Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2611.04962

Nonintegrability theorem on perturbed integrable Hamiltonian dynamical systems with two independent resonances

Nikos Zafeiriadis·Efthymia Meletlidou·Christos Efthymiopoulos·Christos Volos
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we establish a new theorem for nonintegrability of a perturbed, autonomous, two-degree-of-freedom Hamiltonian dynamical system with two independent resonances, based on a criterion which utilizes the well-known theorem of Poincaré and a Lie-series symplectic transformation. We also make certain assumptions and suggestions as to when the theorem holds and when it does not.

Explore related subjects

nlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nikos Zafeiriadis, Efthymia Meletlidou, Christos Efthymiopoulos, Christos Volos. 2026-11-03. Nonintegrability theorem on perturbed integrable Hamiltonian dynamical systems with two independent resonances. https://arxiv.org/abs/2611.04962

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Dynamics and non-integrability of the Swinging Atwood Machine with a massive string: chaos, periodic orbits and resonance structures

Building upon our previous studies on nonlinear variable-length pendulum systems, we investigate the Swinging Atwood Machine with a massive string. In contrast to the classical model, string inertia introduces a configuration-dependent moment of inertia, leading to a modified Hamiltonian structure and substantially richer dynamics. To uncover the global organization of the phase space, we combine Poincaré sections, bifurcation diagrams, and Lyapunov exponent maps with our recently developed numerical framework ,,Lyapunov Refined Maps". This approach provides a unified visualization of periodic, quasi-periodic, chaotic, and terminating motions, revealing intricate resonance networks and high-order periodic structures. We investigate the influence of the string mass, system parameters, and energy by constructing Lyapunov maps in parameter and initial-condition spaces and on fixed-energy surfaces. Liouville integrability is studied within the Morales-Ramis theory. By analyzing the normal variational equations along explicit non-stationary radial solutions and applying the Kovacic algorithm, we prove that the differential Galois group is generically SL(2,C), providing a rigorous obstruction to meromorphic Liouville integrability for every nonzero string mass. Thus, the exceptional integrable case of the classical Swinging Atwood Machine is destroyed by the inclusion of string inertia.

nlin.CD↗
arXiv2026

Modal interactions in strongly nonlinear coupled oscillators with closely spaced modes and nonclassical damping distribution

Linear and nonlinear mechanical structures with closely spaced modes and non-classical (nonproportional) damping distribution are susceptible to modal interactions with energy exchanges between modes. In this work we analyze these interactions in a two-degree-of-freedom (2 DoF) coupled oscillator subject to impulse excitation, utilizing action-angle variables and the method of multiple scales. To show the basic steps of the methodology we start from the linear system, then proceed to analyze modal interactions in the system with strongly nonlinear cubic stiffness nonlinearity. In contrast to the linear case, once the dynamics of the nonlinear system is captured in 1:1 resonance, strongly nonlinear modal interactions occur depending on energy. These regimes as well as attraction to capture, 1:1 capture and escape from capture are analyzed in detail. Our analytical study determines the ranges of parameter values for which nonlinear modal interactions due to resonance capture occur. The analytical predictions are fully verified by direct numerical simulations of the governing equations of motion. Possible extensions of the present work to systems of oscillators with more DoF and other types of dissipative elements, e.g., friction or hysteresis are discussed.

nlin.CD↗
arXiv2026

Symbol sequences from three-rotor coincidences and their word-complexity

In the three-rotor problem, three equally massive point particles move on a circle interacting via attractive pairwise cosine potentials. Rotors can represent superconducting phases of metallic segments in a chain of coupled Josephson junctions. We propose a digitization of the classical dynamics that records successive pair and triple coincidences of rotors using four symbols. Rotor coincidences correspond to boundaries in a disjoint partition of the configuration torus into cells where the rotors are ordered clockwise and anticlockwise. It is shown that isolated rotor coincidences must be crossings. Despite being a rather coarse digitization, we find that replacing trajectories by coincidence symbol sequences captures significant qualitative features of the dynamics through word statistics. Word-complexity $C_n$ measures the diversity of $n$-letter words in the symbol sequence while topological entropy governs asymptotic exponential growth of $C_n$. Sequences from periodic orbits have a word-complexity that saturates at the period. Ultra-high-energy trajectories with irrational `slope' are quasiperiodic. We show that they have zero entropy and $C_n = n+3$ by examining limiting slopes and by a mapping to Sturmian sequences. We examine their grammar rules and propose how their right-special words may be identified. On the other hand, numerical investigation of sequences from globally chaotic orbits leads us to conjecture an exponentially growing word-complexity $C_n = 3 \times 2^{n-1}$ with a topological entropy $\log 2$. We identify grammar rules that imply this conjectured word-complexity and model the resulting sequences as a subshift of finite type, unlike those at ultra-high energies which are not topological Markov chains.

nlin.CD↗