We present chaotic dynamics of flexible curvilinear shallow Euler-Bernoulli beams. The continuous problem is reduced to the Cauchy problem by the finite-difference method of the secondorder accuracy and finite element method (FEM). The Cauchy problem is solved through the fourth-and sixth-order Runge-Kutta methods with respect to time. This preserves reliability of the obtained results. qualitative theory of differential equations. Frequency power spectra using fast Fourier transform, phase and modal portraits, autocorrelation functions, spatiotemporal dynamics of the beam, 2D and 3D Morlet wavelets, and Poincaré sections are constructed. Four first Lyapunov exponents are estimated using the Wolf algorithm. Transitions from regular to chaotic dynamics are detected, illustrated and discussed. Depending on signs of four Lyapunov exponents the chaotic, hyper chaotic, hyper-hyper chaotic, and deep chaotic dynamics is reported. Curvilinear beams are treated as systems with an infinite number of degrees of freedom. Charts of vibration character, elastic-plastic deformations, and stability loss zone versus control parameters of the studied beams are reported.

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Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c(2)) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q(0) and frequency ω(p) of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.

In this paper chaotic vibrations of flexible plates of infinite length are studied. The Kirchhoff–Love hypotheses are used to derive the nondimensional partial differential equations governing the plate dynamics. The finite difference method (FDM) and finite element method (FEM) are applied to validate the numerical results. The numerical analysis includes both standard (time histories, fast Fourier Transform, phase portraits, Poincaré sections, Lyapunov exponents) as well as wavelet-based approaches. The latter one includes the so called Gauss 1, Gauss 8, Mexican Hat and Morlet wavelets. In particular, various plate dynamical regimes including the periodic, quasi-periodic, sub-harmonic, chaotic vibrations as well as bifurcations of the plate are illustrated and studied. In addition, the convergence of numerical results obtained via different wavelets is analyzed.

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