Numerous researches of the Duffing-type oscillator have revealed a big variety of dynamics of this nonlinear system. Nonlinear effects, which result from changes of initial conditions and parameters, can be manifested in various, sometimes seemingly unexpected, ways. The task of studying nonlinear effects has important practical value for engineering dynamical systems working by the Duffing-type oscillator. For the research of nonlinear effects we usually use a method of harmonic balance, analog simulation or a method of direct scanning. Such techniques are a little limited regarding the ability to predict all dynamical characteristics, including bifurcations that lead to occurrence of unstable, subharmonic and chaotic solutions. The given work is devoted to the study of complicated dynamics of the Duffing-type oscillator and the reasons of its occurrence at the change of the system parameters on the basis of the system approach by the method of complete bifurcation analysis. The results are received by direct numerical simulation with the use of software NLO and SPRING. Bifurcation diagrams at the research of dynamics of the Duffing-type oscillator are constructed by means of the parameter continuation approach of steady and unstable periodic solution. The example of the Duffing-type oscillator illustrates high efficiency of the method of complete bifurcation analysis for providing an explanation of the reasons of the birth of regular and chaotic solutions, their coexistence and influence on transient processes. At the systematic approach for certain parameters at which the complete bifurcation analysis specifies the preconditions of the birth of unexpected nonlinear effects, in the investigated system an additional analysis of various periodic and chaotic attractors, forecasting of rare attractors and rare phenomena, studying of scenarios of their birth are carried out.