Analytical Routes to Chaos in Nonlinear Engineering by Albert C. J. Luo

By Albert C. J. Luo

Nonlinear difficulties are of curiosity to engineers, physicists and mathematicians and plenty of different scientists simply because such a lot structures are inherently nonlinear in nature. As nonlinear equations are tough to unravel, nonlinear structures are as a rule approximated through linear equations. This works good as much as a few accuracy and a few variety for the enter values, yet a few fascinating phenomena similar to chaos and singularities are hidden through linearization and perturbation research. It follows that a few points of the habit of a nonlinear approach look typically to be chaotic, unpredictable or counterintuitive. even if one of these chaotic habit may well resemble a random habit, it's totally deterministic.
Analytical Routes to Chaos in Nonlinear Engineering discusses analytical suggestions of periodic motions to chaos or quasi-periodic motions in nonlinear dynamical platforms in engineering and considers engineering functions, layout, and regulate. It systematically discusses advanced nonlinear phenomena in engineering nonlinear platforms, together with the periodically pressured Duffing oscillator, nonlinear self-excited platforms, nonlinear parametric platforms and nonlinear rotor platforms. Nonlinear types utilized in engineering also are offered and a quick heritage of the subject is supplied.

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Analytical Routes to Chaos in Nonlinear Engineering

Nonlinear difficulties are of curiosity to engineers, physicists and mathematicians and lots of different scientists simply because so much structures are inherently nonlinear in nature. As nonlinear equations are tricky to resolve, nonlinear structures are normally approximated via linear equations. This works good as much as a few accuracy and a few diversity for the enter values, yet a few attention-grabbing phenomena equivalent to chaos and singularities are hidden through linearization and perturbation research.

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2 Book Layout This book consists of five chapters. Chapter 1 gave the brief literature review on analytical methods, including perturbation methods, the method of averaging, and generalized harmonic balance methods. Other chapters are briefly summarized as follows. In Chapter 2, analytical bifurcation trees from period-m motions to chaos in periodically forced, Duffing oscillators will be presented. The analytical solutions of period-m motions in Duffing oscillators will be discussed because the Duffing oscillators are extensively applied in structural vibrations and physical problems.

For the kth order Hopf bifurcation of period-m motion, a relation exists as pk ????k = ????k−1 . 117) i. If pk is an irrational number, the kth-order Hopf bifurcation of the period-m motion is called the quasi-period-pk Hopf bifurcation, and the corresponding solution of the kth generalized coordinates is pk -quasi-periodic to the system of the (k − 1)th generalized coordinates. ii. If pk = 2, the kth-order Hopf bifurcation of the period-m motion is called a period-doubling Hopf bifurcation (or a period-2 Hopf bifurcation), and the corresponding solution of the kth generalized coordinates is period-doubling to the system of the (k − 1)th generalized coordinates.

002, and A9 ≤ 10−4 . Thus, for symmetric period-1 motion, the Fourier series solution with three harmonic terms can give a good approximation for Ω > 5. From the quantity level of harmonic response amplitudes, effects of the harmonic terms on the solutions can be observed. The asymmetric period-1 motion with ten harmonic terms has a Hopf bifurcation. Thus a period-2 motion will be formed from such an asymmetric period-1 motion. If this period-2 motion has a Hopf bifurcation, the period-4 motion will appear.

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