In this article, we perform a computational investigation of a nanopore connected to external fluidic reservoirs of asymmetric geometries. The asymmetry between the reservoirs is achieved by changing the cross-sectional areas, and the reservoirs are designated as the micropore reservoir and macropore reservoir. When an electric field is applied, which is directed from the macropore towards the micropore reservoir, we observe local nonequilibrium chaotic current oscillations. The current oscillations originate at the micropore-nanopore interface owing to the local cascade of ions; we refer to this phenomenon as the "avalanche effects." We mathematically quantify chaos in terms of the maximum Lyapunov exponent. The maximum Lyapunov exponent exhibits a monotonic increase with the applied voltage and the macropore reservoir diameter. The temporal power spectra maps of the chaotic currents depict a low-frequency "1/f"-type dynamics for the voltage chaos and "1/f2"-type dynamics for the macropore reservoir chaos. The results presented here offer avenues to manipulate ionic diodes and fluidic pumps. © 2016 American Physical Society.