scholarly journals Lyapunov exponents for Hamiltonian systems under small Lévy-type perturbations

2021 ◽  
Vol 31 (8) ◽  
pp. 081101
Author(s):  
Ying Chao ◽  
Pingyuan Wei ◽  
Jinqiao Duan
Keyword(s):  
Lyapunov Exponents ◽  
Hamiltonian Systems

Probability distributions of local Lyapunov exponents for Hamiltonian systems

1993 ◽  
Vol 47 (5) ◽  
pp. 3158-3173 ◽  
Author(s):  
Concezione Amitrano ◽  
R. Stephen Berry
Keyword(s):  
Lyapunov Exponents ◽  
Hamiltonian Systems ◽  
Probability Distributions

Scale-invariant Lyapunov exponents for classical hamiltonian systems

1985 ◽  
Vol 110 (5) ◽  
pp. 231-234 ◽  
Author(s):  
T.H. Seligman ◽  
J.J.M. Verbaarschot ◽  
M.R. Zirnbauer
Keyword(s):  
Lyapunov Exponents ◽  
Hamiltonian Systems ◽  
Scale Invariant

scholarly journals CHAOTIC DYNAMICS OF N-DEGREE OF FREEDOM HAMILTONIAN SYSTEMS

2006 ◽  
Vol 16 (06) ◽  
pp. 1777-1793 ◽  
Author(s):  
CHRIS ANTONOPOULOS ◽  
TASSOS BOUNTIS ◽  
CHARALAMPOS SKOKOS
Keyword(s):  
Lyapunov Exponents ◽  
Hamiltonian Systems ◽  
Finite Size ◽  
Period Doubling ◽  
Power Laws ◽  
Degree Of Freedom ◽  
Einstein Condensation ◽  
Global Dynamics ◽  
Local And Global Dynamics ◽  
Very High

We investigate the connection between local and global dynamics of two N-degree of freedom Hamiltonian systems with different origins describing one-dimensional nonlinear lattices: The Fermi–Pasta–Ulam (FPU) model and a discretized version of the nonlinear Schrödinger equation related to Bose–Einstein Condensation (BEC). We study solutions starting in the vicinity of simple periodic orbits (SPOs) representing in-phase (IPM) and out-of-phase motion (OPM), which are known in closed form and whose linear stability can be analyzed exactly. Our results verify that as the energy E increases for fixed N, beyond the destabilization threshold of these orbits, all positive Lyapunov exponents Li, i = 1,…, N - 1, exhibit a transition between two power laws, Li ∝ EBk, Bk > 0, k = 1, 2, occurring at the same value of E. The destabilization energy Ec per particle goes to zero as N → ∞ following a simple power-law, Ec/N ∝ N-α, with α being 1 or 2 for the cases we studied. However, using SALI, a very efficient indicator we have recently introduced for distinguishing order from chaos, we find that the two Hamiltonians have very different dynamics near their stable SPOs: For example, in the case of the FPU system, as the energy increases for fixed N, the islands of stability around the OPM decrease in size, the orbit destabilizes through period-doubling bifurcation and its eigenvalues move steadily away from -1, while for the BEC model the OPM has islands around it which grow in size before it bifurcates through symmetry breaking, while its real eigenvalues return to +1 at very high energies. Furthermore, the IPM orbit of the BEC Hamiltonian never destabilizes, having finite-size islands around it, even for very high N and E. Still, when calculating Lyapunov spectra, we find for the OPMs of both Hamiltonians that the Lyapunov exponents decrease following an exponential law and yield extensive Kolmogorov–Sinai entropies per particle h KS /N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

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