To explore memcapacitors and their characteristics in chaotic oscillators, this paper proposes a logarithmic charge-controlled memcapacitor model. Using power-off plot analysis, we show that the memcapacitor possesses continuous non-volatile characteristic. Also, its dynamic route map shows the memcapacitor can rapidly switch from one memcapacitance to another by applying a single voltage pulse. Based on the memcapacitor model, we design a chaotic oscillator, which can exhibit some complex dynamic characteristics, such as chaos, hyperchaos and various coexisting attractors. The multistable coexisting oscillation of the system is further analyzed by using phase portraits, basins of attraction and double-bifurcation diagrams. Symmetric coexistence attractors with infinite homogeneity and heterogeneity are also found, which can evolve into hyperchaos under certain initial conditions. Finally, the chaotic oscillator is verified by numerical simulations and digital signal processor experiments.