455 research outputs found
The boundary of hyperbolicity for Henon-like families
We consider C^{2} Henon-like families of diffeomorphisms of R^{2} and study
the boundary of the region of parameter values for which the nonwandering set
is uniformly hyperbolic. Assuming sufficient dissipativity, we show that the
loss of hyperbolicity is caused by a first homoclinic or heteroclinic tangency
and that uniform hyperbolicity estimates hold uniformly in the parameter up to
this bifurcation parameter and even, to some extent, at the bifurcation
parameter.Comment: 32 pages, 11 figures. Several minor revisions, additional figures,
clarifications of some argument
Conditional exponents, entropies and a measure of dynamical self-organization
In dynamical systems composed of interacting parts, conditional exponents,
conditional exponent entropies and cylindrical entropies are shown to be well
defined ergodic invariants which characterize the dynamical selforganization
and statitical independence of the constituent parts. An example of interacting
Bernoulli units is used to illustrate the nature of these invariants.Comment: 6 pages Latex, 1 black and white and 2 color figures, replacement of
damaged gif file
Ergodic Properties of Invariant Measures for C^{1+\alpha} nonuniformly hyperbolic systems
For an ergodic hyperbolic measure of a
diffeomorphism, there is an full-measured set such
that every nonempty, compact and connected subset of
coincides with the accumulating set of time
averages of Dirac measures supported at {\it one orbit}, where
denotes the space of invariant measures
supported on . Such state points corresponding to a fixed
are dense in the support . Moreover,
can be accumulated by time averages of Dirac
measures supported at {\it one orbit}, and such state points form a residual
subset of . These extend results of Sigmund [9] from uniformly
hyperbolic case to non-uniformly hyperbolic case. As a corollary, irregular
points form a residual set of .Comment: 19 page
Resolution of two apparent paradoxes concerning quantum oscillations in underdoped high- superconductors
Recent quantum oscillation experiments in underdoped high temperature
superconductors seem to imply two paradoxes. The first paradox concerns the
apparent non-existence of the signature of the electron pockets in angle
resolved photoemission spectroscopy (ARPES). The second paradox is a clear
signature of a small electron pocket in quantum oscillation experiments, but no
evidence as yet of the corresponding hole pockets of approximately double the
frequency of the electron pocket. This hole pockets should be present if the
Fermi surface reconstruction is due to a commensurate density wave, assuming
that Luttinger sum rule relating the area of the pockets and the total number
of charge carriers holds. Here we provide possible resolutions of these
apparent paradoxes from the commensurate -density wave theory. To address
the first paradox we have computed the ARPES spectral function subject to
correlated disorder, natural to a class of experiments relevant to the
materials studied in quantum oscillations. The intensity of the spectral
function is significantly reduced for the electron pockets for an intermediate
range of disorder correlation length, and typically less than half the hole
pocket is visible, mimicking Fermi arcs. Next we show from an exact transfer
matrix calculation of the Shubnikov-de Haas oscillation that the usual disorder
affects the electron pocket more significantly than the hole pocket. However,
when, in addition, the scattering from vortices in the mixed state is included,
it wipes out the frequency corresponding to the hole pocket. Thus, if we are
correct, it will be necessary to do measurements at higher magnetic fields and
even higher quality samples to recover the hole pocket frequency.Comment: Accepted version, Phys. Rev. B, brief clarifying comments and updated
reference
Dissipation and criticality in the lowest Landau level of graphene
The lowest Landau level of graphene is studied numerically by considering a
tight-binding Hamiltonian with disorder. The Hall conductance
and the longitudinal conductance are
computed. We demonstrate that bond disorder can produce a plateau-like feature
centered at , while the longitudinal conductance is nonzero in the same
region, reflecting a band of extended states between , whose
magnitude depends on the disorder strength. The critical exponent corresponding
to the localization length at the edges of this band is found to be . When both bond disorder and a finite mass term exist the localization
length exponent varies continuously between and .Comment: 4 pages, 5 figure
Computing Lyapunov spectra with continuous Gram-Schmidt orthonormalization
We present a straightforward and reliable continuous method for computing the
full or a partial Lyapunov spectrum associated with a dynamical system
specified by a set of differential equations. We do this by introducing a
stability parameter beta>0 and augmenting the dynamical system with an
orthonormal k-dimensional frame and a Lyapunov vector such that the frame is
continuously Gram-Schmidt orthonormalized and at most linear growth of the
dynamical variables is involved. We prove that the method is strongly stable
when beta > -lambda_k where lambda_k is the k'th Lyapunov exponent in
descending order and we show through examples how the method is implemented. It
extends many previous results.Comment: 14 pages, 10 PS figures, ioplppt.sty, iopl12.sty, epsfig.sty 44 k
Quantifying chaos: a tale of two maps
In many applications, there is a desire to determine if the dynamics of interest are chaotic or not. Since positive Lyapunov exponents are a signature for chaos, they are often used to determine this. Reliable estimates of Lyapunov exponents should demonstrate evidence of convergence; but literature abounds in which this evidence lacks. This paper presents two maps through which it highlights the importance of providing evidence of convergence of Lyapunov exponent estimates. The results suggest cautious conclusions when confronted with real data. Moreover, the maps are interesting in their own right
Refining Finite-Time Lyapunov Exponent Ridges and the Challenges of Classifying Them
While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model. (C) 2015 AIP Publishing LLC.ONR N000141210665Center for Nonlinear Dynamic
Phase transition in a class of non-linear random networks
We discuss the complex dynamics of a non-linear random networks model, as a
function of the connectivity k between the elements of the network. We show
that this class of networks exhibit an order-chaos phase transition for a
critical connectivity k = 2. Also, we show that both, pairwise correlation and
complexity measures are maximized in dynamically critical networks. These
results are in good agreement with the previously reported studies on random
Boolean networks and random threshold networks, and show once again that
critical networks provide an optimal coordination of diverse behavior.Comment: 9 pages, 3 figures, revised versio
Amplitude death in coupled chaotic oscillators
Amplitude death can occur in chaotic dynamical systems with time-delay
coupling, similar to the case of coupled limit cycles. The coupling leads to
stabilization of fixed points of the subsystems. This phenomenon is quite
general, and occurs for identical as well as nonidentical coupled chaotic
systems. Using the Lorenz and R\"ossler chaotic oscillators to construct
representative systems, various possible transitions from chaotic dynamics to
fixed points are discussed.Comment: To be published in PR
- …