9,061 research outputs found
Chaotic Dynamics of N-degree of Freedom Hamiltonian Systems
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 Schrodinger 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 exhibit a transition between two power laws, occurring at the same
value of E. The destabilization energy E_c per particle goes to zero as N goes
to infinity following a simple power-law. 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. 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, in the thermodynamic limit of fixed
energy density E/N with E and N arbitrarily large.Comment: 29 pages, 10 figures, published at International Journal of
Bifurcation and Chaos (IJBC
Chaos in effective classical and quantum dynamics
We investigate the dynamics of classical and quantum N-component phi^4
oscillators in the presence of an external field. In the large N limit the
effective dynamics is described by two-degree-of-freedom classical Hamiltonian
systems. In the classical model we observe chaotic orbits for any value of the
external field, while in the quantum case chaos is strongly suppressed. A
simple explanation of this behaviour is found in the change in the structure of
the orbits induced by quantum corrections. Consistently with Heisenberg's
principle, quantum fluctuations are forced away from zero, removing in the
effective quantum dynamics a hyperbolic fixed point that is a major source of
chaos in the classical model.Comment: 6 pages, RevTeX, 5 figures, uses psfig, changed indroduction and
conclusions, added reference
Weak and strong chaos in Fermi-Pasta-Ulam models and beyond
We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions. (C) 2005 American Institute of Physics
Emergence of continual directed flow in Hamiltonian systems
We propose a minimal model for the emergence of a directed flow in autonomous
Hamiltonian systems. It is shown that internal breaking of the spatio-temporal
symmetries, via localised initial conditions, that are unbiased with respect to
the transporting degree of freedom, and transient chaos conspire to form the
physical mechanism for the occurrence of a current. Most importantly, after
passage through the transient chaos, trajectories perform solely regular
transporting motion so that the resulting current is of continual ballistic
nature. This has to be distinguished from the features of transport reported
previously for driven Hamiltonian systems with mixed phase space where
transport is determined by intermittent behaviour exhibiting power-law decay
statistics of the duration of regular ballistic periods
Directed transient long-range transport in a slowly driven Hamiltonian system of interacting particles
We study the Hamiltonian dynamics of a one-dimensional chain of linearly
coupled particles in a spatially periodic potential which is subjected to a
time-periodic mono-frequency external field. The average over time and space of
the related force vanishes and hence, the system is effectively without bias
which excludes any ratchet effect. We pay special attention to the escape of
the entire chain when initially all of its units are distributed in a potential
well. Moreover for an escaping chain we explore the possibility of the
successive generation of a directed flow based on large accelerations. We find
that for adiabatic slope-modulations due to the ac-field transient long-range
transport dynamics arises whose direction is governed by the initial phase of
the modulation. Most strikingly, that for the driven many particle Hamiltonian
system directed collective motion is observed provides evidence for the
existence of families of transporting invariant tori confining orbits in
ballistic channels in the high dimensional phase spaces
Cooperative surmounting of bottlenecks
The physics of activated escape of objects out of a metastable state plays a
key role in diverse scientific areas involving chemical kinetics, diffusion and
dislocation motion in solids, nucleation, electrical transport, motion of flux
lines superconductors, charge density waves, and transport processes of
macromolecules, to name but a few. The underlying activated processes present
the multidimensional extension of the Kramers problem of a single Brownian
particle. In comparison to the latter case, however, the dynamics ensuing from
the interactions of many coupled units can lead to intriguing novel phenomena
that are not present when only a single degree of freedom is involved. In this
review we report on a variety of such phenomena that are exhibited by systems
consisting of chains of interacting units in the presence of potential
barriers.
In the first part we consider recent developments in the case of a
deterministic dynamics driving cooperative escape processes of coupled
nonlinear units out of metastable states. The ability of chains of coupled
units to undergo spontaneous conformational transitions can lead to a
self-organised escape. The mechanism at work is that the energies of the units
become re-arranged, while keeping the total energy conserved, in forming
localised energy modes that in turn trigger the cooperative escape. We present
scenarios of significantly enhanced noise-free escape rates if compared to the
noise-assisted case.
The second part deals with the collective directed transport of systems of
interacting particles overcoming energetic barriers in periodic potential
landscapes. Escape processes in both time-homogeneous and time-dependent driven
systems are considered for the emergence of directed motion. It is shown that
ballistic channels immersed in the associated high-dimensional phase space are
the source for the directed long-range transport
Contributions of plasma physics to chaos and nonlinear dynamics
This topical review focusses on the contributions of plasma physics to chaos
and nonlinear dynamics bringing new methods which are or can be used in other
scientific domains. It starts with the development of the theory of Hamiltonian
chaos, and then deals with order or quasi order, for instance adiabatic and
soliton theories. It ends with a shorter account of dissipative and high
dimensional Hamiltonian dynamics, and of quantum chaos. Most of these
contributions are a spin-off of the research on thermonuclear fusion by
magnetic confinement, which started in the fifties. Their presentation is both
exhaustive and compact. [15 April 2016
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