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Period-doubling Transition to Chaos

Period-doubling Transition to Chaos. Eui–Sun Lee Department of Physics Kangwon National University. Strange Attractor [Lorenz, J. Atmos. Sci. 20 , 130 (1963).]. Butterfly Effect : Sensitive Dependence on Initial Conditions

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Period-doubling Transition to Chaos

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  1. Period-doubling Transition to Chaos • Eui–Sun Lee • Department of Physics • Kangwon National University Strange Attractor[Lorenz, J. Atmos. Sci. 20, 130 (1963).] Butterfly Effect: Sensitive Dependence on Initial Conditions ( small cause  large effect )

  2. Attractor and Periodic Orbit In the study of dynamics, some terminology offer us a pretty understanding. 1. Attractor a. Attractor is an invariant set. b. Attractor attracts an open set of initial condition. c. Attractor is minimal.: There is no proper subset that satisfies conditions a and b. Stable fixed points Limit circle 2. Periodic orbit 1D Map : Period-q orbit : Maybe one have interest in a bundle of trajectories seen by projection to the 1dimesional space through the time space.

  3. Investigation of Dynamics • Time series: { t , x } Trajectories vs. time for a given parameter A . • Bifurcation diagram:{ A , x } Plot of attractors by varying the parameter A .

  4. Period-doubling Transition to Chaos • 1D quadratic Map A : a parameter controlling the degree of nonlinearity : a state variable at a discrete time t • Discrete-time system (mapping) The Mapping is useful paradigm for chaos. Period-doubling route to chaos is universal property, proved in many experiments. • A Transition to Chaos occurs through an infinite sequence of period-doubling bifurcation.

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