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This Concept Map, created with IHMC CmapTools, has information related to: a15. Routs to Chaos, System Bifurcates, or Splits into two Trajectories at k = 3 ???? Fundamental new properties of Far-From-Equalibrium Systems Appear, System Bifurcates, or Splits into two Trajectories at k = 3 with Further Bifurcations, At 3.4491 a two point attractor emerges. ???? At 3.5. a four-point attractor emerges., The quasiperiodic route to chaos goes from steady state, to quasiperiodic, to chaotic motion. Each of the various thresholds prior to the chaotic motion is known as a Hopf bifurcation. ???? a2.Chaos, At 3.5644, sixteen x* values appear. ???? Further increases in K yield still further increase in the number of cycles, until there are so many that they result in Chaos, Further increases in K yield still further increase in the number of cycles, until there are so many that they result in Chaos ???? The quasiperiodic route to chaos goes from steady state, to quasiperiodic, to chaotic motion. Each of the various thresholds prior to the chaotic motion is known as a Hopf bifurcation., System Goes Chaotic ???? a2.Chaos, At 3.5441, eight x* values appear. ???? At 3.5644, sixteen x* values appear., Period Doubling Route to Chaos as the control parameter increases Symmetry-breaking transformations develop, period doubling route to chaos ???? At 3.4491 a two point attractor emerges., Further increases in K yield still further increase in the number of cycles, until there are so many that they result in Chaos ???? Areas of Order Interspersed with Disorder, Further Bifurcations lead to period doubling route to chaos, Period Doubling Route to Chaos ???? The logistic equation xt+1 = k xt (1 – xt) is the popular expression of the route to chaos. It is simple, and shows fixed-point period doubling (limit cycles) and chaos, Further increases in K yield still further increase in the number of cycles, until there are so many that they result in Chaos ???? System Goes Chaotic, At 3.5. a four-point attractor emerges. ???? At 3.5441, eight x* values appear., The logistic equation xt+1 = k xt (1 – xt) is the popular expression of the route to chaos. It is simple, and shows fixed-point period doubling (limit cycles) and chaos ???? System Bifurcates, or Splits into two Trajectories at k = 3, a15. Routs to Chaos ???? Period Doubling Route to Chaos, Symmetry-breaking transformations develop ???? System Bifurcates, or Splits into two Trajectories at k = 3