What Is a Strange Attractor?

The attractor theory of the system is a scientific theory about attractors. It is an important part of chaos. The ultimate state of the evolution process, that is, the goal state, has the following characteristics: ultimate polarity, stability, and attractiveness. So what is an attractor? Attractor is a mathematical concept that describes the type of convergence of motion, which exists on the phase plane. In short, the attractor refers to such a set. When time approaches infinity, all orbits of an unsteady flow starting on any bounded set tend to it. Such a set has a very complex geometric structure. Because attractors are inseparable from chaos, it is necessary to understand the nature of the set of attractors in order to better understand the flow they describe, and to reveal the law and structure of chaos.

Strange attractors are a special class of attractors. The phase movements in the attraction area are approaching the attractor. The phase movements inside the attractor are aperiodic and have the following basic characteristics:

1. The system locally has a positive Lyapunov exponent, so that its phase points are separated at an exponential speed at least after being fairly close.

2. Any two-phase point can be arbitrarily approached after a sufficient time.

3. The attractor set has a fractal structure.

Well-known strange attractors include Lorentz attractors describing meteorological behavior, chaotic attractors of logical iterative maps, attractors of two-dimensional Ernon maps, etc.). ^[1]

From phase

Conservative systems have no attractors because the phase volume is always the same.

In general, non-linear systems may have mediocre attractors of various dimensions, such as 0,1,2 ... High-dimensional attractors are most likely to have quasi-periodic motion rather than periodic vibration. However, since the work of Ruelle and Takens, it has become increasingly clear that, in general, it is unlikely that quasi-periodic orbits will become attractors. child. Strange attractors are another important feature of chaotic phenomena in dissipative systems. Simply put, the strange attractor is the phase space (for continuous dynamic systems, at least

In the theory of dynamic systems, repellers are also called sources, and attractors are also called sinks. All meaningful tracks always flow from source to sink. A system in an unstable state of affairs is also settled with the status quo, and has no incentive to change the status quo. But they are not attractive to nearby orbits, but rather repulsive. Once the disturbance leaves the system out of this steady state, the repulsive force will move any orbit away from the steady state. For this reason, unstable nodes, focal points, limit cycles, and toruses are called repellers. The study of exclusion is also an important part of the attractor theory.