  The nonlinear dynamics of a recently derived generalized Lorenz model (Macek
and Strumik, Phys. Rev. E 82, 027301, 2010) of magnetoconvection is studied. A
bifurcation diagram is constructed as a function of the Rayleigh number where
attractors and nonattracting chaotic sets coexist inside a periodic window. The
nonattracting chaotic sets, also called chaotic saddles, are responsible for
fractal basin boundaries with a fractal dimension near the dimension of the
phase space, which causes the presence of very long chaotic transients. It is
shown that the chaotic saddles can be used to infer properties of chaotic
attractors outside the periodic window, such as their maximum Lyapunov
exponent.
