Hostile space weather is one of the principal threats to modern human technology because of the dependency of our lives on satellites. Solar coronal mass ejections, solar flares, and high-speed solar wind streams often cause some sequences of damaging disturbances within the Earth's magnetosphere, in the atmosphere, and even on the Earth's surface. The time-varying Sun, as the main source of space weather causes geomagnetic storms and substorms in many ways with coronal mass ejections (CME) being possibly the most harmful among them. CMEs cause transient magnetized plasma flows as interplanetary consequences, which drive the geomagnetic storms and substorms in the Earth's magnetosphere. It is shown that the cyclic solar activity has chaotic characteristics. It seems that the chaotic modeling of solar and geomagnetic activity indices is an important approach for analysis and prediction of the solar extreme events. This is particularly useful for engineers and designers interested in space weather prediction and its applications. One of the most important tools for eliciting the chaotic trends is the “Lyapunov exponents,” which is a useful measure of the stability of a dynamic system. This paper analyzes the variation of Lyapunov exponents (LE) for three solar and geomagnetic activity indices during CMEs: the sunspot number, disturbance storm time, and proton temperature. The analysis has been carried out using the technique of adaptive LE estimation adopted from previous works; the suitable window for which has been determined via trial and error. It is shown that the LE of these solar and geomagnetic activity indices varies rapidly during CMEs. The variation in LEs creates a pattern as a precursor for the forthcoming CME. This precursor, which is an oscillation in the values of Les, begins several steps sooner than the CME's occurrence. Then, during the CME, the LEs decrease to a small positive or a negative value, which demonstrates that during an anomaly such as a CME the chaotic characteristics of solar and geomagnetic activity indices decrease and solar and geomagnetic activity indices follow more regular dynamics.