Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · nlin/0007017

Symmetry and resonance in periodic FPU chains

Bob Rink
View original source ↗Read source document ↗Export metadata ↓

Abstract

The symmetry and resonance properties of the Fermi Pasta Ulam chain with periodic boundary conditions are exploited to construct a near-identity transformation bringing this Hamiltonian system into a particularly simple form. This `Birkhoff-Gustavson normal form' retains the symmetries of the original system and we show that in most cases this allows us to view the periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville integrable Hamiltonian. According to the KAM theorem this proves the existence of many invariant tori on which motion is quasiperiodic. Experiments confirm this qualitative behaviour. We note that one can not expect it in lower-order resonant Hamiltonian systems. So the FPU chain is an exception and its special features are caused by a combination of special resonances and symmetries.

Explore related subjects

nlin.CDcond-mat.stat-mechmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bob Rink. 2000-07-13. Symmetry and resonance in periodic FPU chains. https://doi.org/10.1007/s002200100428

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

KEEP EXPLORING

Related papers

arXiv2026

Attractors and basins generated by repeated sums of prime factors of natural numbers

Integer maps are discrete dynamical systems defined on the natural numbers. In this paper, we investigate the dynamics of the shifted Alladi-Erd{\H o}s map, a one-parameter family of integer maps in which each composite number is mapped to the sum of its prime factors, while each prime is mapped to $n+A$, where $A \in \mathbb{N}$ is a fixed shift parameter. By systematically exploring the parameter space for $2 \leq A \leq 10^5$, we uncover a rich bifurcation structure characterized by the emergence, disappearance, and reorganization of attractor cycles as the shift parameter varies. We find that although several attractors may coexist for a given value of $A$, for most values of $A$, almost all natural numbers belong to the basins of only two dominant attractors. Using elementary number theoretic arguments, we explain the observed bifurcation diagram. We further characterize the attractor cycles and quantify the distribution of their basin sizes across the parameter space. Our results reveal an unexpectedly rich landscape of arithmetic dynamics arising from a remarkably simple integer map. %and provide a systematic characterization of how attractors and their basins evolve as shift parameter is varied.

nlin.CD↗
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↗