### 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 L_{i}, i = 1, . . . , N - 1, exhibit a transition between two power laws, L_{i} ∝ E^{Bk}, B_{k} > 0, k = 1, 2, occurring at the same value of E. The destabilization energy E_{c} 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 h_{KS}/N ∝ const., in the thermodynamic limit of fixed energy density E/N with E and N arbitrarily large.

Original language | English |
---|---|

Pages (from-to) | 1777-1793 |

Number of pages | 17 |

Journal | International Journal of Bifurcation and Chaos |

Volume | 16 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2006 |

Externally published | Yes |

### Fingerprint

### 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

*International Journal of Bifurcation and Chaos*,

*16*(6), 1777-1793. https://doi.org/10.1142/S0218127406015672

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

Research output: Contribution to journal › Article

*International Journal of Bifurcation and Chaos*, vol. 16, no. 6, pp. 1777-1793. https://doi.org/10.1142/S0218127406015672

}

TY - JOUR

T1 - Chaotic dynamics of N-degree of freedom hamiltonian systems

AU - Antonopoulos, Chris

AU - Bountis, Tassos

AU - Skokos, Charalampos

PY - 2006/6

Y1 - 2006/6

N2 - 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.

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.

KW - Hamiltonian systems

KW - Kolmogorov entropy

KW - Lyapunov spectra

KW - Regular and chaotic behavior

KW - SALI method

KW - Simple periodic orbits

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UR - http://www.scopus.com/inward/citedby.url?scp=33748282546&partnerID=8YFLogxK

U2 - 10.1142/S0218127406015672

DO - 10.1142/S0218127406015672

M3 - Article

VL - 16

SP - 1777

EP - 1793

JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

SN - 0218-1274

IS - 6

ER -