In this study, we examine the magnetospheric chaos and dynamical complexity response to the disturbance storm time (Dst) and solar wind electric field (VBs) during different categories of geomagnetic storm (minor, moderate and major geomagnetic storm). The time series data of the Dst and VBs are analysed for a period of 9 years using non-linear dynamics tools (maximal Lyapunov exponent, MLE; approximate entropy, ApEn; and delay vector variance, DVV). We found a significant trend between each non-linear parameter and the categories of geomagnetic storm. The MLE and ApEn values of the Dst indicate that chaotic and dynamical complexity responses are high during minor geomagnetic storms, reduce at moderate geomagnetic storms and decline further during major geomagnetic storms. However, the MLE and ApEn values obtained from VBs indicate that chaotic and dynamical complexity responses are high with no significant difference between the periods that are associated with minor, moderate and major geomagnetic storms. The test for non-linearity in the Dst time series during major geomagnetic storm reveals the strongest non-linearity features. Based on these findings, the dynamical features obtained in the VBs as input and Dst as output of the magnetospheric system suggest that the magnetospheric dynamics are non-linear, and the solar wind dynamics are consistently stochastic in nature.