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The initial condition as a decaying forcing: ageing and Lyapunov exponents of fractional maps

This paper demonstrates that fractional logistic and Hénon maps can be exactly reformulated as autonomous systems on memory modes to enable efficient computation of Lyapunov exponents, revealing that the initial condition acts as a decaying power-law forcing that induces a persistent, non-trivial memory of the starting state in chaotic regimes.

Original authors: Stefano Fazzino

Published 2026-10-22

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Stefano Fazzino

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of complex systems, scientists often look for simple rules that govern how things change over time. One such rule is the idea of a "map," a mathematical recipe that tells you where a system will be next based on where it is right now. In the simplest versions, the future depends only on the present. But many real-world systems, from the flow of fluids to the spread of diseases, have a memory. Their future depends not just on where they are now, but on where they have been in the past. When this memory fades slowly, like a fading echo that lingers for a long time, the system is described using "fractional" mathematics. These systems are notoriously difficult to study because their long memory means that calculating the next step requires knowing the entire history of the system. This makes them computationally expensive and mathematically opaque, leaving scientists unsure how to measure their chaos or predict their behavior.

A researcher named Stefano Fazzino has now found a way to cut through this complexity, revealing that these memory-laden systems can be understood as simple, self-contained machines. By reimagining the long history of the system not as a list of past events but as a collection of fading "modes," Fazzino showed that these fractional maps are actually autonomous systems. In this new view, the system does not need to look back at its entire past; instead, it carries its history within itself in the form of these modes, which update automatically at every step. This breakthrough allows scientists to simulate these systems for millions of steps on a standard laptop, a task that previously would have taken months or been impossible. More importantly, this new perspective clarifies how the system's starting point influences its future, a question that has puzzled researchers for years.

The study focuses on two specific types of these maps, which are variations of the famous logistic map used to model population growth. In the standard version, the population next year depends only on the current population. In the fractional versions, the new population depends on a weighted average of every single previous year, with older years having less influence but never disappearing completely. Fazzino demonstrated that these two maps, despite their different mathematical origins, share a hidden structure. Their memory kernels—the rules that determine how much weight to give to the past—are actually averages of simple geometric decays. This means the entire infinite history can be replaced by a finite set of about one hundred to two hundred "memory modes." Each mode acts like a separate channel that remembers the past for a specific duration, decaying geometrically over time. By tracking these modes instead of the raw history, the computer cost of running a simulation drops from a quadratic growth, where doubling the time quadruples the work, to a linear growth, where doubling the time simply doubles the work. This efficiency makes it possible to follow the system for up to one hundred million steps, a scale necessary to see long-term behaviors that were previously hidden.

With this powerful new tool, Fazzino investigated a fundamental question: how does the initial condition, the starting value of the system, affect its long-term behavior? In systems without memory, the starting point usually matters only for a brief moment before the system settles into a pattern. In these fractional systems, however, the starting point acts like a decaying force that pushes the system for a very long time. The researchers found that this influence fades slowly, following a power law where the effect decreases as the number of steps increases. For systems that settle into a steady state or a repeating cycle, this means the system's sensitivity to its starting point also fades away, eventually becoming zero. This explains why previous attempts to measure the chaos of these systems sometimes produced confusing negative numbers; those were likely just short-term artifacts of the system slowly forgetting its start.

The most striking discovery occurred in the chaotic regime, where the system behaves unpredictably. Here, the initial condition does not just fade away; it leaves a lasting imprint on the statistics of the chaos. The researchers observed that the rate at which the system expands and contracts—its local expansion rate, a key measure of chaos—remains dependent on the starting point for an astonishingly long time. In simulations with a specific fractional order, this dependence was still clearly visible after ten million iterations, a duration vastly longer than the time it takes for the system's internal correlations to disappear. The further the starting point was from the system's long-term average, the stronger and longer-lasting this drift was. However, if the system is started exactly at its long-term average, this drift vanishes, and the statistics become stationary immediately. This suggests a simple rule for studying these systems: to get a true measure of their chaos, one must either start the simulation at the system's average or account for the age of the simulation, as the results change depending on how long the system has been running.

These findings were not limited to the one-dimensional population models. Fazzino repeated the analysis on a two-dimensional version of the map, known as the Hénon map, and found the exact same behavior. The initial condition acted as a decaying forcing term in both cases, causing the chaotic statistics to age. This confirms that the phenomenon is a fundamental property of fractional systems with power-law memory, rather than a quirk of a specific equation. The study also clarified that along regular, non-chaotic paths, the Lyapunov exponent—a number used to quantify chaos—is exactly zero, meaning the system does not exhibit exponential sensitivity to initial conditions in the traditional sense. The negative values reported in earlier studies were likely finite-time estimates that had not yet converged to this zero limit.

The work resolves a long-standing ambiguity in the field by providing a consistent framework to define and compute Lyapunov exponents for fractional maps without arbitrarily cutting off the memory. It shows that the "ageing" of the system's statistics is not a numerical error but a genuine physical property of systems with long memory. The initial condition is not a forgotten detail but a persistent force that shapes the system's evolution for millions of steps. By identifying this mechanism, the research offers a clear path forward for analyzing complex systems with memory, suggesting that to understand the true nature of chaos in these systems, one must look beyond the immediate behavior and account for the slow, fading echo of where the system began.

Technical Summary: The Initial Condition as a Decaying Forcing in Fractional Maps

Problem Statement
Fractional difference maps, such as the fractional logistic map, are discrete dynamical systems where the new state depends on the entire past orbit through a weighted sum with power-law decaying weights. Unlike integer-order maps, these systems exhibit algebraic convergence to fixed points, bifurcation diagrams that depend on initial conditions and iteration counts, and chaotic regimes that are difficult to characterize quantitatively. A consistent definition and computation of Lyapunov exponents for these systems have been lacking. Existing methods, such as the Jacobian algorithm extended to memory systems, operate on undefined spaces and suffer from computational costs scaling quadratically ( $O(N^2)$ ) with orbit length $N$ , making it impractical to follow the long transients (often $10^5$ to $10^6$ iterations) characteristic of low-order fractional systems. Furthermore, it remains unclear whether finite-time exponent estimates converge and how the initial condition influences chaotic statistics over long timescales.

Methodology
The paper proposes a reformulation of Caputo-type fractional maps (specifically those with falling-factorial and power-law kernels) as autonomous dynamical systems on a space of "memory modes."

Diffusive Representation: The author exploits the fact that the kernels of these maps are moment sequences of positive measures on the unit interval. By expressing the kernel weights as integrals of geometric sequences ( $r^m$ ), the non-autonomous Volterra recurrence is rewritten as an autonomous map on the space $X = \mathbb{R} \times L^2(\beta)$ , where the state consists of the initial condition $\eta$ and a memory function $z(r)$ .
Quadrature and Computational Efficiency: To make the system computationally tractable, the author applies a trapezoidal quadrature in a logarithmic variable to discretize the memory integral. This reduces the infinite-dimensional memory space to a finite set of $M$ modes (typically 100–200). This approximation reproduces the kernel with relative errors below $4 \times 10^{-7}$ over $10^8$ steps. Crucially, this reduces the computational cost of an orbit of length $N$ from $O(N^2)$ to $O(MN)$ , which is linear in $N$ since $M$ is effectively constant for a given precision and order, enabling the simulation of orbits up to $10^8$ steps on standard hardware.
Tangent Map and Exponents: The tangent map is defined on the space of modes. It consists of a multiplication operator (contraction) plus a rank-one operator. This structure allows for the definition of Lyapunov exponents without truncating the memory. The author distinguishes between perturbations of the memory (subspace $\delta \eta = 0$ ) and perturbations of the initial condition ( $\delta \eta \neq 0$ ).
Analytical Derivation of Initial Condition Effects: By inverting the convolution relation, the author derives an exact expression showing that the initial condition $x_0$ acts as a forcing term proportional to $(x_0 - \xi)n^{-\nu}$ , where $\xi$ is an arbitrary constant (chosen as the time average $\bar{x}$ ) and $\nu$ is the fractional order.

Key Contributions and Results

Autonomous Formulation: The paper establishes that fractional maps are equivalent to autonomous maps on a space of memory modes. This provides a rigorous setting for defining Lyapunov exponents, resolving the ambiguity of the underlying state space in previous studies.
Zero Exponents for Regular Orbits: In regular regimes (orbits attracted to fixed points or periodic sinks), the author proves that the sensitivity to the initial condition, $\partial x_n / \partial x_0$ , decays as $n^{-\nu}$ . Consequently, the Lyapunov exponent is exactly zero. The negative exponents reported in previous literature are identified as finite-time artifacts of this algebraic decay, not true exponential contraction.
Ageing of Chaotic Statistics: In the chaotic regime, the local expansion rate and other statistical properties retain a memory of the initial condition for timescales far exceeding the correlation time (up to $10^7$ $1 0^{7}$ iterations). The deviation of these statistics from their stationary values decays algebraically as $n^{-p}$ $n^{- p}$ , where the exponent $p$ $p$ is close to the fractional order $\nu$ $ν$ .
- The amplitude of this drift is proportional to the distance $|x_0 - \bar{x}|$ .
- Orbits initiated close to the time average $\bar{x}$ do not exhibit this drift and yield stationary statistics immediately.
Validation on Hénon Map: The findings regarding the decaying forcing and ageing of statistics are confirmed in a two-dimensional fractional Hénon map, demonstrating that the results are not specific to the logistic nonlinearity.
Numerical Efficiency: The modal representation allows for the computation of exponents and long transients with linear cost, a significant improvement over previous quadratic methods.

Significance and Claims
The paper claims to provide a consistent quantitative framework for analyzing chaos in discrete fractional systems. By identifying the initial condition as a decaying forcing, the author explains the "ageing" phenomenon where chaotic statistics drift over time. This resolves the open problem of how to define and compute Lyapunov exponents in these systems:

For regular orbits: The maximal exponent is zero.
For chaotic orbits: The exponent depends on the initial condition and the orbit length (age). To obtain a stationary value, one must either start the orbit near the time average $\bar{x}$ or report the exponent alongside the age of the orbit.

The author emphasizes that the space of memory modes is the natural setting for such analysis, revealing that the dependence on initial conditions and the algebraic convergence of exponents are intrinsic properties of dynamics with power-law memory, rather than numerical artifacts. The work suggests that the "ageing" of chaotic statistics is a fundamental feature of fractional systems, observable in discrete time where continuous fractional oscillators (which are non-chaotic at low orders) could not demonstrate it.

Technical Summary: The Initial Condition as a Decaying Forcing in Fractional Maps

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