Discrete mappings with an explicit discrete Lyapunov function related to integrable mappings

Daisuke Takahashi; Hironori Inoue; Junta Matsukidaira

REVIEW

Discrete mappings with an explicit discrete Lyapunov function related to integrable mappings

Hironori Inoue, Daisuke Takahashi, Junta Matsukidaira

Exactly Solvable and Integrable Systems · Submitted 1 Mar 2006 · 14 references

Pith's verdict: Clean construction, correct algebra, but the Lyapunov convergence theorem is conditional on an unproven inequality that can fail near the fixed point.

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arXiv IDnlin/0603005 (version 1)

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CitationDaisuke Takahashi, Hironori Inoue, Junta Matsukidaira. "Discrete mappings with an explicit discrete Lyapunov function related to integrable mappings." arXiv:nlin/0603005 (2006). https://pith.science/paper/ZAKXJYJH

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Editor's note

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Short version: this paper has a genuinely new construction—take a QRT map, add a damping term, and its conserved quantity becomes a discrete Lyapunov function—and the core algebra is correct. But the main convergence claim is conditional on an inequality that is only half-proven, and in one of the paper's own examples it can fail.

The identity (19)–(20) is the real result: the modified map is chosen so that h_{n+1} − h∞ = κ_n (h_n − h∞) with κ_n explicit, so if κ_n stays bounded away from 1 the trajectory is forced toward the level set h = h∞. That is a clean, useful mechanism, and the paper shows it survives the differential and ultradiscrete limits. The differential Lyapunov identity (39) is a direct and sound calculation. The idea of carrying a Lyapunov function through the ultradiscrete limit is nice, though the max-plus treatment is more heuristic than the rest.

The problem is condition (21), which is load-bearing. For class I the paper derives an upper bound on α_n but never proves the required lower bound ε < α_n (x_{n+1} − x_{n−1})²/D(...). That is not a mere gap: the example in Fig. 1 has a fixed point x* ≈ −0.165 with h(x*, x*) ≈ −0.157, far from h∞ = 5, and a stable manifold; orbits on that manifold have (x_{n+1} − x_{n−1})² → 0, so κ_n → 1 and h_n converges to −0.157, not to 5. The text's 'f₂² ≥ 4' observation does not control the denominator in α_n. For class II the paper explicitly omits the proof of (21), and the differential, ultradiscrete, and coupled-map sections all inherit that omission.

So the paper is honest, readable, and the machinery is worth knowing, but the stated 'any trajectory' claim is too strong. It deserves a serious referee—the construction is novel and reproducible—but a revision should prove the lower bound, or state the theorem only for orbits where the displacement term stays bounded below, and supply the class II proof.

I'd use this as a citation for the construction, not for the convergence theorem.

Referee report

Significance

The central idea is attractive and the algebraic identity (19)-(20) is a clean and checkable calculation. If the convergence claim were fully established, the paper would provide a useful method for constructing nonintegrable discrete systems with explicit Lyapunov functions from integrable ones, and for obtaining differential and ultradiscrete limits that inherit the Lyapunov property. The numerical examples are suggestive, and the possible applications to controlled coupled systems are interesting. However, the main convergence theorem is not proven in the claimed generality, and a concrete counterexample in the paper's own class I example appears to invalidate the statement that every trajectory approaches h_∞. The class II proof is explicitly omitted, and the later sections depend on it. Thus the contribution, as it stands, is a conditional construction rather than a general theorem.

Major objections (2)

[Section 2, Eq. (21) and Eq. (27)] The paper states that if condition (21) holds, then any trajectory gets close to the attractor h_n = h_∞, and it asserts that for class I the choice (27)-(28) ensures (21). However, the argument after Eq. (27) only proves the upper bound on α_n; the lower bound ε < α_n (x_{n+1}-x_{n-1})^2 / (D(x_{n-1},x_n)D(x_n,x_{n+1})) is never established. The remark that f_2(x)^2 = (x^2+2x+3)^2 ≥ 4 bounds α_n itself, not the product with the squared displacement. For the Fig. 1 parameters (a00,a01,a02,a11,a12,a22) = (1,2,3,1,1,0) and h_∞ = 5, the fixed point x* ≈ -0.1649 satisfies h(x*,x*) ≈ -0.157 ≠ 5, and its linearization has a stable eigenvalue. Consequently there are non-fixed orbits on the stable manifold for which x_{n+1} - x_{n-1} → 0, κ_n → 1, and h_n converges to h(x*,x*) rather than to h_∞. This directly contradicts the claimed convergence to h_∞ for this example, so the central claim of Section 2 is not correct as stated.
[Section 3, page 9, before Fig. 4] For the class II mapping (31), the proof that condition (21) holds under the assumptions x_n > 0 and (34) is omitted: the text says 'The proof is complicated and is omitted here.' This omission is load-bearing because the differential limit (Section 3.1), the ultradiscrete limit (Section 3.2), and the coupled mappings (Section 4.3) all rely on the class II convergence property. Without a proof or a precise statement of additional hypotheses under which (21) is guaranteed, the results of Sections 3 and 4 remain conditional. A complete proof would need to address the behavior near fixed points and stable manifolds, where the displacement in (21) can vanish.

Minor objections (3)

[Abstract] There is a typo in the abstract: 'disc rete' should be 'discrete'.
[Section 2, paragraph after Eq. (28)] The sentence 'Note that f_2(x)^2 = (x^2+2x+3)^2 ≥ 4 and the condition ε < α_n in (27) is satisfied in this case' is misleading, because the condition in (21) involves α_n times a squared displacement, not α_n alone. This should be clarified, especially in light of the fixed-point issue raised in the major comments.
[References] Reference [14] is listed as 'private communications with Ryogo Hirota'. It would be more appropriate to cite a published work or a preprint if one exists; otherwise, the derivation in Section 3.1 should be self-contained or acknowledged as a private communication in the text without a numbered reference.

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The paper

The paper claims

A modified QRT mapping carries an explicit discrete Lyapunov function—the QRT conserved quantity itself—that converges monotonically to a chosen level and draws every orbit to an attractor.

The reading

The paper tries to show that integrable QRT mappings can be modified, without destroying their explicit structure, so that their conserved quantity becomes a discrete Lyapunov function. In the modified maps, the quantity $h_n = h(x_{n-1},x_n)$ satisfies the one-step contraction formula $h_{n+1} - h_\infty = \kappa_n(h_n - h_\infty)$ with $\kappa_n \in [0,1)$, so $h_n$ moves monotonically toward the preassigned value $h_\infty$ and every orbit is pulled to the invariant curve $h_n = h_\infty$. Because the same function is conserved in the unmodified QRT case, the construction connects integrable dynamics, where orbits stay on level curves, to nonintegrable dynamics, where those level curves become attractors. The paper further claims that this Lyapunov structure survives differential and ultradiscrete limits and several practical variants, so the asymptotic behavior remains explicitly readable. A sympathetic reader would care because explicit Lyapunov functions for discrete maps are rare, and here one is inherited directly from an integrability structure.

What carries the argument

The central object is the QRT conserved quantity $h_n = N(x_{n-1},x_n)/D(x_{n-1},x_n)$, defined by two symmetric $3\times 3$ matrices $A$ and $B$. Its new role is enforced by the identity $h_{n+1}-h_\infty = \kappa_n(h_n-h_\infty)$, where $\kappa_n = 1 - \alpha_n (x_{n+1}-x_{n-1})^2 / (D(x_{n-1},x_n)D(x_n,x_{n+1}))$; the ratio $\kappa_n$ measures how much of the distance $h_n-h_\infty$ survives one step. The construction works because the modified map is engineered so that the numerator's difference $h_{n+1}-h_n$ becomes proportional to $-\alpha_n (x_{n+1}-x_{n-1})^2$ after division by $D(x_{n-1},x_n)D(x_n,x_{n+1})$, and choosing $\alpha_n$ so that $\kappa_n$ stays in $[0,1)$ turns the conserved quantity into a contracting clock. The paper makes the contraction check tractable by restricting to class I and class II QRT matrices, where $D(x,y)$ becomes $1$ or $xy$ respectively and the sufficient conditions on $\alpha_n$ or on the related parameter $\beta_n$ become rational inequalities. This is the machinery that carries all subsequent differential, ultradiscrete, and coupled-system statements.

Extended reading notes

Core claim

For the second-order mapping obtained by adding the term $-\alpha_n x_{n-1}(h_n-h_\infty)$ to the QRT form, the paper's central discovery is the identity $$h_{n+1}-h_\infty = \left(1 - \alpha_n \frac{(x_{n+1}-x_{n-1})^2}{D(x_{n-1},x_n)D(x_n,x_{n+1})}\right)(h_n-h_\infty).$$ Whenever $0 \le \kappa_n < 1-\varepsilon$ for all $n$, $h_n$ decreases or increases monotonically and converges to $h_\infty$; hence $h_n$ is an explicit discrete Lyapunov function and the curve $h_n = h_\infty$ is an attractor. The paper proves the strict contraction condition for special choices of the QRT matrices: a class I case with a rational sufficient bound on $\alpha_n$, and a class II case where positivity of the orbit and nonnegativity of the coefficients are assumed to imply the contraction condition. It then derives a continuum equation and an ultradiscrete piecewise-linear equation as limits of the class II map, each carrying an explicit Lyapunov function obtained from the same $h_n$, and demonstrates applications with adjusted, probabilistic, and coupled coefficients.

Load-bearing premise

The load-bearing premise is that along every orbit the contraction factor $\kappa_n$ stays between 0 and 1, which the paper asserts for its main class of examples from positivity and coefficient assumptions but without giving the proof, and verifies for the other class only by example.

Editorial extensions

If this is right

Under the stated conditions, every trajectory of the modified map approaches the attractor $h_n = h_\infty$ monotonically, rather than wandering over the whole level set of the QRT conserved quantity.
The differential limit (36) has an explicit Lyapunov function $\tilde h(y,y')$ whose time derivative is non-positive, so solutions converge to the limit cycle $\tilde h = \tilde h_\infty$.
The ultradiscrete limit (41) maps points in a region onto the attractor $H_n = H_\infty$ in finitely many steps, although additional attractors can appear because ultradiscretization changes strict inequalities into non-strict ones.
Parameter adjustment, probabilistic coefficients, and coupled mappings preserve monotone convergence as long as the chosen $\alpha_n$ or $\beta_n$ stays within the stated bounds.
The modified maps are nonintegrable because they are not reversible, yet they retain an explicit quantity that controls asymptotic behavior, illustrating a controlled bridge from integrable to nonintegrable discrete dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The construction suggests that any reversible integrable second-order map with an explicit rational conserved quantity could be perturbed in the same way, turning that quantity into a Lyapunov function; the paper lists this generalization as a future problem.
The class II convergence proof is omitted in the text, so a reader extending the result to new parameters should treat condition (21) as the hypothesis to check rather than as a theorem.
If the finite-step convergence observed in the ultradiscrete case holds generically, ultradiscrete maps of this type could serve as cellular-automaton models where the attractor is reached after a bounded number of updates, with possible applications in discrete optimization or control.
The coupled-system example shows a master-slave type arrangement where one map's Lyapunov function controls another map's convergence speed, which could be tested as a model of coordinated relaxation or synchronization.

Original abstract

We propose discrete mappings of second order that have a discrete analogue of Lyapunov function. The mappings are extensions of the integrable Quispel-Roberts-Thompson (QRT) mapping, and a discrete Lyapunov function of the mappings is identical to an explicit conserved quantity of the QRT mapping. Moreover we can obtain a differential and an ultradiscrete limit of the mappings preserving the existence of Lyapunov function. We also give applications of a mapping with an adjusted parameter, a probabilistic mapping and coupled mappings.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.62 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We propose discrete mappings of second order that have a discrete analogue of Lyapunov function. The mappings are extensions of the integrable Quispel-Roberts-Thompson (QRT) mapping, and a discrete Lyapunov function of the mappings is identical to an explicit conserved quantity of the QRT mapping. Moreover we can obtain a differential and an ultradiscrete limit of the mappings preserving the existence of Lyapunov function. We also give applications of a mapping with an adjusted parameter, a probabilistic mapping and coupled mappings.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We propose discrete mappings of second order that have a discrete analogue of Lyapunov function.	contribution 1.00	0.94	0.37	asserted 1.00
s1	The mappings are extensions of the integrable Quispel-Roberts-Thompson (QRT) mapping, and a discrete Lyapunov function of the mappings is identical to an explicit conserved quantity of the QRT mapping.	result 0.70	0.80	0.44	asserted 1.00
s2	Moreover we can obtain a differential and an ultradiscrete limit of the mappings preserving the existence of Lyapunov function.	result 0.58	0.93	0.15	asserted 0.84
s3	We also give applications of a mapping with an adjusted parameter, a probabilistic mapping and coupled mappings.	contribution 0.96	0.92	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

not examined

full text
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levels run

P abstract

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S sections
T body sentences
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Solutions to (24) with αn defined by (28) for (a00, a01, a02, a11, a12, a22) = (1, 2, 3, 1, 1, 0), h∞ = 5 and p = 1. Points (xn−1, xn) (1 ≤ n ≤ 1000) in the phase plane are plotted. (a) (x0, x1) = (3, 2), (b) (x0, x1) = (−0.2, −0.3). (a) (b)

**Figure 2.** Solutions to (24) with αn defined by (28) for p = 0.02. Parameters other than p are the same as in

**Figure 3.** Evolution of hn for Figs. 2 (a) and (b). 7

**Figure 4.** Solutions to (31) from (x0, x1) = (5, 1) and (2, 1) with (a00, a01, a02, a11, a12, a22) = (1, 0, 0, 0, 1, 2), h∞ = 6, βn = 100. Points (xn−1, xn) (1 ≤ n ≤ 1000) in the phase plane are plotted. Moreover, there is a remarkable feature about the mapping (31). Differential and ultradiscrete (piecewise-linear) equations are obtained by two kinds of limiting procedure. 3.1 Differential equation obtained from (31) With regard to a differential equation, we use the following transformation including a small parameter δ. xn = e y(nδ+ξ0) , a02 = c02, a11 = −2c02 + δ 2 c11, a00 = δ 2 c0

**Figure 5.** Solutions to (36) for (c00, c01, c02, c11, c12, c22) = (2, 1, 1, 0, 1, 1), eh∞ = 8, βe = 0.2 and (a) y(0) = 1, y ′ (0) = 0, (b) y(0) = y ′ (0) = 0. Both solutions are calculated numerically by the fourth-order Runge-Kutta method. Using (36), we can show d dt( eh − eh∞) = −4c 2 02 eβ e2y (y ′ ) 2 ( eh − eh∞). (39) This means eh converges to eh∞ and eh is a Lyapunov function to (36)

**Figure 6.** Solutions to (41) with (A00, A01, A02, A11, A12, A22) = (13, 71, 11, 17, 43, 23), H∞ = 100. (a) (x0, x1) = (800, 700), (b) (−13, −12). Note that aij = 0 corresponds to Aij → −∞ and the terms including Aij in (42) are eliminated from max functions

**Figure 7.** Evolution of points (x, y) = (5i, 5j) (−40 ≤ i, j ≤ 40) by the mapping (41). All parameters are the same as in

**Figure 8.** Mapping (24) with αn of (28) adjusted by p = c0 + (1−c0) sin2 (c1(hn −h∞)). Parameters and initial data are the same as in

**Figure 9.** Probabilistic mapping (24) with αn of (28) using probabilistic p defined by (44) with c = 0.5 and M = 30. Other parameters are the same as in

**Figure 10.** Evolution of the discrete Lyapunov function (a)

**Figure 11.** Solutions in a phase plane for (a) x (1) n and (b) x (2) n . Mappings and initial data are the same as in

Figures reproduced from arXiv:nlin/0603005 by the authors.

References

Works cited (14)

[1]

Hirota R. Nonlinear partial diﬀerence equations. I. A diﬀe rence analogue of the Korteweg-de Vries equation, J Phys Soc Jpn;43;1424-1433

View on Pith
[2]

Hirota R. Nonlinear partial diﬀerence equations. II. Dis crete-time Toda equation, J Phys Soc Jpn;43;2074-2078

View on Pith
[3]

Hietarinta J, Nijhoﬀ F W, Satsuma J. Symmetries and Integ rability of diﬀerence equations:special issue dedicated to the subject of the SID E IV meeting, J Phys A: Math Gen 2005;34;10337-10744. 16

View on Pith 2005
[4]

Noumi M, Yamada Y. Aﬃne Weyl groups, discrete dynamical s ystems and Painlev´ e equations, Comm Math Phys 1998;199;281-295

View on Pith 1998
[5]

Kajiwara K, Masuda T, Noumi M, Ohta Y, Yamada Y. Hypergeom etric solutions to the q-Painlev´ e equations, Int Math Res Not 200 4;47;2497-2521

View on Pith
[6]

Robinson C. Dynamical systems: stability, symbolic dynamics, and chao s, 2nd edition, CRC Press (Boca Raton, FL, 1999)

View on Pith 1999
[7]

Quispel G R W, Roberts J A G, Thompson C J, Integrable mappi ngs and soliton equations, Phys Lett A 1988;126;419-421

View on Pith 1988
[8]

Quispel G R W, Roberts J A G, Thompson C J, Integrable mappi ngs and soliton equations II, Physica D 1989;34;183-192

View on Pith 1989
[9]

Ramani A, Carstea A S, Grammaticos B, Ohta Y. On the autono mous limit of discrete Painlev´ e equations, Physica A 2002;305;437-444

View on Pith 2002
[10]

Willox R, Grammaticos B, Ramani A. A study of the antisym metric QRT mappings, J Phys A: Math Gen 2005;38;5227-5236

View on Pith 2005
[11]

Tokihiro T, Takahashi D, Matsukidaira J, Satsuma J. Fro m soliton equations to integrable cellular automata through a limiting procedu re, Phys Rev Lett 1996;76;3247-3250

View on Pith 1996
[12]

Matsukidaira J, Satsuma J, Takahashi D, Tokihiro T, Tor ii M. Toda-type cellular autoamaton and its N -soliton solution, Phys Lett A 1997;225;287-295

View on Pith 1997
[13]

Grammaticos B, Ohta Y, Ramani A, Takahashi D, Tamizhman i K M, Cellular automata and ultra-discrete Painlev´ e equations, Phys Lett A 1997;226;53-58

View on Pith 1997
[14]

private communications with Ryogo Hirota. 17

View on Pith

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Hirota R. Nonlinear partial diﬀerence equations. I. A diﬀe rence analogue of the Korteweg-de Vries equation, J Phys Soc Jpn;43;1424-1433

View on Pith

[2] [2]

Hirota R. Nonlinear partial diﬀerence equations. II. Dis crete-time Toda equation, J Phys Soc Jpn;43;2074-2078

View on Pith

[3] [3]

Hietarinta J, Nijhoﬀ F W, Satsuma J. Symmetries and Integ rability of diﬀerence equations:special issue dedicated to the subject of the SID E IV meeting, J Phys A: Math Gen 2005;34;10337-10744. 16

View on Pith 2005

[4] [4]

Noumi M, Yamada Y. Aﬃne Weyl groups, discrete dynamical s ystems and Painlev´ e equations, Comm Math Phys 1998;199;281-295

View on Pith 1998

[5] [5]

Kajiwara K, Masuda T, Noumi M, Ohta Y, Yamada Y. Hypergeom etric solutions to the q-Painlev´ e equations, Int Math Res Not 200 4;47;2497-2521

View on Pith

[6] [6]

Robinson C. Dynamical systems: stability, symbolic dynamics, and chao s, 2nd edition, CRC Press (Boca Raton, FL, 1999)

View on Pith 1999

[7] [7]

Quispel G R W, Roberts J A G, Thompson C J, Integrable mappi ngs and soliton equations, Phys Lett A 1988;126;419-421

View on Pith 1988

[8] [8]

Quispel G R W, Roberts J A G, Thompson C J, Integrable mappi ngs and soliton equations II, Physica D 1989;34;183-192

View on Pith 1989

[9] [9]

Ramani A, Carstea A S, Grammaticos B, Ohta Y. On the autono mous limit of discrete Painlev´ e equations, Physica A 2002;305;437-444

View on Pith 2002

[10] [10]

Willox R, Grammaticos B, Ramani A. A study of the antisym metric QRT mappings, J Phys A: Math Gen 2005;38;5227-5236

View on Pith 2005

[11] [11]

Tokihiro T, Takahashi D, Matsukidaira J, Satsuma J. Fro m soliton equations to integrable cellular automata through a limiting procedu re, Phys Rev Lett 1996;76;3247-3250

View on Pith 1996

[12] [12]

Matsukidaira J, Satsuma J, Takahashi D, Tokihiro T, Tor ii M. Toda-type cellular autoamaton and its N -soliton solution, Phys Lett A 1997;225;287-295

View on Pith 1997

[13] [13]

Grammaticos B, Ohta Y, Ramani A, Takahashi D, Tamizhman i K M, Cellular automata and ultra-discrete Painlev´ e equations, Phys Lett A 1997;226;53-58

View on Pith 1997

[14] [14]

private communications with Ryogo Hirota. 17

View on Pith