Chaotic dynamics of size dependent Timoshenko beams with functionally graded properties along their thickness

•Dynamics of the size-dependent FG Timoshenko beams with the von Kármán nonlinearity are studied.•The reference line simplifying the governing equations is introduced.•The results are validated through Lyapunov and wavelet spectra.•Influence of size and material grading coefficients on vibration characteristics is given.•Transition from regular to chaotic beam vibrations coincides with the Ruelle-Takens-Newhouse scenario. Chaotic dynamics of microbeams made of functionally graded materials (FGMs) is investigated in this paper based on the modified couple stress theory and von Kármán geometric nonlinearity. We assume that the beam properties are graded along the thickness direction. The influence of size-dependent and functionally graded coefficients on the vibration characteristics, scenarios of transition from regular to chaotic vibrations as well as a series of static problems with an emphasis put on the load-deflection behavior are studied. Our theoretical/numerical analysis is supported by methods of nonlinear dynamics and the qualitative theory of differential equations supplemented by Fourier and wavelet spectra, phase portraits, and Lyapunov exponents spectra estimated by different algorithms, including Wolf’s, Rosenstein’s, Kantz’s, and neural networks. We have also detected and numerically validated a general scenario governing transition into chaotic vibrations, which follows the classical Ruelle-Takens-Newhouse scenario for the considered values of the size-dependent and grading parameters.