Fractional order convection-diffusion systems show chaos under exponential cubic B-spline discretization

The dynamical behavior of nonlinear fractional convection-diffusion equations (CDEs) is very rich and extends beyond their applicability as transport models, but it has yet to be explored numerically. In this paper, it is demonstrated that a time- and space-fractional nonlinear convection-diffusion equation exhibits a transition from stable equilibrium to deterministic chaos its nonlinear convection parameter increases by employing an exponential cubic B-spline scheme for this said problem. The Caputo fractional derivative is discretized using the L1 formula, whereas the fractional spatial operators are approximated using the exponential spline method. Using phase portraits, bifurcation diagrams, and Lyapunov exponent spectra, we identify three different dynamical behaviors: a stable regime, periodic behavior, and finally chaos, arising from the increasing value of the nonlinear convection parameter. Therefore, this work establishes the nonlinear convection parameter as a true bifurcation parameter and demonstrates the possibility of complex nonlinear dynamics in fractional convection-diffusion systems.