The Fractional Gray Lotka-Volterra Model (FGLVM) is introduced and used for modeling the transaction counts of three cryptocurrencies, namely, Bitcoin, Litecoin, and Ripple. The 2-dimensional study is on Bitcoin and Litecoin, while the 3-dimensional study is on Bitcoin, Litecoin, and Ripple. Dataset from 28 April 2013 to 10 February 2018 provides forecasting values for Bitcoin and Litecoin through the 2-dimensional FGLVM study, while dataset from 7 August 2013 to 10 February 2018 provides forecasting values of Bitcoin, Litecoin, and Ripple through the 3-dimensional FGLVM study. Forecasting values of cryptocurrencies for the n-dimensional FGLVM study, n={2,3} along 100 days of study time, are displayed. The graph and Lyapunov exponents of the 2-dimensional Lotka-Volterra system using the results of FGLVM reveal that the system is a chaotic dynamical system, while the 3-dimensional Lotka-Volterra system displays parabolic patterns in spite of the chaos indicated by the Lyapunov exponents. The mean absolute percentage error indicates that 2-dimensional FGLVM has a good accuracy for the overall forecasting values of Bitcoin and a reasonable accuracy for the last 300 forecasting values of Litecoin, while the 3-dimensional FGLVM has a good accuracy for the overall forecasting values of Bitcoin and a reasonable accuracy for the last 300 forecasting values of both Litecoin and Ripple. Both 2- and 3-dimensional FGLVM analyses evoke a future constant trend in transacting Bitcoin and a future decreasing trend in transacting Litecoin and Ripple. Bitcoin will keep relatively higher transaction counts, with Litecoin transaction counts everywhere superior to that of Ripple.

Topics
Lyapunov exponent, Chaos theory, Dynamical systems, Fractional calculus
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