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In recent years, a considerable quantity of work has been done analysing nonlinear systems in bi-dimensional parameter space. As a result, noticeable periodic windows, such as resonance tongues and shrimp-shaped structures, have been identified embedded in the chaotic regions for different types of systems. In this paper, we characterise, using Lyapunov exponents, the dynamics of autoparametric oscillators identifying interesting properties for periodic windows in parameter space, such as period-adding and
Fibonacci-type sequences. In addition, we determine multistability associated with fractal basin boundaries.