First Lyapunov value and bifurcation behavior of specific class of three-dimensional systems

This paper presents a study of the behaviour of a special class of 3D dynamic systems (i.e. RHS of the third-order equation is a cubic polynomial), using Lyapunov-Andronov's theory. Considering the general case, we find a new analytical formula for the first Lyapunov's value at the boundary of stability. It enables one to study in detail the bifurcation behaviour (and the route to chaos, in particular) of dynamic systems of the above type. We check the validity of our analytical results on the first Lyapunov's value by studying the route to chaos of two 3D dynamic systems with proved chaotic behaviour. These are Chua's and Rucklidge's systems. Considering their route to chaos, we find new results.