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This Concept Map, created with IHMC CmapTools, has information related to: b3. Order Within Chaos, b3. Order within Chaos: ???? Chaos looks erratic; however, it is not just a sea of disorder., A second kind of order within chaos is the route to chaos, such as period-doubling and intermittency. ???? The period doubling goes on in the same cascading way as in the pre-chaotic domain. With the period-three attractor at k = 3.83 in the logistic equation, for example, slight increase in k bring bifurcations to cycle periods of 6, 12, 24, etc., Increases in a given (control) parameter, such as k in the logistic equation, do not necessarily bring increased degrees of chaos. Each systematic increase in the control parameter, however, no matter how slight, does bring about a different trajectory. ???? Most orderly features of chaos depend strictly on the control parameter. For a given value of that parameter, the order develops spontaneously, with out external cause., One orderly peculiarity within chaos is called windows. They are zones of k values for which iterations from any xo produce a periodic attractor, instead of a chaotic one. ???? The most prominent window generated on the average personal computer using the logistic equation is at k ~ 3.83 where a period-three cycle turns up., Most orderly features of chaos depend strictly on the control parameter. For a given value of that parameter, the order develops spontaneously, with out external cause. ???? The order lasts only over a brief range of the control parameter., b3. Order within Chaos: ???? Order within chaos is the rule rather than the exception., Chaos looks erratic; however, it is not just a sea of disorder. ???? Increases in a given (control) parameter, such as k in the logistic equation, do not necessarily bring increased degrees of chaos. Each systematic increase in the control parameter, however, no matter how slight, does bring about a different trajectory., A fifth orderly feature of chaos is the fractal structure. ???? b4. Fractals, One orderly peculiarity within chaos is called windows. They are zones of k values for which iterations from any xo produce a periodic attractor, instead of a chaotic one. ???? A second kind of order within chaos is the route to chaos, such as period-doubling and intermittency., The most prominent window generated on the average personal computer using the logistic equation is at k ~ 3.83 where a period-three cycle turns up. ???? A second kind of order within chaos is the route to chaos, such as period-doubling and intermittency., A second kind of order within chaos is the route to chaos, such as period-doubling and intermittency. ???? Any periodicity within the chaotic regime usually lasts for just a brief range of k values. Most windows are so narrow that they only turn up under high magnification., Within the chaotic regime, increases in the control parameter often lead to some type of regularity, or kind of order. ???? The order lasts only over a brief range of the control parameter., Increases in a given (control) parameter, such as k in the logistic equation, do not necessarily bring increased degrees of chaos. Each systematic increase in the control parameter, however, no matter how slight, does bring about a different trajectory. ???? Within the chaotic regime, increases in the control parameter often lead to some type of regularity, or kind of order., Order within chaos is the rule rather than the exception. ???? Most orderly features of chaos depend strictly on the control parameter. For a given value of that parameter, the order develops spontaneously, with out external cause., Regardless of where the trajectories begin, the same chaotic attractor develops. ???? All this regularity indicates that order is a basic ingredient of chaos. Some people look upon chaos as order disguised as disorder. Some think that pure chaos sets in only when the control parameter is at a maximum, such as k = 4 in the logistic equation., A second kind of order within chaos is the route to chaos, such as period-doubling and intermittency. ???? Intermittency within chaos works in a way opposite from that within the pre-chaotic domain. Rather than moving from order too disorder, it now moves from disorder to order., The fourth orderly structure of chaos is zones of relatively greater popularity on each chaotic attractor. These are zones a chaotic system is more likely to visit during its evolution. ???? A fifth orderly feature of chaos is the fractal structure., A third structural aspect or underlying framework of chaos is the chaotic, or strange, attractor. ???? The fourth orderly structure of chaos is zones of relatively greater popularity on each chaotic attractor. These are zones a chaotic system is more likely to visit during its evolution., A second kind of order within chaos is the route to chaos, such as period-doubling and intermittency. ???? A third structural aspect or underlying framework of chaos is the chaotic, or strange, attractor., Chaos looks erratic; however, it is not just a sea of disorder. ???? Order within chaos is the rule rather than the exception.