Chaos in the Hamiltonian mean field model

Abstract

We study the dynamical properties of the canonical ordered phase of the Hamiltonian mean-field (HMF) model, in which $N$ particles, globally-coupled via pairwise attractive interactions, form a rotating cluster. Using a combination of numerical and analytical arguments, we first show that the largest Lyapunov exponent remains strictly positive in the infinite-size limit, converging to its asymptotic value with $1/\ln N$ corrections. We then elucidate the scaling laws ruling the behavior of this asymptotic value in the critical region separating the ordered, clustered phase and the disordered phase present at high energy densities. We also show that the full spectrum of Lyapunov exponents consists of a bulk component converging to the (zero) value taken by a test oscillator forced by the mean field, plus subextensive bands of $\mathcal{O}(\ln N)$ exponents taking finite values. We finally investigate the robustness of these results by studying a "2D" extension of the HMF model where each particle is endowed with 4 degrees of freedom, thus allowing the emergence of chaos at the level of single particle. Altogether, these results illustrate the subtle effects of global (or long-range) coupling, and the importance of the order in which the infinite-time and infinite-size limits are taken: for an infinite-size HMF system represented by the Vlasov equation, no chaos is present, while chaos exists and subsists for any finite system size.

References in corpus (7)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Characterizing dynamics with covariant Lyapunov vectors
• Entrainment transition in populations of random frequency oscillators
• Hyperbolicity and the effective dimension of spatially-extended dissipative systems
• Hyperbolic decoupling of tangent space and effective dimension of dissipative systems
• Lyapunov analysis captures the collective dynamics of large chaotic systems
• Extensive and Sub-Extensive Chaos in Globally-Coupled Dynamical Systems

Cited by in corpus (16)

• Intermittent chaotic chimeras for coupled rotators
• Chimera states in coupled Kuramoto oscillators with inertia
• Dynamics and Statistics of the Fermi--Pasta--Ulam $\beta$--model with different ranges of particle interactions
• Sensitivity to initial conditions of a $d$-dimensional long-range-interacting quartic Fermi-Pasta-Ulam model: Universal scaling
• Critical fluctuations and slowing down of chaos
• Diverging fluctuations of the Lyapunov exponents
• $N$-body chaos and the continuum limit in numerical simulations of self-gravitating systems, revisited
• Origin and scaling of chaos in weakly coupled phase oscillators
• Lyapunov Exponent and Criticality in the Hamiltonian Mean Field Model
• Dynamics of fully coupled rotators with unimodal and bimodal frequency distribution
• A robust balancing mechanism for spiking neural networks
• Importance-sampling computation of statistical properties of coupled oscillators
• Quantum fluctuations inhibit symmetry breaking in the HMF model
• Classical Goldstone modes in Long-Range Interacting Systems
• Contribution of individual degrees of freedom to Lyapunov vectors in many-body systems
• Hard-core collisional dynamics in the hamiltonian mean-field model