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

For the numerous various 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 virtually periodic vibrations is without doubt one of the most important achievements of recent technological know-how. An in-depth examine of the idea and alertness of non-linear technological know-how will surely switch one's conception of diverse non-linear phenomena and legislation significantly, including its nice results on many components of program. because the vital subject material of non-linear technological know-how, bifurcation concept, singularity conception and chaos conception have built swiftly long ago or 3 many years. they're now advancing vigorously of their purposes to arithmetic, physics, mechanics and lots of technical components around the globe, and they'll be the most topics of our predicament. This e-book is worried with purposes of the equipment of dynamic platforms and subharmonic bifurcation concept within the learn of non-linear dynamics in engineering. It has grown out of the category notes for graduate classes on bifurcation conception, chaos and alertness concept of non-linear dynamic platforms, supplemented with our most up-to-date result of medical learn and fabrics from literature during this box. The bifurcation and chaotic vibration of deterministic non-linear dynamic structures are studied from the perspective of non-linear vibration.

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From lemma 4 and r ~ ro( x), it follows that r = ro( x). That implies that (0 (x) is a closed orbit. 16). If the semi-positive orbit of r(x) is bounded, then ro(x)satisfies one of the following: (1) ro(x) is an equilibrium point; (2) ro(x) is a closed orbit; and (3) ro(x) is the union of fixed points and the orbits connecting them. The point on the orbits tends to one of the above fixed points as t ~ +00 and t ~ -00. The same conclusions can be given for a(x). Note 1. A closed simple curve formed of some fixed points and orbits connecting them is called a singular closed orbit.

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. n. Suppose yet) =e,AC is a solution of the IVP. For linear differential equation y'= Ay, its solution can be expressed in 40 Bifurcation and Chaos in Engineering the form of flow

1 Linear Flows and Algebra Characteristic of e A Assume flow