By Jr. Arthur E. Bryson, Yu-Chi Ho
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Additional resources for Applied Optimal Control: Optimization, Estimation, And Control
W h a t characterizes t h e c h a o t i c s y s t e m is a hypersensitivity to initial conditions over its t i m e history to s u c h a n e x t e n t that output results are either unpredictable or nearly so. Coupled w i t h unpredictability is the loss of knowledge about the initial conditions in a s y s t e m that is chaotic; information entropy increases with t i m e in chaotic systems. I n f o r m a t i o n entropy arises from the fact t h a t the initial conditions of a n y system, linear or nonlinear, c a n be k n o w n only to a certain extent.
1 and r = 3 . 2 , after a certain n u m b e r of iterations Xn + l flip-flops between the values of 0 . 7 9 9 9 4 5 5 5 and 0 . 5 1 3 0 4 4 6 indefinitely,- this is a n e x a m p l e of period-2 behavior (Fig. 2 - 9 C ) . Period-4 behavior (Fig. 2 - 9 D ) arises at r = 3 . 4 4 9 4 8 9 7 3 , and the result oscillates through t h e sequence (again XQ = 0 . 1 ) : 0 . 8 5 0 5 6 4 6 , 0 . 4 3 8 4 4 5 6 , 31 CHAOTIC AND RANDOM SIGNALS Ν (C) Fig. 2 - 9 Behavior of the logistic equation: (A) R < 1, (B) 1 < R < 3, ( C ) period 2 (R > 3), (D) period 4, and (E) chaotic.
2 - 1 2 . T h e waveform in Fig. 2 - 1 2 A is a c o n t i n u o u s voltage function of t i m e , V(t)} in this case, a triangle waveform is seen. If t h e signal is sampled b y a n o t h e r signal, p(t), w i t h frequency Fs and sampling period Τ = l/Fs, as s h o w n in Fig. 2 - 1 2 B , and t h e n later reconstructed, t h e waveform m a y l o o k s o m e t h i n g like Fig. 2 - 1 2 C . W h i l e this m a y b e sufficiently representative of t h e waveform for m a n y purposes, it would b e reconstructed w i t h greater fidelity if t h e sampling frequency (Fs) is increased.
Applied Optimal Control: Optimization, Estimation, And Control by Jr. Arthur E. Bryson, Yu-Chi Ho