{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T19:12:26Z","timestamp":1719861146557},"reference-count":41,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2006,6]]},"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\u2013Pasta\u2013Ulam (FPU) model and a discretized version of the nonlinear Schr\u00f6dinger equation related to Bose\u2013Einstein 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<\/jats:sub>, i = 1,\u2026, N - 1, exhibit a transition between two power laws, Li<\/jats:sub> \u221d EBk<\/jats:sub><\/jats:sup>, Bk<\/jats:sub> > 0, k = 1, 2, occurring at the same value of E. The destabilization energy Ec<\/jats:sub> per particle goes to zero as N \u2192 \u221e following a simple power-law, Ec<\/jats:sub>\/N \u221d N-\u03b1<\/jats:sup>, with \u03b1 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\u2013Sinai entropies per particle h KS <\/jats:sub>\/N \u221d const., in the thermodynamic limit of fixed energy density E\/N with E and N arbitrarily large. <\/jats:p>","DOI":"10.1142\/s0218127406015672","type":"journal-article","created":{"date-parts":[[2006,8,31]],"date-time":"2006-08-31T07:05:57Z","timestamp":1157007957000},"page":"1777-1793","source":"Crossref","is-referenced-by-count":31,"title":["CHAOTIC DYNAMICS OF N-DEGREE OF FREEDOM HAMILTONIAN SYSTEMS"],"prefix":"10.1142","volume":"16","author":[{"given":"CHRIS","family":"ANTONOPOULOS","sequence":"first","affiliation":[{"name":"Department of Mathematics, and Center for Research and Applications of Nonlinear Systems (CRANS), University of Patras, GR\u201326500, Rio, Patras, Greece"}]},{"given":"TASSOS","family":"BOUNTIS","sequence":"additional","affiliation":[{"name":"Department of Mathematics, and Center for Research and Applications of Nonlinear Systems (CRANS), University of Patras, GR\u201326500, Rio, Patras, Greece"}]},{"given":"CHARALAMPOS","family":"SKOKOS","sequence":"additional","affiliation":[{"name":"Department of Mathematics, and Center for Research and Applications of Nonlinear Systems (CRANS), University of Patras, GR\u201326500, Rio, Patras, Greece"},{"name":"Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4, GR\u201311527, Athens, Greece"},{"name":"Department of Applications of Informatics and Management in Finance, Technological Institute of Mesologhi, GR\u201330200, Mesologhi, Greece"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","volume-title":"Handbook of Mathematical Functions","author":"Abramowitz M.","year":"1965"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.76.1607"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02128236"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/BF02128237"},{"key":"rf5","first-page":"211","volume":"3","author":"Benettin G.","journal-title":"Physica D"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1063\/1.1855036"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(83)90236-1"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.69.046604"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/0370-1573(79)90023-1"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04917-4"},{"key":"rf12","volume-title":"An Introduction to the Theory of Functions of a Complex Variable","author":"Copson E. 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