clarification needed] This is often the case when the dynamics on it are chaotic, but strange nonchaotic attractors also exist. Since the basin of attraction contains an open set containing A {\displaystyle A} , every point that is sufficiently close to A {\displaystyle A} is attracted to A {\displaystyle A} . For example, in physics, one frequency may dictate the rate at which a planet orbits a star while a second frequency describes the oscillations in the distance between the two bodies. The point x 0 is also a limit set, as trajectories converge to it; the point x 1 is not a limit set. Attracting period-3 cycle and its immediate basin of attraction for a certain parametrization of the Julia set, which iterates the function f( z)= z 2+ c.

starting values of 0 < x < 1 {\displaystyle 0
In addition, physical dynamic systems with at least one fixed point invariably have multiple fixed points and attractors due to the reality of dynamics in the physical world, including the nonlinear dynamics of stiction, friction, surface roughness, deformation (both elastic and plasticity), and even quantum mechanics. In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, [2] for a wide variety of starting conditions of the system. The parameters of a dynamic equation evolve as the equation is iterated, and the specific values may depend on the starting parameters.

The limit cycle of an ideal pendulum is not an example of a limit cycle attractor because its orbits are not isolated: in the phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit, so the former orbit is not attracting. The difference with the clock pendulum is that there, energy is injected by the escapement mechanism to maintain the cycle. If you think you’re not worthy enough, you may be struggling with imposter syndrome, which is a psychological phenomenon that many successful people deal with.

In the case of a marble on top of an inverted bowl, even if the bowl seems perfectly hemispherical, and the marble's spherical shape, are both much more complex surfaces when examined under a microscope, and their shapes change or deform during contact.

Until the 1960s, attractors were thought of as being simple geometric subsets of the phase space, like points, lines, surfaces, and simple regions of three-dimensional space. As soon as I allow the Universe to replace my fear-based beliefs with new perceptions, I receive a miracle. Debt can feel like drowning, but it can bring us financial knowledge and control as we pay things off. This is illustrated by the logistic map, which depending on its specific parameter value can have an attractor consisting of 1 point, 2 points, 2 n points, 3 points, 3×2 n points, 4 points, 5 points, or any given positive integer number of points. The attractor(s) for any value of the parameter r {\displaystyle r} are shown on the ordinate in the domain 0 < x < 1 {\displaystyle 0

In a discrete-time system, an attractor can take the form of a finite number of points that are visited in sequence. In the case of a marble on top of an inverted bowl (a hill), that point at the top of the bowl (hill) is a fixed point (equilibrium), but not an attractor (unstable equilibrium).

If we regard the evolution of a dynamical system as a series of transformations, then there may or may not be a point which remains fixed under each transformation.