Modeling nonlinear variable-order fractional chaotic systems using the Caputo-Fabrizio operator and radial basis function neural networks

Why unpredictable systems matter

From the weather and the stock market to brain activity and laser light, many systems in nature and technology behave in ways that look random but are actually governed by strict rules. This behavior is known as chaos. The article explores a new way to model such chaotic systems when they have a kind of "memory" of their past, and shows how a specialized type of neural network can learn and predict their wild motions with remarkable accuracy. Understanding and taming this kind of behavior can improve secure communications, control engineering, and signal processing.

Adding memory to chaos

Classic mathematical models of chaos use ordinary differential equations that treat the future as depending only on the present state. In reality, many systems remember what happened earlier: a material that has been stressed, an electronic component that has aged, or a biological rhythm shaped by past cycles. To capture this, researchers use "fractional" calculus, which lets the strength of this memory be tuned continuously between no memory and long memory. This paper goes a step further by letting that memory strength vary over time instead of staying fixed, creating what are called variable-order chaotic systems. Such models better reflect situations where memory gradually builds up, fades, or oscillates.

A smoother way to describe memory

The authors choose a particular mathematical tool, the Caputo–Fabrizio operator, to express this changing memory. Unlike some traditional formulations that involve sharp, singular kernels and can cause numerical headaches, this operator uses a smooth exponential kernel. That makes the equations easier and more stable to solve on a computer, especially for systems where only short- to medium-term memory is important. The team compares this choice with other popular operators and finds that, for their purposes, Caputo–Fabrizio strikes a balance: it retains the essential memory effects that shape chaotic motion while cutting computational cost and avoiding stiffness that can derail simulations.

Two ways a system can remember

To see how changing memory affects chaos, the researchers study a three-variable dynamical system whose trajectories trace looped, butterfly-like shapes in space. They test two scenarios for how the memory strength evolves. In the first, memory gradually strengthens over time, mimicking devices or circuits that become more history-dependent as they age. In the second, memory fluctuates periodically, echoing rhythmic biological or feedback-driven processes. For each case they simulate the system over a long time, examine the spread of values of the three variables, reconstruct the hidden geometric structure of the motion in "phase space," and compute Lyapunov exponents that measure how sensitively nearby trajectories diverge. They find that stronger memory generally intensifies chaotic behavior, while weaker memory dampens it, revealing a tight link between history and instability.

Teaching a neural network to follow chaos

Solving these memory-rich equations directly can be demanding, so the authors turn to an artificial intelligence approach. They employ radial basis function neural networks, a type of network particularly good at fitting smooth, nonlinear functions. Using simulated time series from their fractional variable-order system as training data, they configure networks with thousands of hidden units and train them to reproduce the system’s three state variables. Careful design choices—how the centers and widths of the radial functions are set, how data are split between training and testing, and how error is measured—allow the networks to approximate the chaotic trajectories with extremely small discrepancies, down to error levels near the limits of numerical precision.

What this means for real-world applications

The study shows that allowing a chaotic system’s memory to change over time produces models that more closely mimic complex, real-world behavior than traditional constant-order or memory-free equations. At the same time, the use of radial basis function neural networks turns these heavy mathematical descriptions into efficient, data-driven surrogates that can be evaluated quickly. For a non-specialist, the main takeaway is that the researchers have built a flexible and accurate toolkit for describing and predicting erratic signals that depend on their past history. Such tools could ultimately make it easier to design secure communication schemes, robust control strategies, and advanced signal-processing methods that take full advantage of, rather than fall victim to, chaos.

Citation: Sawar, S., Ayaz, M., Aldhabani, M.S. et al. Modeling nonlinear variable-order fractional chaotic systems using the Caputo-Fabrizio operator and radial basis function neural networks. Sci Rep 16, 7912 (2026). https://doi.org/10.1038/s41598-026-39288-8

Keywords: chaotic systems, fractional calculus, variable-order dynamics, neural networks, nonlinear modeling