Deautonomising the Lyness mapping

This paper demonstrates that while the standard Lyness mapping is deautonomisable only for the $N=2$ case, its derivative form allows for deautonomisation for arbitrary $N$, revealing novel secular dependencies and a new realization of the full-deautonomisation principle through the growth of late singularity confinement conditions.

The Gist

Imagine you are a gardener trying to grow a very specific, magical plant. This plant follows strict rules: every day, its height depends on its height from a few days ago and a constant amount of water you give it. In the world of mathematics, this plant is called the Lyness mapping.

For a long time, mathematicians knew this plant was "integrable." In plain English, this means the plant is perfectly predictable and stable; it doesn't grow into a chaotic mess, and you can write down a formula to know exactly where it will be in the future.

The authors of this paper, Grammaticos, Ramani, and Willox, decided to play a game with this plant. They asked: "What happens if we stop giving it a constant amount of water and instead change the amount of water every single day?"

In math terms, they wanted to turn a "constant" rule into a "changing" (non-autonomous) rule. This process is called deautonomisation. Usually, when you change the rules of a stable system, it breaks and becomes chaotic. The goal was to see if they could tweak the changing rules just right so the plant stayed stable and predictable.

Here is the story of their discovery, broken down into simple parts:

1. The Standard Approach: Only One Size Fits All

First, they tried to change the water rules for the plant in its standard form.

• The Result: They found that for the simplest version of the plant (where the rules depend on the last 2 days), they could successfully change the water rules and keep the plant stable.
• The Problem: When they tried this for more complex versions of the plant (depending on 3, 4, or more days), the plant immediately died. The changing water made it chaotic. It seemed impossible to make the complex versions work with changing rules.

2. The Secret Ingredient: The "Derivative" Form

Just when they were about to give up on the complex plants, they realized they were looking at the plant the wrong way. They decided to look at the rate of change of the plant's growth (the "derivative" form) instead of just the height itself.

• The Breakthrough: When they applied the changing water rules to this new perspective, something magical happened. All versions of the plant, no matter how complex (N=3, N=4, N=100), could be made stable!
• The Analogy: Imagine trying to balance a broom on your hand. If you try to balance it by looking at the tip, it's hard. But if you look at the handle and adjust your hand based on the handle's movement, suddenly, you can balance even the tallest, wobbliest broom. They found the "handle" of the Lyness mapping.

3. The Double-Exponential Surprise

When they looked closely at the simplest plant (N=2) using this new method, they found something they had never seen before.

• The Old Way: Usually, when you change rules over time, the change follows a single pattern, like a simple exponential curve (e.g., doubling every day).
• The New Discovery: In this specific case, the rules required two different exponential patterns happening at the same time. It's like the plant needed water that was both doubling every day AND halving every day simultaneously.
• The Magic Trick: Because these two patterns were so specific, they could actually blend them together to create a rule where the water amount changed linearly (adding a fixed amount every day) instead of exponentially. It was like discovering that a complex, high-tech engine could actually run on simple, straight-line logic.

4. The "Late" Confusion and the Mystery of Growth

The most fascinating part of the paper involves a concept called "Singularity Confinement."

• The Metaphor: Imagine the plant hits a snag (a "singularity") where it looks like it's about to explode or vanish. In a stable system, this snag gets "confined"—it resolves itself after a few steps, and the plant continues growing normally.
• The Late Confusion: The authors looked at cases where the snag took much longer to resolve (a "late" confinement). Usually, when a system is unstable, the math explodes into chaos.
• The Surprise: They found a system of equations that was non-linear and non-integrable (meaning it looked chaotic and unsolvable). However, when they watched how fast the numbers in this chaotic system grew, they discovered a hidden rhythm.
• The Lesson: Even though the system looked messy, the speed at which the numbers grew revealed the exact "stability score" (dynamical degree) of the original, perfect system. It's like listening to the static on a radio and realizing the static itself contains the perfect melody of the song underneath.

The Big Picture

This paper teaches us a profound lesson about how we understand complex systems:

1. Perspective Matters: Sometimes, a problem looks impossible until you change your point of view (looking at the derivative instead of the value).
2. Chaos Hides Order: Even in systems that look broken or chaotic (the "late confinement" case), there is a hidden mathematical fingerprint (the growth rate) that tells us exactly how the system behaves.
3. The "Full-Deautonomisation" Principle: The authors confirm that by carefully breaking the rules of a system and seeing how it tries to fix itself, we can uncover deep truths about its stability.

In short, they took a rigid, predictable mathematical garden, tried to make the weather change, found that the garden would die—until they looked at the garden from a different angle, where they discovered that any weather pattern could be survived, provided you knew the secret language of the plants.

Technical Summary

1. Problem Statement

The paper investigates the Lyness mapping, a family of $N$-th order discrete integrable systems defined by the recurrence relation:
$$x_{n+N}x_n = a + x_{n+1} + \dots + x_{n+N-1}$$
The primary objective is to perform deautonomisation on this system. Deautonomisation involves promoting the constant parameters (specifically $a$) to functions of the independent variable $n$ ($a \to a_n$) while preserving the system's integrability.

The authors aim to determine:

1. Whether the standard form of the Lyness mapping can be deautonomised for arbitrary order $N$.
2. Whether the derivative form of the mapping allows for deautonomisation for higher orders.
3. How the singularity confinement criterion and the full-deautonomisation principle apply to these systems, particularly in cases involving "late" confinement where the dynamical degree is not immediately obvious.

2. Methodology

The authors employ a combination of discrete integrability criteria:

• Singularity Confinement: Analyzing the behavior of the mapping when initial conditions lead to singularities (e.g., division by zero). In integrable systems, these singularities must disappear after a finite number of iterations.
• Full-Deautonomisation: Using the confinement conditions to derive the functional form of the time-dependent parameters ($a_n$, $p_n$, $q_n$).
• Dynamical Degree Analysis: Calculating the growth rate of the degrees of the iterates ($d_n$) to determine the dynamical degree $\lambda = \lim_{n\to\infty} d_n^{1/n}$. For integrable systems, $\lambda=1$; for non-integrable ones, $\lambda > 1$.
• Diophantine Integrability: Using the growth of the arithmetic height of rational iterates (Halburd's method) as a verification tool.
• Bilinear Formalism: Linking the nonlinear mapping to the Hirota-Miwa equation via $\tau$-functions to prove integrability.

3. Key Contributions and Results

A. Failure of Standard Deautonomisation for $N \geq 3$

• Case $N=2$: The standard Lyness mapping ($x_{n+2}x_n = a_n + x_{n+1}$) can be successfully deautonomised. The confinement condition yields $a_n = \kappa \lambda^n \phi_3(n)\phi_2(n)$, leading to a known $q$-discrete Painlevé equation.
• Case $N \geq 3$: The authors demonstrate that attempting to deautonomise the standard form (4) for $N \geq 3$ fails. The singularity pattern becomes unconfined (singularities do not disappear) unless the parameter $a_n$ remains constant.
  • Result: The standard Lyness mapping is not deautonomisable for $N > 2$ while preserving integrability.
  • Non-integrable Dynamics: For $N > 2$ with arbitrary $a_n$, the dynamical degree is found to be the largest root of $k^{2N} - k^{2N-1} - k^N + 1 = 0$, which is strictly greater than 1.

B. Success via the Derivative Form

The authors introduce the derivative form of the Lyness mapping (an $(N+1)$-th order system):
$$x_{n+N}(1 + x_n) = x_{n+1}(1 + x_{n+N+1})$$
They generalize this to a non-autonomous form:
$$x_{n+N}(p_n + x_n) = x_{n+1}(q_{n+N+1} + x_{n+N+1})$$

• Result: By applying the confinement criterion to $p_n$ and $q_n$, they successfully derive integrable deautonomisations for arbitrary $N$.
• Constraints: The confinement requires $q_n = p_n$. The resulting function is $p_n = \kappa \lambda^n \phi_N(n)$.
• Integrability: These systems can be integrated once to recover a lower-order mapping (related to discrete Painlevé equations) and are proven to be reductions of the Hirota-Miwa equation, confirming their integrability for all $N$.

C. Novelty in the $N=2$ Derivative Case

The deautonomisation of the derivative form for $N=2$ yields a unique result not previously observed:

• Dual Exponential Dependence: The confinement conditions for $p_n$ and $q_n$ lead to a solution involving two distinct exponential terms within the same parameter:
  $$q_{n+1} = \phi_4(n)(\sigma \phi_3(n)\lambda^n + \mu \lambda^{-3n})$$
• Additive Limit: Due to the presence of these two exponentials, the authors show that taking the limit $\lambda \to 1$ (with appropriate scaling of constants) transforms the multiplicative dependence into an additive linear dependence ($n$) without changing the functional form of the equation. This provides a new realization of a linearly dependent non-autonomous system.

D. Late Confinement and Non-Linear Dynamical Degrees

The paper explores "late" confinement scenarios (where singularities are confined after more iterations than the standard pattern).

• Standard Case: Usually, confinement conditions are linear or multiplicative, and the dynamical degree is the root of a characteristic polynomial.
• The $N=2$ Derivative Surprise: For the late confinement of the $N=2$ derivative mapping, the confinement conditions result in a system of two coupled, non-linear, non-integrable equations (Eqs. 37 and 38).
• New Realisation of Full-Deautonomisation:
  • The dynamical degree is not given by the root of a linear characteristic equation.
  • Instead, the dynamical degree emerges from the growth of the solutions to these non-linear confinement conditions themselves.
  • By analyzing the degree growth of the iterates of the confinement system, the authors calculated a dynamical degree of $\approx 1.425$, which perfectly matches the value predicted by the characteristic equation of the late confinement pattern ($k^9 - k^7 - k^6 - k^3 - k^2 + 1 = 0$).
  • This confirms that the full-deautonomisation principle holds even when the confinement conditions themselves form a complex, non-integrable system.

4. Significance

1. Extension of Integrability: The paper resolves the issue of deautonomising higher-order Lyness mappings by shifting focus from the standard form to the derivative form, providing a method to construct integrable equations of arbitrary order.
2. Validation of Full-Deautonomisation: The study of the $N=2$ derivative case provides a profound validation of the full-deautonomisation principle. It demonstrates that the principle is robust enough to extract the dynamical degree even from the growth of solutions to non-linear, non-integrable constraint equations.
3. New Classes of Equations: The discovery of a system with dual exponential dependence that admits an additive limit expands the known landscape of discrete Painlevé-type equations.
4. Theoretical Insight: The work bridges the gap between singularity analysis, degree growth, and the Hirota-Miwa equation, reinforcing the connection between discrete integrability and the geometry of the solution space.

In conclusion, the authors successfully demonstrate that while the standard Lyness mapping resists deautonomisation for $N>2$, its derivative form offers a rich family of integrable non-autonomous systems. Furthermore, the analysis of late confinement in these systems reveals a novel mechanism where the dynamical degree is encoded in the growth of non-linear constraints, offering a deeper understanding of discrete integrability.