The classic Vallis model, proposed in 1986, links the system's chaotic behavior to El Niño. However, integer-order derivatives do not capture the memory and hereditary properties of oceanic processes. Fractional derivatives can describe systems where the current state depends on the entire history. This work sits alongside studies of quantum states and chaos in other systems, but here the fractional version of the Vallis model is examined in detail for all values of the parameter p for the first time.
The authors consider the fractional Vallis system with Caputo derivatives of order α. For stability analysis, they use the eigenvalues of the Jacobian matrix and the Matignon criterion. Numerical simulations are performed using the new predictor-corrector method (NPCM) proposed by Daftardar-Gejji and coauthors. Bifurcation diagrams are constructed with respect to the parameter p and the fractional order α, and the largest Lyapunov exponents (LLE) are computed using Wolf's algorithm. For the incommensurate system, the orders α1, α2, α3 are varied individually.
For the commensurate system with B=150, C=4, the chaotic region in p narrows as α decreases: at α=1 chaos is observed for |p|≤2.45, at α=0.985 for |p|≤1.5, at α=0.975 only for a narrow range near p=0, and at α=0.965 chaos disappears entirely. For B=105, C=4 and p=0.35, the critical order below which chaos vanishes is α=0.975, that is, a total order of 2.925. For the incommensurate system with α2=α3=1, chaos persists down to α1=0.71 (total order 2.71); when varying α2, the critical value is 0.942 (total 2.942); when varying α3, it is 0.948 (total 2.948). The largest Lyapunov exponent for α=0.98 and p=0.35 is 0.4181, confirming chaos.
The work shows that the fractional order acts as an effective parameter controlling chaos in a climate model. Unlike neighboring studies, where fractional systems were examined only for isolated parameter values, here a complete analysis is performed for all p, and exact critical orders are found. This provides a tool for understanding how ocean memory can suppress or amplify El Niño.
The approach could be extended to more realistic El Niño models that include time delays and spatial structure. It would be interesting to investigate the effect of fractional order on the synchronization of multiple climate oscillators and on the predictability of the phenomenon.
The results may be useful in climatology for assessing the sensitivity of El Niño models to memory effects, and in dynamical systems theory for understanding the role of fractional derivatives in chaos.
Verify the obtained critical orders for other parameter sets and compare with observational El Niño data. Investigate synchronization of the fractional Vallis system with more complex climate models.
In the theory of fractional dynamical systems, the exact connection between the fractional derivative order and the physical mechanisms of memory remains an open question. This work provides quantitative benchmarks for climate applications but does not reveal the microscopic nature of ocean memory.
The Vallis model is obtained from the Lorenz system by adding just one parameter p, which describes the average influence of equatorial winds on the ocean. At p=0 and B=1, it coincides exactly with the Lorenz system.