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Forced oscillations of flexible plates with a longitudinal, time dependent load acting on one plate side are investigated. Regular (harmonic, subharmonic and quasi-periodic) and irregular (chaotic) oscillations appear depending on the system parameters as well as initial and boundary conditions. In order to achieve highly reliable results, an effective algorithm has been applied to convert a problem of finding solutions to the hybrid type partial differential equations (the so-called von Kármán form) to that of the ordinary differential equations (ODEs) and algebraic equations (AEs). The obtained equations are solved using finite difference method with the approximations 0(h4) and 0(h2) (in respect to the spatial coordinates). The ODEs are solved using the Runge–Kutta fourth order method, whereas the AEs are solved using either the Gauss or relaxation methods. The analysis and identification of spatio-temporal oscillations are carried out by investigation of the series wij(t), wt,ij(t), phase portraits wt,ij (wij) and wtt,ij(wt,ij, wij) and the mode portraits in the planes wx,ij(wij), wy,ij (wij) and in the space wxx(wx,ij,wij), FFT as well as the Poincaré sections and pseudo-sections.

The first part of the paper was aimed at analyzing the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method [1], Rosenstein method [2], Kantz method [3], method based on the modification of a neural network [4, 5], and the synchronization method [6, 7]) for the classical problems governed by difference and differential equations (Hénon map [8], hyper-chaotic Hénon map [9], logistic map [10], Rössler attractor [11], Lorenz attractor [12]) and with the use of both Fourier spectra and Gauss wavelets [13]. It was shown that a modification of the neural network method [4, 5] makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyper-chaos, hyper hyper-chaos and deep chaos [14-16]. Different algorithms for computation of Lyapunov exponents were validated by comparison with the known dynamical systems spectra of the Lyapunov exponents. The carried out analysis helps comparatively estimate the employed methods in order to choose the most suitable/optimal one to study different kinds of dynamical systems and different classes of problems in both this and the next paper parts.

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