not appear repeatedly, but presents a complex and unpredictable pattern. This feature enables the butterfly model to simulate chaotic phenomena in the real world, such as weather system and ecosystem.
2. Sensitive dependence: The state of the butterfly model is very sensitive to the initial conditions, that is, a small initial value difference will lead to completely different trajectories of the system. This characteristic is called "butterfly effect", which means that in chaotic systems, small changes in initial conditions may lead to huge long-term effects.
3. Fractal structure: The trajectory of butterfly model has fractal structure, that is, similar complex patterns can be seen at any scale. Fractal structure is a remarkable feature of chaotic system, which shows that chaotic phenomena are self-similar and scale-free. By studying the fractal structure of butterfly model, we can better understand the essence and law of chaos.
The Origin of Butterfly Model
1. Butterfly model, also known as Lorentz attractor, is a chaos theory model proposed by American meteorologist Edward Lorenz in 1963. The inspiration of this model comes from some problems he found in the process of studying weather forecast.
At that time, Lorenz was trying to simulate the weather system with a computer in order to predict the future weather more accurately. However, he found that even if the initial conditions are very small, it may lead to great changes in the prediction results. This phenomenon is called "butterfly effect", which means that in chaotic systems, small changes in initial conditions may lead to huge long-term effects.
3. In order to describe this phenomenon vividly, Lorenz proposed a simplified mathematical model, namely the butterfly model. The model includes three interrelated equations, which describe the changes of temperature, humidity and wind speed in the atmosphere with time. By observing the evolution of this model.
4. Lorenz found a strange phenomenon: even under the same initial conditions, the evolution trajectory of the model will show a complex and unpredictable pattern. This mode is called Lorentz attractor, which has a fractal structure, that is, similar complex patterns can be seen at any scale.
5. The butterfly model lays the foundation for the development of chaos theory. It reveals some basic characteristics of chaos, such as aperiodicity, sensitive dependence and fractal structure. These characteristics enable the butterfly model to simulate many complex systems in the real world, such as weather system, ecosystem and financial system.