An **attractor** is a concept used primarily in mathematics and physics, particularly in the study of dynamical systems. It refers to a set of values toward which a system tends to evolve over time. Here are some key points about attractors: 1. **Types of Attractors**: - **Fixed Point Attractors**: These are single points in state space. If the system's state is near this point, it will eventually converge to it.
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An attractor is an invariant set approached by nearby trajectories of a dynamical system. A useful precise convention is a compact invariant set that is Lyapunov stable and attracts some open neighborhood: starting sufficiently close keeps the trajectory close for all future times, and the distance to the set tends to zero. Definitions differ over the stability, uniformity and minimality conditions, so those conditions should be stated when they affect an argument.
For the discrete dynamical system on , the fixed point zero is an attractor. If , then ; if is small, every later value is at least as small. Its basin of attraction is exactly . The other fixed point, one, is invariant but repelling: its derivative is two, and arbitrarily close starting values can move away.
An attracting limit cycle is an attractor consisting of a whole repeating orbit. A trajectory can approach that set while continuing to move around it, so attraction to a set does not mean convergence to one stationary point. A strange attractor can have more complicated geometry; attracting nearby trajectories alone does not imply chaos.