Download PDF by Guanrong Chen, Xinghuo Yu: Chaos Control: Theory and Applications

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By Guanrong Chen, Xinghuo Yu

ISBN-10: 3540404058

ISBN-13: 9783540404057

Chaos regulate refers to purposefully manipulating chaotic dynamical behaviors of a few advanced nonlinear platforms. There exists no related keep an eye on theory-oriented publication out there that's dedicated to the topic of chaos keep watch over, written by way of keep an eye on engineers for keep watch over engineers. World-renowned prime specialists within the box supply their cutting-edge survey in regards to the broad learn that has been performed over the past few years during this topic. the recent expertise of chaos keep an eye on has significant effect on novel engineering functions akin to telecommunications, energy structures, liquid blending, net expertise, high-performance circuits and units, organic platforms modeling just like the mind and the center, and choice making. The e-book isn't just geared toward energetic researchers within the box of chaos regulate concerning keep watch over and structures engineers, theoretical and experimental physicists, and utilized mathematicians, but in addition at a normal viewers in similar fields.

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Read e-book online Chaos Control: Theory and Applications PDF

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Example text

When boundary conditions are present in a time-dependent nonlinear PDE, analysis becomes very complicated and, to our knowledge, not too much work is available in the literature. But for a special class of PDEs, namely, the wave equation, one can utilize wave reflection on the boundary to analyze or even “manipulate” chaotic behavior. This study actually complements the type of work mentioned in the preceding paragraph [12, 15–17] where, as we mentioned earlier, boundary conditions are for the most part either not included or not regarded as important in the models.

The energy associated with vibration at time t is E(t) = 1 2 1 wx2 (x, t) + 0 1 2 w (x, t) dx. c2 t (5) The objective of the stabilization problem is to find a feedback law for u(t) in (3) such that lim E(t) = 0. (6) t→∞ A simple choice of the feedback law is the negative velocity feedback: u(t) = −αwt (1, t), t > 0, α > 0, α = 1/c, (7) under the assumption that the velocity wt (1, t) at x = 1 can be observed and be fedback. Substituting (7) into (3), we obtain the so-called viscous damping boundary condition wx (1, t) + αwt (1, t) = 0, t > 0.

88 time units. For initial conditions, we choose v0 (x) ≡ 0, 0 ≤ x ≤ 1, 46 G. Chen et al. and a C 2 -spline for u0 :  (x − x1 )3   , x 1 ≤ x ≤ x2 ,    h3   3 2  3(x − x2 ) 3(x − x2 ) 3(x − x2 )   1 + , x 2 ≤ x ≤ x3 , − +  2  h3 h h 1 3 3 u0 (x) = 3(x − x4 ) 3(x − x4 ) 3(x − x4 ) 1 − 12  , x 3 ≤ x ≤ x4 , + + 2  h3 h h    3  (x5 − x)   , x 4 ≤ x ≤ x5 ,   h3   0, elsewhere, j 1 j = 1, 2, 3, 4, 5. h = , xj = , 6 6 The profiles of u and v, as well as the gradient wx , wt , are plotted in Figs.

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Chaos Control: Theory and Applications by Guanrong Chen, Xinghuo Yu


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