Abstract
The book is devoted to study the nonlinear phenomena exhibited by the size-dependent structural members including bifurcations and chaotic processes, and hence this chapter provides an overview of one of the main tools for identifying the nonlinear dynamics of these objects. Namely, the concept of Lyapunov exponents is briefly revisited, which allows us to distinguish between regular (periodic or quasi-periodic) and chaotic vibrations of the size-dependent beams, plates and shells studied in this book. In particular, the methods of Benettin, Wolf, Rosenstein, Kantz based on Jacobian estimation and the neural network method are presented and discussed. As noted above, an important issue in solving problems of nonlinear dynamics, especially at the nano-level, is the question of the reliability of chaotic oscillations. This problem was first identified by René Lozi in 2013. In this monograph, in order to obtain reliable results, it is proposed to achieve a coincidence not only of the basic functions during chaotic oscillations, but also of their second derivatives with respect to time.
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Awrejcewicz, J., Krysko, A.V., Zhigalov, M.V., Krysko, V.A. (2021). Lyapunov Exponents and Methods of Their Analysis. In: Mathematical Modelling and Numerical Analysis of Size-Dependent Structural Members in Temperature Fields. Advanced Structured Materials, vol 142. Springer, Cham. https://doi.org/10.1007/978-3-030-55993-9_3
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