Routes to chaos in continuous mechanical systems. Part 3: The Lyapunov exponents, hyper, hyper-hyper and spatial-temporal chaos

Third part of the paper is devoted to analysis of the hyper, hyper-hyper and spatial-temporal chaos of continuous mechanical systems using the Lyapunov exponents. The constructed algorithms for the Lyapunov exponents' computation allowed detecting and analysing novel phase transitions from chaos through hyper chaos to hyper-hyper chaos. In addition, a novel characteristic "maximal deflection versus excitation amplitude" has been introduced to study stability properties of the investigated continuous systems. It should be emphasized that the latter characteristic yields results in full agreements with those obtained via the Lyapunov exponents' spectrum estimation. The introduced methods and tools of analysis allowed detecting the Sharkovskii windows of periodicity in all continuous mechanical systems investigated in this paper. Finally, the approach to study the space-temporal chaos exhibited by shell structural-members is also proposed.