Abstract
We use the weight δI, deduced from the estimation of Lyapunov vectors, in order to characterise regions in the kinetic (x, v) space with particles that most contribute to chaoticity. For the paradigmatic model, the cosine Hamiltonian mean field model, we show that this diagnostic highlights the vicinity of the separatrix, even when the latter hardly exists.
Highlights
Lyapunov exponents are part of the fundamental characterisation of dynamical systems [30, 32]
The cosine Hamiltonian mean field model, we show that this diagnostic highlights the vicinity of the separatrix, even when the latter hardly exists
They measure the rate of divergence between nearby trajectories due to small perturbation in the initial conditions, viz. they determine the chaoticity of dynamical systems
Summary
Lyapunov exponents are part of the fundamental characterisation of dynamical systems [30, 32] They measure the rate of divergence between nearby trajectories due to small perturbation in the initial conditions, viz. For any finite N , the empirical measure and the single-point in Gibbs’ 6N -dimensional phase space provide equivalent information on the system microscopic state, and their evolutions under the equations of motion are equivalent. This is the cornerstone of the mean-field derivation of the Vlasov equation in the N → ∞ limit for smooth interparticle interactions [17, 39].
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