Evaluating floquet theory solutions for periodic coefficient systems with applications to Ebro basin reservoir management

Introduction

In arid and semi-arid river basins, water managers must balance competing demands irrigation, hydropower, urban supply, environmental flows against inflows that oscillate with the seasons [1] . The Ebro, Spain's longest river draining entirely within the country, supplies water to over three million hectares of irrigated land, and its reservoirs collectively hold some 7,500 hm³ of storage [2] . Managing this storage optimally requires models that respect the periodic nature of inflows, a feature that classical steady-state approaches ignore [3] . Floquet theory provides the natural mathematical language for such problems. Developed in the 1880s for linear differential equations with periodic coefficients, it reduces the stability question to the eigenvalues of a monodromy matrix the so-called Floquet multipliers [4] . If all multipliers lie inside the unit circle, the system is stable; if any escape, solutions grow without bound [5] . Despite its elegance, Floquet analysis has seen limited use in hydrology, where simpler regression and ARIMA models dominate forecasting practice [6] . This research pursued two objectives. First, we performed a systematic numerical evaluation of Floquet multipliers for periodic coefficient systems across a range of periods and coefficient structures [4,7] . Second, we recast the seasonal reservoir mass balance as a periodic linear system and applied Floquet stability analysis to storage data from five Ebro Basin reservoirs spanning 2012-2022 [2,8] . The computational work was carried out at the Surat Department of Algebraic Studies between April 2021 and September 2023.

Theoretical Framework

Consider the system x'(t) = A(t)x(t) where A(t + T) = A(t) for all t. Floquet's theorem states that the fundamental matrix can be written as Φ(t) = P(t)e^{Bt}, where P(t) is T-periodic and B is a constant matrix [4] . The Floquet multipliers ρ_i are the eigenvalues of the monodromy matrix M = Φ(T), and the system's asymptotic stability is determined entirely by whether |ρ_i| < 1 for all i [5] . In our reservoir application, the state vector x contains storage levels, and A(t) encodes seasonal inflowoutflow rates that repeat annually with T = 12 months [7] .

Materials and Methods

Materials

This research was conducted at the Department of Probability and Statistics, Surat Department of Algebraic Studies, Surat, Gujarat, India, between April 2021 and September 2023.

Numerical computations used MATLAB R2022b on a workstation with 32 GB RAM [4] . For the hydrological application, monthly reservoir storage data (hm³) were obtained from the Confederación Hidrográfica del Ebro public database for five reservoirs: Mequinenza, Canelles, Yesa, Barasona, and El Grado, covering January 2012 to December 2022 [2] . Monthly inflow and regulated outflow records accompanied each storage series. Precipitation and temperature data from AEMET weather stations provided climatic context [8] .

Methods

The numerical Floquet analysis discretised each period T into 1,000 steps and computed the monodromy matrix via successive multiplication of local transition matrices [5] . Multiplier magnitudes were extracted as eigenvalues of M, and convergence was verified by doubling the step count and checking that multipliers changed by less than 10⁻⁵ [7] . For the reservoir application, the monthly mass balance S(t+1) = S(t) + I(t) -O(t) -E(t) was linearised about the mean annual cycle, where I, O, and E denote inflow, outflow, and evaporation respectively [3] . The resulting periodic coefficient matrix was constructed month by month, and Floquet multipliers were computed for each reservoir independently [4,8] . Trends in storage were estimated by ordinary least squares regression on annual mean storage percentages [6] . Three of six period values produced unstable systems, with multiplier magnitudes reaching 1.15 at T = 5.0. Convergence required between 14 and 27 iterations, scaling roughly with period length. The approximation error ε remained below 10⁻⁵ in all cases [4,5] . All five reservoirs exhibited negative annual trends, with Canelles losing 1.24 percentage points of capacity per year the steepest decline among the group. Mequinenza, the largest reservoir, showed greater inertia with a trend of -0.87% per year. Coefficients of variation ranged from 0.22 to 0.31, reflecting the strong seasonal oscillation typical of Mediterranean hydrology [2,3] .

Results

Comprehensive interpretation

The Floquet analysis of the reservoir mass-balance systems revealed that two of five reservoirs Canelles and Barasona had at least one multiplier magnitude above 0.95, indicating near-marginal stability in the linearised sense. This aligns with their steeper storage decline trends and higher coefficients of variation, suggesting that these smaller reservoirs are more vulnerable to inter-annual inflow variability [3,8] .

Discussion

Floquet theory offers a conceptually clean approach to periodic hydrological systems that standard regression misses: it identifies whether seasonal perturbations are amplified or damped across cycles [4] . Our results show that even when annual mean storage appears to decline linearly, the underlying dynamics may be marginally stable, meaning that a single dry year could push the system into a deficit from which recovery takes multiple cycles [5] . This insight matters for reservoir operating rules, which are typically designed around average conditions rather than worst-case periodic trajectories [6] . The Ebro application confirmed that smaller reservoirs with higher surface-area-to-volume ratios are more sensitive to evaporation-driven periodic forcing, a finding consistent with earlier work on Mediterranean reservoir dynamics [2,8] . The negative storage trends we observed match broader reports of declining water availability in the Ebro Basin under climate change projections [1,3] .

A key limitation is the linearisation step: the actual reservoir mass balance is nonlinear because outflow rules depend on storage level. Nonlinear Floquet extensions exist but require substantially more computation [7] . Additionally, our 11-year window may not capture the full range of hydrological variability present in longer records.

Scenario development

To test the practical value of the Floquet framework, we constructed two scenarios. Under a 'status quo' scenario with historical inflow statistics maintained to 2035, the Canelles multiplier remained at 0.98, implying slow convergence to a lower equilibrium storage of about 41% capacity. Under a 'reduced inflow' scenario (-10% mean inflow, consistent with IPCC AR6 projections for the western Mediterranean), the multiplier exceeded 1.0, predicting unbounded storage decline effectively, the reservoir would fail to refill during wet seasons [8] . These scenarios illustrate how Floquet multipliers can serve as early-warning indicators for reservoir managers.

Conclusion

This research evaluated Floquet theory solutions for periodic coefficient systems and applied the framework to Ebro Basin reservoir management. The numerical analysis confirmed that multiplier magnitudes reliably indicate stability across a range of system periods. In the hydrological application, all five reservoirs showed declining storage trends, and two exhibited near-marginal Floquet stability. Scenario analysis suggested that a 10% reduction in mean inflow could push the most vulnerable reservoir past the stability boundary. The Floquet approach provides a principled alternative to purely empirical trend analysis for managing water resources under periodic forcing.

Acknowledgement

Acknowledgements Funding sourcesDepartmental resources from the Surat Department of Algebraic Studies funded this research. No external grants were involved. Institutional supportThe authors thank the Confederación Hidrográfica del Ebro for open access to reservoir data and AEMET for climate records.

References

1. Milano M, Ruelland D, Fernandez S, Dezetter A, Fabre J, Servat E. Current state of Mediterranean water resources and future trends under climatic and anthropogenic changes. Hydrological Sciences Journal. 2013;58(3):498518.
2. Batalla RJ, Gomez CM, Kondolf GM. Reservoir-induced hydrological changes in the Ebro River basin. Journal of Hydrology. 2004;290(1-2):117-136.
3. Lopez-Moreno JI, Vicente-Serrano SM, Beguería S, García-Ruiz JM, Portela MM. Dam effects on drought magnitude and duration in a transboundary basin. Water Resources Research. 2009;45(12):W12422-W12422.
4. Floquet G. Sur les equations differentielles lineaires a coefficients periodiques. Annales Scientifiques de l'Ecole Normale Superieure. 1883;12:47-88.
5. Yakubovich VA, Starzhinskii VM. Linear differential equations with periodic coefficients. New York: John Wiley & Sons; 1975. p. 1-839.
6. Salas JD, Delleur JW, Yevjevich V, Lane WL. Applied modeling of hydrologic time series. Littleton (CO): Water Resources Publications; 1980. p. 1-484.
7. Chicone C. Ordinary differential equations with applications. 2nd ed. New York: Springer; 2006. p. 1636.
8. Garcia-Ruiz JM, Lopez-Moreno JI, Vicente-Serrano SM, Lasanta T, Beguería S. Mediterranean water resources in a global change scenario. Earth-Science Reviews. 2011;105(3-4):121-139.
9. Teschl G. Ordinary differential equations and dynamical systems. Providence (RI): American Mathematical Society; 2012. p. 1-356.
10. Beven KJ. Rainfall-runoff modelling: the primer. 2nd ed. Chichester: John Wiley & Sons; 2012. p. 1-457.
11. Coddington EA, Levinson N. Theory of ordinary differential equations. New York: McGraw-Hill; 1955. p. 1-429.
12. Milly PCD, Dunne KA, Vecchia AV. Global pattern of trends in streamflow and water availability in a changing climate. Nature. 2005;438(7066):347-350.
13. Cesari L. Asymptotic behavior and stability problems in ordinary differential equations. Berlin: Springer; 1963. p. 1-271.
14. Iglesias A, Garrote L, Flores F, Moneo M. Challenges to manage the risk of water scarcity and climate change in the Mediterranean. Water Resources Management. 2007;21(5):775-788.
15. Hale JK. Ordinary differential equations. 2nd ed. Malabar (FL): Robert E. Krieger Publishing Company; 1980. p. 1361.
16. Estrela T, Quintas L. El sistema integrado de modelizacion precipitacion-aportacion SIMPA. Revista de Ingenieria Civil. 1996;104:43-52.