Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2502.21293

On the role of 5-wave resonances in the nonlinear dynamics of the Fermi-Pasta-Ulam-Tsingou lattice

Tiziana Comito·Matteo Lotriglia·Miguel D. Bustamante
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the dynamics of the $(α+β)$ Fermi-Pasta-Ulam-Tsingou lattice (FPUT lattice, for short) for an arbitrary number $N$ of interacting particles, in regimes of small enough nonlinearity so that a Birkhoff-Gustavson type of normal form can be found using tools from wave-turbulence theory. Specifically, we obtain the so-called Zakharov equation for $4$-wave resonant interactions and its extension to $5$-wave resonant interactions by Krasitskii, but we introduce an important new feature: even the generic terms in these normal forms contain $resonant$ $interactions$ $only$, via a $unique$ canonical transformation. The resulting normal forms provide an approximation to the original FPUT lattice that possesses a significant number of exact quadratic conservation laws, beyond the quadratic part of the Hamiltonian. We call the new equations "exact-resonance evolution equations" and examine their properties: (i) Heisenberg representation's slow evolution allows us to implement numerical methods with large time steps to obtain relevant dynamical information, such as Lyapunov exponents. (ii) We introduce tests, such as convergence of the normal form transformation and truncation error verification, to successfully validate our exact-resonance evolution equations. (iii) The systematic construction of new quadratic invariants (via the resonant cluster matrix) allows us to use finite-time Lyapunov exponent calculations to quantify the level of nonlinearity at which the original FPUT lattice is well approximated by the exact-resonance evolution equations. We show numerical experiments in the case $N=9$, but the theory and numerical methods are valid for arbitrary values of $N$. We conclude that, when $3$ divides $N$, at small enough nonlinearity the FPUT lattice's dynamics and nontrivial hyperchaos are governed by $5$-wave resonant interactions.

Explore related subjects

nlin.CDnlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tiziana Comito, Matteo Lotriglia, Miguel D. Bustamante. 2025-02-28. On the role of 5-wave resonances in the nonlinear dynamics of the Fermi-Pasta-Ulam-Tsingou lattice. https://arxiv.org/abs/2502.21293

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

KEEP EXPLORING

Related papers

arXiv2026

Reservoir Computing with a single Josephson junction

Physical reservoir computing exploits the nonlinear dynamics of a physical system to perform information processing tasks. Josephson junctions (JJs), as nonlinear superconducting devices with rich dynamical behavior, represent promising yet relatively unexplored candidates for reservoir computing. In this work, we demonstrate for the first time that a single Josephson junction can be employed as a reservoir computing substrate without the use of an explicit delay loop. Using numerical simulations, we analyze the reservoir performance in different dynamical regimes and show that optimal performance is achieved when the JJ operates in a stable yet responsive regime. Despite the absence of delayed feedback, the JJ exhibits sufficient memory through its intrinsic dynamics to achieve good performance on a chaotic time series prediction task. The underlying mechanism is analogous, at the dynamical level, to that of a driven nonlinear pendulum, highlighting the generality of the approach to other nonlinear oscillators. In addition, we explore an alternative input masking approach based on continuous modulation, highlighting its compatibility with practical implementations. These results establish Josephson junctions as a viable and efficient platform for reservoir computing and open the way to ultrafast, low-dissipation hardware realizations.

nlin.CD↗
arXiv2026

Risk-Sensitive Learning in Population Games under Extreme Events: Bifurcations and Chaotic Dynamics

Inspired by nonequilibrium phenomena in game dynamics and behavioral evidence on the impact of extreme events on decision making, we investigate the nonlinear dynamics of a discrete-time multiagent learning rule in population congestion games under extreme events affecting one of the actions. The population state, following a risk-sensitive variant of the Multiplicative Weights Update (MWU), is coupled with a belief variable capturing the agents perceived risk and updated through an adaptive expectation rule. We perform a two-parameter bifurcation analysis with respect to the agents controlled parameters, identifying regions of qualitatively distinct behavior. Equilibria are studied first from both game-theoretic and dynamical perspectives. The resulting two-dimensional system exhibits complex behavior, including multi-stability among fixed points, invariant curves, periodic and chaotic attractors. Despite this complexity, the attractors can be grouped into distinct families, while the Cesàro averages of the trajectories are shown to converge to the stationary equilibrium. The incorporation of risk associated with the extreme event leads to new dynamical phenomena: attracting invariant curves arise and give rise to phase-locking Arnold tongues, within which the dynamics is qualitatively similar. In this setting, codimension-two resonances are identified as organizing centers, both within individual tongues and along the bifurcation curves associated with the fixed-point family. Chaotic attractors emerge and are destroyed through Feigenbaum cascades and forward or reverse boundary crises, with interior and merging crises also observed, along with transient chaos and narrow periodic windows. For each qualitatively distinct region, representative phase portraits and the associated basins of attraction are examined.

nlin.CD↗
arXiv2026

Intermittency-induced transitions in fast-slow dynamical systems

Intermittent dynamics are ubiquitous in the Earth system and often arise from the interaction of processes evolving on different time scales. In this work, we investigate how intermittent bursts in a fast forcing system propagate to and reshape the dynamics of a slower response system that would otherwise settle onto a quasi-stationary or weakly oscillatory regime. We address this question in two coupled models of increasing complexity: a low-dimensional Lorenz-63 system and the spatially extended Kuramoto-Sivashinsky equation. Across both systems, intermittency in the forcing progressively reshapes the attractor of the slow response system and drives it into different regimes. Using the Wasserstein distance, we show that increasing the frequency of intermittent events progressively displaces the response attractor from its unperturbed counterpart, up to a limit beyond which this deviation saturates. We then show that varying the forcing intensity and the time-scale separation between the forcing and response systems drives distinct regime transitions, which we characterize through the variance of ensemble maxima, the power spectra of both systems, and extreme value statistics. Finally, we examine how the type of intermittency in the forcing system affects synchronization between the two systems through local phase locking, showing that specific transitions in the phase-locking behavior are tied to the underlying intermittency regime, and that the response delay scales exponentially with the time-scale separation.

nlin.CD↗