By Professor Yushu Chen PhD, Professor Andrew Y. T. Leung DSc,PhD,CEng,FRAes,MIStructE,MHKIE (auth.)
For the numerous diverse deterministic non-linear dynamic structures (physical, mechanical, technical, chemical, ecological, fiscal, and civil and structural engineering), the invention of abnormal vibrations as well as periodic and nearly periodic vibrations is without doubt one of the most important achievements of contemporary technology. An in-depth research of the speculation and alertness of non-linear technological know-how will surely swap one's notion of various non-linear phenomena and legislation significantly, including its nice results on many components of program. because the vital material of non-linear technology, bifurcation concept, singularity concept and chaos conception have built quickly some time past or 3 a long time. they're now advancing vigorously of their purposes to arithmetic, physics, mechanics and plenty of technical parts around the globe, and they'll be the most topics of our trouble. This e-book is anxious with purposes of the equipment of dynamic structures and subharmonic bifurcation idea within the examine of non-linear dynamics in engineering. It has grown out of the category notes for graduate classes on bifurcation concept, chaos and alertness concept of non-linear dynamic platforms, supplemented with our newest result of medical examine and fabrics from literature during this box. The bifurcation and chaotic vibration of deterministic non-linear dynamic platforms are studied from the point of view of non-linear vibration.
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Extra resources for Bifurcation and Chaos in Engineering
Or rather, for any given E > 0 , there exists 0 > 0 such that there exists the solution y(t) satisfying y(to) defined on the same interval and Iy(t) - YCt)l:S E for any Yo to satisfy IYo - Yol:S 0, to :S t :S T. The continuous dependence on the initial value shows that with small changes of the initial value the motion state of the system is close to the specific motion. As there is always some error in the initial value of practical systems, the continuous dependence on the initial value of the solution ensures that the system can work according to the specific law of motion.
13) with periodic coefficients is reducible in the sense of Liapunov. Proof: Suppose ~t) is the fundamental matrix and A is the non-singular matrix such that ~t + T) = ~t)A. Let A = e TB • Suppose that Pj = u j + iP j are the eigenvalues of A, or the Floquet multipliers which are related to the eigenvalues Aj of B Aj = ~[ln~u~ + P~ + i arct~:~)] (j = 1,2, ... n) 29 Dynamical Systems, Ordinary Differential Equations and Stability of Motion Aj (j Let = 1,2, ... , n) are called the characteristic exponents corresponding to periodic T.
Y = rsinewe ~ ~ ~ (b) Fig. 10 y x get the following equation dr 2 r'=r(l-r2) or -=2r 2(l-r 2 ). Ifwe dt let it: = r2 , we can get the logistic equation as in example 2. 11 Dynamical Systems, Ordinary Differential Equations and Stability of Motion 19 The solution for r is similar to that in example 2, and the solution 8 for 8 = t + 8 . 0). y) =~. x 2 + y2 > 1 . x 2 + y2 S 1. 1 Ifx is an equilibrium point of the trajectory passing this point, then x is the coCa) -limit point set. 2 Every point in a closed trajectory is the coCa) -limit point of itself, so the closed orbit is the co( a) -limit point set itself.
Bifurcation and Chaos in Engineering by Professor Yushu Chen PhD, Professor Andrew Y. T. Leung DSc,PhD,CEng,FRAes,MIStructE,MHKIE (auth.)