Abstract
In this chapter we consider the modified Lorenz-Malkus water wheel model. Within a novel approach we take into account friction features on the rim of the water wheel formalized by strong nonlinearities. Namely, the dry friction (within the Coulomb model) and hysteresis friction (within the Bouc-Wen model and the Dahl model) are considered. The dynamic characteristics such as fixed points, Lyapunov characteristic exponents, bifurcation diagrams, are presented and discussed. Detailed analysis of a 2-dimensional Lorenz-Malkus system (where the third coordinate is supposed to be constant) is also presented and discussed. Namely we show the bifurcation process where two saddles and stable node birth from a saddle. It is shown that the static friction (formalized within the Coulomb model) leads to stabilization of the system at the origin independent on the value of the friction coefficient. At the same time we show that using certain parameters within the Bouc-Wen and Dahl models, chaotic behaviour can be controlled. A way to use the modified Lorenz-Malkus system with certain parameters as a “natural” pseudo-random numbers generator is also discussed.
