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This Concept Map, created with IHMC CmapTools, has information related to: b1. Attractor, A basin of attraction is that portion of state space from which all trajectories gravitate to the attractor. ???? Trajectories might originate anywhere within the attractor’s basin of attraction. They then gravitate toward the attractor over time., The periodic attractor acts as a limit or goal for any trajectories that originate within the basin of attraction. For that reason, a periodic attractor is often called a limit cycle. ???? Periodic, or limit cycle, attractors (but not point or toroidal attractors) can also appear in the chaotic domain., NON-CHAOTOC attractors (k less than 3) are of three types; point; periodic; and toroidal. They consist of regular predictable trajectories. That is, prediction of long-term evolutions or recurrent behavior can be quite accurate, even far into the future. ???? The periodic attractor acts as a limit or goal for any trajectories that originate within the basin of attraction. For that reason, a periodic attractor is often called a limit cycle., Two neighboring trajectories, once “on” their attractors, stay a calculably close distance from one another as they evolve. ???? If the chosen starting point x0 already lies on the attractor, then the trajectory stays there forever and never leaves the attractor., The quadratic equation and the quadratic map (xt+1 = k – xt2) and corresponding chaos reveal several interesting generalities. ???? The logistic equation’s chaotic trajectory, goes to many sites on its attractor. It does show some favoritism for some points give a certain parameter, and it also becomes periodic at some parameter values, CHAOTIC or “STRANGE” attractors (k 3.57 to 4) arise only after the onset of chaos. They take interesting and complex shapes in phase space, and there is no way of predicting their long-term evolution. ???? The curious shapes of attractors are the distorted versions of the basic line, surface, or volume of the trajectory of the system in phase space., b1. Attractor ???? The most fundamental concept in chaos analysis is that of the attractor., If the chosen starting point x0 already lies on the attractor, then the trajectory stays there forever and never leaves the attractor. ???? Once that steady-state condition arrives, the point attractor is independent of time. It stays fixed, and the system no longer evolves. In phase space the system is static., On a standard phase space plot, an attractor is a summary of a system’s long-term behavior. It is a compact, global picture of all of a system’s various possible steady states. ???? A phase space is an abstract mathatical space in which coordinates represent the variables necessary to specify the phase (state) of a dynamical system at any time in its progression., CHAOTIC or “STRANGE” attractors (k 3.57 to 4) arise only after the onset of chaos. They take interesting and complex shapes in phase space, and there is no way of predicting their long-term evolution. ???? Descriptions of chaos as “random-like” behavior are mostly justified. Chaotic time-series not only look uncorrelated and unsystematic, they often pass every statistical test for randomness. To that extent, chaotic data are both random and deterministic., The quadratic equation and the quadratic map (xt+1 = k – xt2) and corresponding chaos reveal several interesting generalities. ???? 2. A random-like or even chaotic evolution does not have to be the result of a random operation. Instead, it can arise by design., Trajectories might originate anywhere within the attractor’s basin of attraction. They then gravitate toward the attractor over time. ???? Two neighboring trajectories, once “on” their attractors, stay a calculably close distance from one another as they evolve., An attractor is a dynamical system’s steady state or set of stable or equilibrium conditions. ???? On a standard phase space plot, an attractor is a summary of a system’s long-term behavior. It is a compact, global picture of all of a system’s various possible steady states., A steady state is a state that doesn’t change with time, or the state toward which the system’s behavior becomes asymptotic to the attractor as time goes to infinity. It is like the system’s ID card. ???? Once that steady-state condition arrives, the point attractor is independent of time. It stays fixed, and the system no longer evolves. In phase space the system is static., CHAOTIC or “STRANGE” attractors (k 3.57 to 4) arise only after the onset of chaos. They take interesting and complex shapes in phase space, and there is no way of predicting their long-term evolution. ???? Sensitivity to initial conditions means that the dynamics of the chaotoc system have a divergent aspect. Thus some property is driving the trajectories away from each other, yet they remain bounded within a domain. There is a compensatory convergence which balances the divergence resulting in a strange attractor., CHAOTIC or “STRANGE” attractors (k 3.57 to 4) arise only after the onset of chaos. They take interesting and complex shapes in phase space, and there is no way of predicting their long-term evolution. ???? A chaotic attractor is an attractor that shows extreme sensitivity to initial conditions., Periodic, or limit cycle, attractors (but not point or toroidal attractors) can also appear in the chaotic domain. ???? The basic data for many limit cycles plot as a loop (or circle) in standard phase space., The quadratic equation and the quadratic map (xt+1 = k – xt2) and corresponding chaos reveal several interesting generalities. ???? 5. Period doubling followed by irregular fluctuations in some cases indicates that those fluctuations are chaotic., The quadratic equation and the quadratic map (xt+1 = k – xt2) and corresponding chaos reveal several interesting generalities. ???? 1. Systems governed by physical laws or by deterministic equations can produce regular results under some conditions but irregular or disorderly results under others., CHAOTIC or “STRANGE” attractors (k 3.57 to 4) arise only after the onset of chaos. They take interesting and complex shapes in phase space, and there is no way of predicting their long-term evolution. ???? A chaotic attractor is a complex phase space surface to which the trajectory is asymptotic in time and on which it wanders chaotically.