In this paper we observed the global dynamics and the occurrence of a certain bifurcation for the corresponding values of a certain rational difference equation of the second order with analyzed quadratic terms. The analysis of the local stability of the unique equilibrium point, as well as the unique periodic solution of period two, was performed in detail. The constraint of the equations on both sides for the corresponding values of the parameters is proved and on this basis the global stability is analyzed. The existence of Neimark-Sacker bifurcation with respect to the arrangement of equilibrium points has been proven. Thus, the basins of attraction have been determined in full for all the positive values of the parameters and all the positive initial conditions.

In this paper we will look at the one system of ODE and analyze it. We aim to determine the points of equilibrium; examine their character and establish the existence of a bifurcation for the corresponding parameter value. A detailed analysis of local stability was performed for all values of the given parameter. For a certain value of the parameter, the existence of supercritical Hopf bifurcation of the observed system of differential equations has been proved. Also, the existence of a limit cycle that is always stable has been proved.

The main objective of this research was to propose a light and practical design solution for electric bike front drive with bottom bracket electric motor. The initial design needs to be redesigned so it can enable simultaneous use of the electric drive and pedal drive, with integration of the front gear shifter. After gathering the basic information linked to the problem and inspecting the initial design solution, the assets and flaws have been identified. The CAD models of the considered possible solutions were developed into FEM models which were used for structural analysis in CAD/CAE software system CATIA. On the basis of the FEM analysis and additional criteria, the optimal solution was chosen, and structural optimization, based on FEM model, was performed. A prototype was manufactured and a mounting process in a place of the initial design was performed. Afterwards, electric bike with mounted prototype was tested under real conditions.