4.6 Application of the Principal Component Analysis (PCA)
111
can be used only in the case of small beam deflections (w ≤ 0.25(2h)). Comparison
of the geometrically linear and nonlinear problem, taking into account the beams
contact interaction, yields the conclusion that the function of displacements u(x, t)
implies the increase in the chaotic components of the signal.
References
1. Krysko, A.V., Awrejcewicz, J., Papkova, I.V., Saltykova, O.A., Krysko, A.V.: On reliability
of chaotic dynamics of two Euler-Bernoulli beams with a small clearance. Int. J. Non-Linear.
Mech. 104, 8–18 (2018)
2. Lozi, R.: Can we trust in numerical computations of chaotic solutions of dynamical systems.
In: Letellier, Ch., Gilmore, R. (eds.) Topology and Dynamics of Chaos in Celebration of Robert
Gilmore’s 70th Birthday, pp. 63–98. World Scientific, Singapore (2013)
3. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Reading
(1989)
4. Banks, J., Brooks, J., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Am. Math. Month.
99(4), 332–334 (1992)
5. Knudsen, C.: Chaos without periodicity. Am. Math. Mon. 101, 563–565 (1994)
6. Gulick, D.: Encounters with Chaos. McGraw-Hill, New York (1992)
7. Awrejcewicz, J., Krysko, V.A., Papkova, I.V., Krysko, A.V.: Deterministic Chaos in OneDimensional Continuous Systems. World Scientific, Singapore (2016)
8. Süli, E., Mayers, D.: An Introduction to Numerical Analysis. Cambridge University Press,
Cambridge (2003)
9. Fehlberg, E.: Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. NASA Tech. Rep. 315 (1969)
10. Fehlberg, E.: Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit
Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing 6(1–2),
61–71 (1970)
11. Cash, J.R., Karp, A.H.: A variable order Runge-Kutta method for initial value problems with
rapidly varying right-hand sides. ACM Trans. Math. Soft. 16, 201–222 (1990)
12. Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl.
Math. 6(1), 19–26 (1980)
13. Haar, A.: Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69(3), 331–371 (1910)
14. Meyer, Y.: Ondelettes, fonctions splines et analyses graduees. Lecture Notes for University,
Torino (1986)
15. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys.
27, 1271–1283 (1986)
16. Grossman, A., Morlet, S.: Decomposition of Hardy functions into square separable wavelets
of constant shape. SIAM J. Math. Anal. 15(4), 723–736 (1984)
17. Meyer, Y.: Ondelettes et functions splines. Ecole Polytech. Paris, Tech. Rep. Semin. EDP
(1986)
18. Meyer, Y.: Wavelets: Algorithms and Applications. SIAM, Philadelphia (1993)
19. Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series. Phys.
Lett. A 185, 77–87 (1994)
20. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a
time series. Phys. D 16, 285–317 (1985)
21. Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D 65, 117–134 (1993)
22. Kantor, B.Y., Bohatyrenko, T.L.: The method for solving contact problems in the nonlinear
theory of shells. Rep. Acad. Sci. Ukr. SSR A(1), 18–21 (1986)
111
can be used only in the case of small beam deflections (w ≤ 0.25(2h)). Comparison
of the geometrically linear and nonlinear problem, taking into account the beams
contact interaction, yields the conclusion that the function of displacements u(x, t)
implies the increase in the chaotic components of the signal.
References
1. Krysko, A.V., Awrejcewicz, J., Papkova, I.V., Saltykova, O.A., Krysko, A.V.: On reliability
of chaotic dynamics of two Euler-Bernoulli beams with a small clearance. Int. J. Non-Linear.
Mech. 104, 8–18 (2018)
2. Lozi, R.: Can we trust in numerical computations of chaotic solutions of dynamical systems.
In: Letellier, Ch., Gilmore, R. (eds.) Topology and Dynamics of Chaos in Celebration of Robert
Gilmore’s 70th Birthday, pp. 63–98. World Scientific, Singapore (2013)
3. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Reading
(1989)
4. Banks, J., Brooks, J., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Am. Math. Month.
99(4), 332–334 (1992)
5. Knudsen, C.: Chaos without periodicity. Am. Math. Mon. 101, 563–565 (1994)
6. Gulick, D.: Encounters with Chaos. McGraw-Hill, New York (1992)
7. Awrejcewicz, J., Krysko, V.A., Papkova, I.V., Krysko, A.V.: Deterministic Chaos in OneDimensional Continuous Systems. World Scientific, Singapore (2016)
8. Süli, E., Mayers, D.: An Introduction to Numerical Analysis. Cambridge University Press,
Cambridge (2003)
9. Fehlberg, E.: Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. NASA Tech. Rep. 315 (1969)
10. Fehlberg, E.: Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit
Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing 6(1–2),
61–71 (1970)
11. Cash, J.R., Karp, A.H.: A variable order Runge-Kutta method for initial value problems with
rapidly varying right-hand sides. ACM Trans. Math. Soft. 16, 201–222 (1990)
12. Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl.
Math. 6(1), 19–26 (1980)
13. Haar, A.: Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69(3), 331–371 (1910)
14. Meyer, Y.: Ondelettes, fonctions splines et analyses graduees. Lecture Notes for University,
Torino (1986)
15. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys.
27, 1271–1283 (1986)
16. Grossman, A., Morlet, S.: Decomposition of Hardy functions into square separable wavelets
of constant shape. SIAM J. Math. Anal. 15(4), 723–736 (1984)
17. Meyer, Y.: Ondelettes et functions splines. Ecole Polytech. Paris, Tech. Rep. Semin. EDP
(1986)
18. Meyer, Y.: Wavelets: Algorithms and Applications. SIAM, Philadelphia (1993)
19. Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series. Phys.
Lett. A 185, 77–87 (1994)
20. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a
time series. Phys. D 16, 285–317 (1985)
21. Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D 65, 117–134 (1993)
22. Kantor, B.Y., Bohatyrenko, T.L.: The method for solving contact problems in the nonlinear
theory of shells. Rep. Acad. Sci. Ukr. SSR A(1), 18–21 (1986)