Lyapunov exponents of fractional-order systems: definition from the diffusive representation, ageing of memory and effect of memory truncation

In the study of complex systems, from the swirling patterns of weather to the firing of neurons, scientists often look for a signature of chaos. This signature is a measure of how quickly two nearly identical starting points drift apart over time. If they separate rapidly, the system is chaotic and unpredictable in the long run; if they stay close, the system is stable. For decades, researchers have used a specific mathematical tool to calculate this rate of separation, known as the Lyapunov exponent. This tool works beautifully for systems that behave like standard clocks, where the future depends only on the present moment. However, many real-world materials and processes, such as the way rubber stretches or how electricity moves through certain batteries, do not follow these simple rules. They possess a "memory," meaning their current state is influenced by everything that happened to them in the past. When scientists try to apply the standard tools to these memory-rich systems, the results have been confusing and often contradictory.

A new study by independent researcher Stefano Fazzino addresses this confusion by redefining how we measure chaos in systems with memory. The paper focuses on a specific type of mathematical model called a fractional-order system, which is used to describe processes where the past weighs heavily on the present. Unlike standard systems, these models do not just look at the current speed of change; they look at the entire history of the system, stretching back to the very beginning. Fazzino argues that because of this deep connection to the past, the standard way of calculating chaos fails. He proposes a new method that treats the system's memory not as a vague background influence, but as a vast collection of hidden internal states, each relaxing at its own speed. By viewing the system through this expanded lens, he is able to apply the rigorous mathematical rules of chaos theory that were previously unavailable.

The core discovery of this work is that the way we have been calculating chaos in these memory-based systems has been fundamentally flawed. When researchers use standard computer codes to estimate how fast these systems become chaotic, they often get negative numbers for the rates of separation, suggesting the system is shrinking or stabilizing. Fazzino's analysis shows that these negative numbers are not true measures of the system's long-term stability. Instead, they represent the contraction rates of the observable on short time scales or are spurious artifacts created when the memory is truncated. In reality, for these fractional-order systems, the true spectrum of separation rates consists of positive values indicating chaos, a zero exponent associated with the flow, and an essential band where all other exponents vanish. The negative values reported by existing software do not reflect the intrinsic stability of the system; they are merely the speed at which the calculation method forgets the history it was supposed to keep.

To reach this conclusion, Fazzino transformed the problem into a different mathematical space. Instead of trying to track a single point moving through time, he imagined the system as a continuous stream of internal filters, or "modes," that store information about the past. Some of these filters hold onto information for a very short time, while others hold it for an incredibly long time. This approach revealed a phenomenon called "ageing." Because the system never truly forgets its starting point, the memory of that initial condition grows stronger over time, even if the visible behavior of the system seems stable. This means the system is not invariant; it changes its character as it gets older. The study proves that to define chaos correctly, one must look at the system's behavior after it has settled into a steady rhythm, rather than trying to measure it from the very first moment it starts.

The paper also tested how different ways of simplifying the memory affect the results. In computer simulations, it is often necessary to cut off the memory after a certain amount of time to save computing power. Fazzino found that where you make this cut matters immensely. If you cut off the memory in the part of the equation that describes the rate of change, the results remain accurate, and the system behaves as expected. However, if you cut off the memory in the part that sums up the history, the system changes completely. In these cases, the chaotic behavior can disappear entirely, or the system can be forced into a stable state that it would never reach in reality. The study shows that a memory window of just one hundred time units can destroy the chaotic nature of a system that should be chaotic, while a window of five hundred units can reduce the measured chaos by half. This suggests that many previous studies using these shortcuts may have been studying a different, less chaotic system than the one they intended to model.

Furthermore, the research clarifies why the standard "zero" exponent, which usually indicates a system moving along a stable path, was missing from previous calculations. In a system with memory, the direction of time itself acts as a special path that does not grow or shrink. The new method captures this, showing that the system does indeed have a zero exponent, but only when the memory is treated correctly. The study also suggests that in the linear versions of these systems, the visible parts of the system do not shrink exponentially fast as they do in standard systems; instead, they fade away very slowly, following a power law. This slow fading explains why the negative exponents seen in older computer codes were actually just short-term measurements of this slow fade, rather than true indicators of stability.

The implications of these findings are significant for anyone modeling complex materials or devices. The study provides a new, reliable way to calculate the Lyapunov exponents for these systems, confirming that the positive exponents are real and robust, while the negative ones are largely mathematical artifacts or short-term contraction rates. It warns that using fixed memory windows in certain types of calculations can lead to false conclusions about whether a system is chaotic or stable. The author recommends using a specific approach that keeps the memory of the rate of change intact, or using the new method based on the internal modes, to ensure accuracy. While the mathematical proof for the existence of a stable, long-term state in these chaotic systems remains a hypothesis, the numerical evidence strongly supports the new definition. The work does not just offer a new calculation; it offers a corrected understanding of how memory shapes the unpredictability of the world around us.