Global-best-guided electric eel foraging optimizer for robust parameter identification of Lorenz and memristive chaotic systems

Why this matters for real‑world chaos

From weather patterns to brain activity and power grids, many systems around us behave in ways that look random but actually follow hidden rules. These are called chaotic systems, and learning their rules precisely is crucial for secure communication, medical devices, and advanced electronics. This paper introduces a new way to uncover those hidden rules using a virtual swarm of “electric eels” that search for the best explanation of the data, achieving a level of precision far beyond existing methods.

When predictability hides inside apparent randomness

Chaotic systems sit at the edge between order and disorder. Small changes in their starting conditions can lead to completely different outcomes, which makes them powerful models of nature but also very hard to reverse engineer. To understand or control such systems, scientists often need to estimate a handful of key parameters—numbers that define how the system evolves over time. Traditional math tools struggle here because the landscape of possible answers is full of deceptive valleys and peaks, where a search can easily get stuck. Over the past two decades, researchers have increasingly turned to swarm‑like optimization methods, where many candidate solutions wander and learn together, as a more reliable way to solve these thorny inverse problems.

A digital swarm inspired by electric eels

The study builds on a recently proposed optimization method modeled on how electric eels forage. In this digital ecosystem, each “eel” represents a candidate solution—one particular guess at the system’s parameters. The swarm cycles through four behaviors: interacting with one another, resting in promising zones, hunting around attractive spots, and migrating toward new regions. These phases help keep the swarm diverse early on, when broad exploration is needed, and more focused later, when it is time to refine the best guesses. The authors’ main innovation is to gently layer a form of global learning on top of these behaviors, so that the swarm can share what it has collectively discovered without losing its variety.

Soft guidance from the best performers

The enhanced method, called global‑best‑guided electric eel foraging optimization (g‑EEFO), introduces a carefully controlled influence from the best solution found so far. After an eel finishes one of its four natural behaviors, its position is nudged slightly in the direction of the current best performer, using a rule borrowed from particle‑swarm methods. Crucially, this nudge is weak and temporary, and its strength changes over time according to an “energy” factor. Early in the search, the influence is small, allowing wide exploration; later, it grows, helping the eels converge on a common high‑quality solution. In this way, global information becomes a soft bias rather than a rigid pull, preserving the rich movement patterns that make the original algorithm powerful.

Putting the method to the test

To see how well g‑EEFO works, the authors apply it to two classic testbeds. The first is the famous Lorenz system, often used as a toy model of atmospheric convection and known for its butterfly‑shaped trajectory. The second is a more intricate electronic circuit that includes a “memristor,” a component whose resistance depends on its past, giving the system memory and making its behavior even more irregular. In both cases, the researchers generate synthetic time series from known parameters, then challenge several algorithms—including the original eel method and four recent competitors—to recover those parameters from the data. All methods are run under identical conditions, and their performance is compared using error measures, convergence curves, statistical tests, and how closely the recovered parameters match the true values.

Near‑perfect recovery of hidden rules

The results are striking. For the Lorenz system, g‑EEFO reduces the average error in the reconstructed behavior to around 10⁻²⁶, many orders of magnitude better than all rival methods, and with extremely small variation between runs. For the more demanding memristive circuit, it again outperforms competitors by several orders of magnitude while remaining remarkably stable. In practical terms, the recovered parameters are almost indistinguishable from the true ones, showing that the algorithm can reliably tease out the governing rules of both a well‑studied chaotic model and a more complex electronic system. Because the method does not depend on the specific equations and its extra computational cost is modest, the authors argue that it can be readily extended to other chaotic and even higher‑dimensional systems.

What this means going forward

To a non‑specialist, the key message is that the authors have found a way to let a digital swarm learn from its best members without collapsing into groupthink. By combining rich, nature‑inspired movement patterns with gentle global guidance, their g‑EEFO method can uncover the hidden rules behind seemingly erratic data with unprecedented accuracy and reliability. This makes it a promising tool for fields that depend on precise models of complex behavior, from secure communication schemes based on chaos to next‑generation electronic circuits and advanced control of unstable processes.

Citation: Izci, D., Ekinci, S., Ökten, İ. et al. Global-best-guided electric eel foraging optimizer for robust parameter identification of Lorenz and memristive chaotic systems. Sci Rep 16, 8579 (2026). https://doi.org/10.1038/s41598-026-39729-4

Keywords: chaotic systems, metaheuristic optimization, swarm intelligence, parameter identification, memristive circuits