By Professor Yushu Chen PhD, Professor Andrew Y. T. Leung DSc,PhD,CEng,FRAes,MIStructE,MHKIE (auth.)

For the numerous varied 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 technological know-how. An in-depth learn of the idea and alertness of non-linear technology will surely swap one's belief of diverse non-linear phenomena and legislation significantly, including its nice results on many components of software. because the very important material of non-linear technology, bifurcation conception, singularity concept and chaos concept have constructed speedily long ago or 3 a long time. they're now advancing vigorously of their purposes to arithmetic, physics, mechanics and lots of technical parts around the globe, and they're going to be the most matters of our quandary. This publication is worried with purposes of the tools of dynamic platforms and subharmonic bifurcation thought 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 idea of non-linear dynamic platforms, supplemented with our most modern result of medical study 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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16) as a flow for t E ;e. 18) it satisfies the properties of groups and initial condition yeO) = e OA x = x. Substituting y = e' A x into eq. +--dt dt 2! (n-I)! t"-2 ] =A [ 1+tA+---A"-z+... x=A[e'A]x (n - 2)! thus, y' = Ay is verified. -A"+ ... 2! 3! 19) n! g. 1 = [~ ~l' and A O = 1, O! = I. In order to prove the convergence we introduce an operator norm of A IIAII= suplAxl. Ixl =1 It is not difficult to prove that the scalar operational series ellA I converges for any real II All. The conclusion can be proved by the comparison test principle, but we shall not give the process of proof.

23). It is easy to see that there is no fixed point in Chapter 2 Calculation of Flows In this chapter, the calculations of flows are discussed. 1 introduces divergence of flows and Bendixson's negative criterion. 2. 3 discusses linearly hyperbolic flows and their classification. 4 deals with calculations of local flows of non-linear system. Finally, we introduce the stable manifold theorem. 1 Divergence of Flows and Bendixson's Negative Criterion To determine variations of flows with time in terms of differential equations without solving the equations, we have to study the divergence of flows.

E B 2! n! +~(A + Br +... n! By using binomial theorem and AB = BA , we have e A+B n! ,-, + ... k. k. =c~ ~; )C~o ~:) =eAe B (3) Let B So = - A in part (2), then e A- B = eO = I and e A - A = I. e- A = (eAr' . Lemma can be generalized to e' A for 1 E iIi!. 11+A +... 2! n! + _ _ A n +_A n+ I+... dl (n-l)! n! n-I n (n-l)! n! +-1--A n - 1+~An+ ... = Ae,A So e'A satisfies the ordinary differential equation Y'= AY, where Y = yet) is a column matrix. If I = 0 , then e' A = eO A = eO = I. So e' A is a fundamental matrix.

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