Chaotic dynamics of N-degree of freedom hamiltonian systems

Chris Antonopoulos, Tassos Bountis, Charalampos Skokos

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

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, E c/N ∝ N, 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 hKS/N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

Original languageEnglish
Pages (from-to)1777-1793
Number of pages17
JournalInternational Journal of Bifurcation and Chaos
Volume16
Issue number6
DOIs
Publication statusPublished - Jun 2006
Externally publishedYes

Fingerprint

Hamiltonians
Degrees of freedom (mechanics)
Chaotic Dynamics
Bose-Einstein Condensation
Hamiltonian Systems
Bose-Einstein condensation
Orbits
Degree of freedom
Orbit
Lyapunov Exponent
Periodic Orbits
Motion
Power Law
Energy
Eigenvalue
Nonlinear Lattice
Lyapunov Spectrum
Decrease
Period-doubling Bifurcation
Global Dynamics

Keywords

  • Hamiltonian systems
  • Kolmogorov entropy
  • Lyapunov spectra
  • Regular and chaotic behavior
  • SALI method
  • Simple periodic orbits

ASJC Scopus subject areas

  • Modelling and Simulation
  • Engineering(all)
  • General
  • Applied Mathematics

Cite this

Chaotic dynamics of N-degree of freedom hamiltonian systems. / Antonopoulos, Chris; Bountis, Tassos; Skokos, Charalampos.

In: International Journal of Bifurcation and Chaos, Vol. 16, No. 6, 06.2006, p. 1777-1793.

Research output: Contribution to journalArticle

Antonopoulos, Chris ; Bountis, Tassos ; Skokos, Charalampos. / Chaotic dynamics of N-degree of freedom hamiltonian systems. In: International Journal of Bifurcation and Chaos. 2006 ; Vol. 16, No. 6. pp. 1777-1793.
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AB - 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, E c/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 hKS/N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

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