Dynamics of phases and chaos in models of locally coupled conservative or dissipative oscillators

Sergey P. Kuznetsov; Vyacheslav P. Kruglov

arxiv: 1906.10451 · v1 · pith:WHNEOJ23new · submitted 2019-06-25 · 🌊 nlin.CD

Dynamics of phases and chaos in models of locally coupled conservative or dissipative oscillators

Vyacheslav P. Kruglov , Sergey P. Kuznetsov This is my paper

Pith reviewed 2026-05-25 16:08 UTC · model grok-4.3

classification 🌊 nlin.CD

keywords Hamiltonian latticeTopaj-Pikovsky modelinvariant manifoldsnonlinear Schrödinger equationphase oscillatorsreversibilitychaosdissipative models

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The pith

A Hamiltonian lattice of locally coupled oscillators has invariant manifolds with dynamics exactly equivalent to the Topaj-Pikovsky phase model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a Hamiltonian model of an oscillator lattice with local coupling that represents spatial modes of the nonlinear Schrödinger equation with a periodic tilted potential. This Hamiltonian system exhibits the reversibility property of the Topaj-Pikovsky phase oscillator lattice. It contains invariant manifolds on which the dynamics are identical to those of the Topaj-Pikovsky model. Numerical simulations illustrate the resulting complex dynamics, and two dissipative models close to the Topaj-Pikovsky system are introduced. The equivalence connects conservative lattice dynamics to a known phase model.

Core claim

The Hamiltonian system manifests reversibility of the Topaj-Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with dynamics exactly equivalent to the Topaj-Pikovsky model.

What carries the argument

Invariant manifolds in the Hamiltonian oscillator lattice that carry dynamics exactly equivalent to the Topaj-Pikovsky phase model.

If this is right

The Hamiltonian lattice reproduces the reversibility of the Topaj-Pikovsky phase oscillator lattice.
Dynamics on the invariant manifolds are identical to the Topaj-Pikovsky model.
Numerical simulations of the Hamiltonian system reveal complex phase dynamics and chaos.
Two dissipative models are proposed that remain close to the Topaj-Pikovsky system.
The lattice describes spatial modes of the nonlinear Schrödinger equation with periodic tilted potential.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The manifold reduction may let known results on phase-oscillator chaos transfer directly to certain Hamiltonian systems.
Similar exact reductions could appear in other discretizations of wave equations with external potentials.
Numerical exploration of the proposed dissipative models could show how dissipation alters the reversible chaotic regimes.
The construction supplies a concrete route to embed phase models inside conservative lattices while preserving key invariants.

Load-bearing premise

The lattice model accurately captures spatial modes of the nonlinear Schrödinger equation with periodic tilted potential.

What would settle it

Numerical integration of the nonlinear Schrödinger equation with tilted potential whose spatial modes deviate from the lattice dynamics or fail to reproduce Topaj-Pikovsky behavior on the manifolds.

Figures

Figures reproduced from arXiv: 1906.10451 by Sergey P. Kuznetsov, Vyacheslav P. Kruglov.

**Figure 1.** Figure 1: Phase portraits of system (8) composed of N = 3 oscillators at parameter values β = 0, ε = 0.39, ω1 = −1, ω2 = 0, ω3 = 1. (a) Phase portrait for fixed populations I1 = I2 = I3 = 1/2 and initial phases for all trajectories φ2−φ1 = π−(φ3−φ2). (b,c,d) Phase portraits for unfixed populations I1 = I3 = 1/2+ 0.01, I2 = 1/2 − 0.02, panel (b) shows dynamics of phase shifts, (c) shows evolution of populations of fi… view at source ↗

**Figure 2.** Figure 2: Lyapunov exponents vs. ε on the invariant torus for N = 3. β = 0, ω1 = −1, ω2 = 0, ω3 = 1. dimensional stable manifold and phase volume on the invariant torus is overall contracting. Nevertheless in the neigborhood of the invariant torus Ij = 1/2 there are two directions of expansion and contraction that compensate contraction of phase volume on invariant torus. At ε > 0.6 invariant torus Ij = 1/2 has two … view at source ↗

**Figure 3.** Figure 3: Phase portraits in Poincaré cross-section by surface φ3 − φ2 = π/2 of system (8) composed of N = 4 oscillators. Populations I1 = I2 = I3 = I4 = 1/2 are fixed and initial phases for all trajectories satisfy φ2 − φ1 = π − (φ4 − φ3), φ3 − φ2 = π/2. Parameter values β = 0, ω1 = −1.5, ω2 = −0.5, ω3 = 0.5, ω4 = 1.5. At ε = 0.19 most of trajectories are regular (a). At ε = 0.39 some of trajectories are chaotic, b… view at source ↗

**Figure 4.** Figure 4: Lyapunov exponents vs. ε on the invariant torus for N = 4. β = 0, ω1 = −1.5, ω2 = −0.5, ω3 = 0.5, ω4 = 1.5. Initial condition for all values of ε was I1 = I2 = I3 = I4 = 1/2, φ1 = −π/2, φ2 = −π/3, φ3 = π/6, φ4 = π. This initial condition belongs to the invariant set of involution [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Phase portraits in Poincaré cross-section by surface φ3 − φ2 = π/2 of system (8) composed of N = 4 oscillators. Populations are not fixed. Initial conditions for all trajectories satisfy I1 = 1/2 + 0.01, I2 = 1/2−0.01, I3 = 1/2−0.01, I4 = 1/2 + 0.01, φ2 −φ1 = π −(φ4 −φ3), φ3 −φ2 = π/2. Parameter values ε = 0.19, β = 0, ω1 = −1.5, ω2 = −0.5, ω3 = 0.5, ω4 = 1.5. Panel (a) shows dynamics of phases, panel (b) … view at source ↗

**Figure 6.** Figure 6: Phase portraits in Poincaré cross-section by surface φ3 − φ2 = π/2 of system (8) composed of N = 4 oscillators. Populations are not fixed. Initial conditions for all trajectories satisfy I1 = 1/2 + 0.01, I2 = 1/2−0.01, I3 = 1/2−0.01, I4 = 1/2 + 0.01, φ2 −φ1 = π −(φ4 −φ3), φ3 −φ2 = π/2. Parameter values ε = 0.39, β = 0, ω1 = −1.5, ω2 = −0.5, ω3 = 0.5, ω4 = 1.5. Panel (a) shows dynamics of phases, panel (b) … view at source ↗

**Figure 7.** Figure 7: Lyapunov exponents vs. ε for N = 4 with non-constant populations. β = 0, ω1 = −1.5, ω2 = −0.5, ω3 = 0.5, ω4 = 1.5. Initial condition for all values of ε was I1 = 1/2 + 0.01, I2 = 1/2 − 0.01, I3 = 1/2 − 0.01, I4 = 1/2 + 0.01, φ1 = −π/2, φ2 = −π/3, φ3 = π/6, φ4 = π. 4 Dissipative models close to Topaj – Pikovsky system We propose two dissipative models close to Topaj – Pikovsky system. The first is a lattice… view at source ↗

read the original abstract

We discuss Hamiltonian model of oscillator lattice with local coupling. Model describes spatial modes of nonlinear Schr\"{o}dinger equation with periodic tilted potential. The Hamiltonian system manifests reversibility of Topaj - Pikovsky phase oscillator lattice. Furthermore, the Hamiltonian system has invariant manifolds with dynamics exactly equivalent to the Topaj - Pikovsky model. We demonstrate the complexity of dynamics with results of numerical simulations. We also propose two dissipative models close to Topaj - Pikovsky system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper establishes exact equivalence on invariant manifolds between its Hamiltonian lattice and the Topaj-Pikovsky phase model, and adds two new dissipative variants.

read the letter

The main advance here is the construction of invariant manifolds in a Hamiltonian oscillator lattice that carry dynamics identical to the Topaj-Pikovsky model, plus the introduction of two dissipative models built around the same setup. The reversibility match and the link to spatial modes of the nonlinear Schrödinger equation with tilted potential are the concrete pieces that go beyond prior work on the phase model alone. Numerical examples of complex behavior are included to illustrate the point. This is useful for people already working on coupled conservative and dissipative oscillators, as it gives a conservative counterpart and some extensions without adding free parameters. The modeling choice for the lattice seems reasonable on its face and the stress-test found no internal contradictions in the stated claims. The soft spot is that the abstract supplies no derivation steps, error bounds, or simulation parameters, so the exactness of the manifold reduction cannot be checked from the given text. If the full paper supplies those steps and they hold, the equivalence is a solid technical result; otherwise it stays at the level of a plausible reduction. This is a specialist paper. Readers focused on exact reductions in nonlinear lattices or on phase models will get something from it, while broader audiences will not. It is worth sending for peer review so that experts can verify the manifold calculations and the new models against the equations.

Referee Report

2 major / 0 minor

Summary. The paper introduces a Hamiltonian lattice model with local coupling that is claimed to describe spatial modes of the nonlinear Schrödinger equation with periodic tilted potential. It asserts that the system manifests the reversibility of the Topaj-Pikovsky phase oscillator lattice and possesses invariant manifolds on which the dynamics are exactly equivalent to the Topaj-Pikovsky model. Numerical simulations are presented to illustrate dynamical complexity, and two dissipative models approximating the Topaj-Pikovsky system are proposed.

Significance. If the exact manifold equivalence is established, the work would link Hamiltonian reductions of the NLS equation to a known phase-oscillator model, offering a conservative embedding that could clarify reversibility and chaos in synchronization problems. The proposal of nearby dissipative models is a useful extension. Credit is due for identifying an invariant-manifold reduction, but the absence of explicit steps limits assessment of whether the result is parameter-free or exact as stated.

major comments (2)

[Abstract] Abstract and main text: the central claim that invariant manifolds have dynamics 'exactly equivalent' to the Topaj-Pikovsky model is asserted without any derivation, reduction steps, or verification that reversibility is preserved under the mapping; this is load-bearing and requires explicit construction of the manifolds and the restricted vector field.
[Numerical simulations] Numerical simulations section: results demonstrating complexity are reported but no integration method, time step, lattice size, initial conditions, or error controls are supplied, preventing assessment of whether the observed behavior supports the equivalence claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed reading and useful suggestions. We agree that the manuscript requires additional explicit material on both the manifold reduction and the numerical methods, and we will incorporate these in a revised version.

read point-by-point responses

Referee: [Abstract] Abstract and main text: the central claim that invariant manifolds have dynamics 'exactly equivalent' to the Topaj-Pikovsky model is asserted without any derivation, reduction steps, or verification that reversibility is preserved under the mapping; this is load-bearing and requires explicit construction of the manifolds and the restricted vector field.

Authors: We accept that the current text asserts the existence of the invariant manifolds and the exact equivalence without supplying the explicit construction or the restricted vector field. In the revision we will add a dedicated section deriving the manifolds from the Hamiltonian lattice, restricting the vector field to them, and verifying that the resulting dynamics coincide with the Topaj-Pikovsky equations while preserving reversibility. revision: yes
Referee: [Numerical simulations] Numerical simulations section: results demonstrating complexity are reported but no integration method, time step, lattice size, initial conditions, or error controls are supplied, preventing assessment of whether the observed behavior supports the equivalence claim.

Authors: We agree that the numerical section lacks the necessary technical details. The revised manuscript will specify the integrator, time step, lattice sizes, initial conditions, and any error-control or conservation checks employed in the simulations. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central claim is that the Hamiltonian lattice model (derived from spatial modes of the NLS equation with periodic tilted potential) possesses invariant manifolds whose dynamics are exactly equivalent to the Topaj-Pikovsky model, along with shared reversibility. This equivalence is asserted as a direct structural property of the equations rather than a fitted quantity, self-definition, or reduction to prior self-citations. No load-bearing step reduces by construction to its inputs; the derivation chain remains independent of the target result and is supported by explicit model construction plus numerical evidence. The cited Topaj-Pikovsky model originates from external authors, introducing no self-citation circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract supplies no explicit free parameters, invented entities, or detailed axioms beyond the domain statement that the model describes spatial modes of the nonlinear Schrödinger equation.

axioms (1)

domain assumption Model describes spatial modes of nonlinear Schrödinger equation with periodic tilted potential
Invoked to justify the Hamiltonian lattice construction

pith-pipeline@v0.9.0 · 5605 in / 1011 out tokens · 74019 ms · 2026-05-25T16:08:49.224368+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the Hamiltonian system has invariant manifolds with dynamics exactly equivalent to the Topaj – Pikovsky model
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

involution R : Ij ↦→ IN−j+1, ψj ↦→ π − ψN−j

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

and Pikovsky, A., Reversibility vs

Topaj, D. and Pikovsky, A., Reversibility vs. synchroni zation in oscillator lattices, Physica D: Nonlinear Phenomena, 2002, vol. 170, no. 2, pp. 118–130

work page 2002
[2]

S., Gonchenko, S

Gonchenko, A. S., Gonchenko, S. V., Kazakov, A. O. and Tur aev, D. V., On the phenomenon of mixed dynamics in Pikovsky – Topaj system of coupled rotators, Physica D: Nonlinear Phenomena , 2017, vol. 350, pp. 45–57

work page 2017
[3]

Roberts, J. A. G. and Quispel, G. R. W., Chaos and time-rev ersal symmetry. Order and chaos in re- versible dynamical systems, Physics Reports, 1992, vol. 216, no. 2-3, pp. 63–177

work page 1992
[4]

Borisov, A. V. and Mamaev, I. S., Strange attractors in ra ttleback dynamics, Physics-Uspekhi, 2003, vol. 46, no. 4. pp. 393. 10

work page 2003
[5]

Thommen, Q., Garreau, J. C. and Zehnlé, V., Classical cha os with Bose-Einstein condensates in tilted optical lattices, Physical review letters , 2003, vol. 91, no. 21, pp. 210405

work page 2003
[6]

and Timme, M., Kuramoto dynamics in Hamilto nian systems Physical Review E

Witthaut, D. and Timme, M., Kuramoto dynamics in Hamilto nian systems Physical Review E. , 2014, vol. 90, no. 3, pp. 032917

work page 2014
[7]

A., Borisov, A

Bizyaev, I. A., Borisov, A. V. and Kuznetsov S. P, The Chap lygin sleigh with friction moving due to periodic oscillations of an internal mass, Nonlinear Dynamics, 2018, vol. 95, no. 1, pp. 699–714

work page 2018
[8]

and Zanette, D

Hampton, A. and Zanette, D. H., Measure synchronization in coupled Hamiltonian systems, Physical review letters, 1999, vol. 83, no. 11, pp. 2179

work page 1999
[9]

E., Njah, A

Vincent, U. E., Njah, A. N. and Akinlade, O., Measure sync hronization in a coupled Hamiltonian system associated with Nonlinear Schrödinger Equation, Modern Physics Letters B. 2005, vol. 19, no. 15, pp. 737–742. 11

work page 2005

[1] [1]

and Pikovsky, A., Reversibility vs

Topaj, D. and Pikovsky, A., Reversibility vs. synchroni zation in oscillator lattices, Physica D: Nonlinear Phenomena, 2002, vol. 170, no. 2, pp. 118–130

work page 2002

[2] [2]

S., Gonchenko, S

Gonchenko, A. S., Gonchenko, S. V., Kazakov, A. O. and Tur aev, D. V., On the phenomenon of mixed dynamics in Pikovsky – Topaj system of coupled rotators, Physica D: Nonlinear Phenomena , 2017, vol. 350, pp. 45–57

work page 2017

[3] [3]

Roberts, J. A. G. and Quispel, G. R. W., Chaos and time-rev ersal symmetry. Order and chaos in re- versible dynamical systems, Physics Reports, 1992, vol. 216, no. 2-3, pp. 63–177

work page 1992

[4] [4]

Borisov, A. V. and Mamaev, I. S., Strange attractors in ra ttleback dynamics, Physics-Uspekhi, 2003, vol. 46, no. 4. pp. 393. 10

work page 2003

[5] [5]

Thommen, Q., Garreau, J. C. and Zehnlé, V., Classical cha os with Bose-Einstein condensates in tilted optical lattices, Physical review letters , 2003, vol. 91, no. 21, pp. 210405

work page 2003

[6] [6]

and Timme, M., Kuramoto dynamics in Hamilto nian systems Physical Review E

Witthaut, D. and Timme, M., Kuramoto dynamics in Hamilto nian systems Physical Review E. , 2014, vol. 90, no. 3, pp. 032917

work page 2014

[7] [7]

A., Borisov, A

Bizyaev, I. A., Borisov, A. V. and Kuznetsov S. P, The Chap lygin sleigh with friction moving due to periodic oscillations of an internal mass, Nonlinear Dynamics, 2018, vol. 95, no. 1, pp. 699–714

work page 2018

[8] [8]

and Zanette, D

Hampton, A. and Zanette, D. H., Measure synchronization in coupled Hamiltonian systems, Physical review letters, 1999, vol. 83, no. 11, pp. 2179

work page 1999

[9] [9]

E., Njah, A

Vincent, U. E., Njah, A. N. and Akinlade, O., Measure sync hronization in a coupled Hamiltonian system associated with Nonlinear Schrödinger Equation, Modern Physics Letters B. 2005, vol. 19, no. 15, pp. 737–742. 11

work page 2005