lecnotes12 - 12 H´ enon attractor The chaotic phenomena of...

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Unformatted text preview: 12 H´ enon attractor The chaotic phenomena of the Lorenz equations may be exhibited by even simpler systems. We now consider a discrete-time, 2-D mapping of the plane into itself. The points in R 2 are considered to be the the Poincar´ e section of a flow in higher dimensions, say, R 3 . The restriction that d > 2 for a strange attractor does not apply, because maps generate discrete points; thus the flow is not restricted by continuity (i.e., lines of points need not be parallel). 12.1 The H´ enon map The discrete time, 2-D mapping of H´ enon is X k +1 = Y k + 1 − X k 2 Y k +1 = λX k • controls the nonlinearity. • λ controls the dissipation. Pictorially, we may consider a set of initial conditions given by an ellipse: X Y 124 Now bend the elllipse, but preserve the area inside it (we shall soon quantify area preservation): X ’ Y ’ Map T 1 : X ∗ = X Y ∗ = 1 − X 2 + Y Next, contract in the x-direction ( λ < 1) | | X ’’ Y ’’ Map T 2 : X ∗∗ = λX ∗ Y...
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This note was uploaded on 02/24/2012 for the course MECHANICAL 12.006J taught by Professor Danielrothman during the Fall '06 term at MIT.

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lecnotes12 - 12 H´ enon attractor The chaotic phenomena of...

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