239 research outputs found
Analytical Proof of Space-Time Chaos in Ginzburg-Landau Equations
We prove that the attractor of the 1D quintic complex Ginzburg-Landau
equation with a broken phase symmetry has strictly positive space-time entropy
for an open set of parameter values. The result is obtained by studying chaotic
oscillations in grids of weakly interacting solitons in a class of
Ginzburg-Landau type equations. We provide an analytic proof for the existence
of two-soliton configurations with chaotic temporal behavior, and construct
solutions which are closed to a grid of such chaotic soliton pairs, with every
pair in the grid well spatially separated from the neighboring ones for all
time. The temporal evolution of the well-separated multi-soliton structures is
described by a weakly coupled lattice dynamical system (LDS) for the
coordinates and phases of the solitons. We develop a version of normal
hyperbolicity theory for the weakly coupled LDSs with continuous time and
establish for them the existence of space-time chaotic patterns similar to the
Sinai-Bunimovich chaos in discrete-time LDSs. While the LDS part of the theory
may be of independent interest, the main difficulty addressed in the paper
concerns with lifting the space-time chaotic solutions of the LDS back to the
initial PDE. The equations we consider here are space-time autonomous, i.e. we
impose no spatial or temporal modulation which could prevent the individual
solitons in the grid from drifting towards each other and destroying the
well-separated grid structure in a finite time. We however manage to show that
the set of space-time chaotic solutions for which the random soliton drift is
arrested is large enough, so the corresponding space-time entropy is strictly
positive
Bifurcation to Chaos in the complex Ginzburg-Landau equation with large third-order dispersion
We give an analytic proof of the existence of Shilnikov chaos in complex
Ginzburg-Landau equation subject to a large third-order dispersion
perturbation
Six degree-of-freedom analysis of hip, knee, ankle and foot provides updated understanding of biomechanical work during human walking
Measuring biomechanical work performed by humans and other animals is critical for understanding muscle–tendon function, jointspecific contributions and energy-saving mechanisms during locomotion. Inverse dynamics is often employed to estimate jointlevel contributions, and deformable body estimates can be used to study work performed by the foot. We recently discovered that these commonly used experimental estimates fail to explain whole-body energy changes observed during human walking. By re-analyzing previously published data, we found that about 25% (8 J) of total positive energy changes of/about the body’s center-of-mass and \u3e30% of the energy changes during the Push-off phase of walking were not explained by conventional joint- and segment-level work estimates, exposing a gap in our fundamental understanding of work production during gait. Here, we present a novel Energy-Accounting analysis that integrates various empirical measures of work and energy to elucidate the source of unexplained biomechanical work. We discovered that by extending conventional 3 degree-of-freedom (DOF) inverse dynamics (estimating rotational work about joints) to 6DOF (rotational and translational) analysis of the hip, knee, ankle and foot, we could fully explain the missing positive work. This revealed that Push-off work performed about the hip may be \u3e50% greater than conventionally estimated (9.3 versus 6.0 J, P=0.0002, at 1.4 m s−1 ). Our findings demonstrate that 6DOF analysis (of hip– knee–ankle–foot) better captures energy changes of the body than more conventional 3DOF estimates. These findings refine our fundamental understanding of how work is distributed within the body, which has implications for assistive technology, biomechanical simulations and potentially clinical treatment
The dynamics of interacting multi-pulses in the one-dimensional quintic complex Ginzburg-Landau equation
We formulate an effective numerical scheme that can readily, and accurately,
calculate the dynamics of weakly interacting multi-pulse solutions of the
quintic complex Ginzburg-Landau equation (QCGLE) in one space dimension. The
scheme is based on a global centre-manifold reduction where one considers the
solution of the QCGLE as the composition of individual pulses plus a remainder
function, which is orthogonal to the adjoint eigenfunctions of the linearised
operator about a single pulse. This centre-manifold projection overcomes the
difficulties of other, more orthodox, numerical schemes, by yielding a
fast-slow system describing 'slow' ordinary differential equations for the
locations and phases of the individual pulses, and a 'fast' partial
differential equation for the remainder function. With small parameter
where is a constant and is the
pulse separation distance, we write the fast-slow system in terms of
first-order and second-order correction terms only, a formulation which is
solved more efficiently than the full system. This fast-slow system is
integrated numerically using adaptive time-stepping. Results are presented here
for two- and three-pulse interactions. For the two-pulse problem, cells of
periodic behaviour, separated by an infinite set of heteroclinic orbits, are
shown to 'split' under perturbation creating complex spiral behaviour. For the
case of three pulse interaction a range of dynamics, including chaotic pulse
interaction, are found. While results are presented for pulse interaction in
the QCGLE, the numerical scheme can also be applied to a wider class of
parabolic PDEs.Comment: 33 page
Complexity for extended dynamical systems
We consider dynamical systems for which the spatial extension plays an
important role. For these systems, the notions of attractor, epsilon-entropy
and topological entropy per unit time and volume have been introduced
previously. In this paper we use the notion of Kolmogorov complexity to
introduce, for extended dynamical systems, a notion of complexity per unit time
and volume which plays the same role as the metric entropy for classical
dynamical systems. We introduce this notion as an almost sure limit on orbits
of the system. Moreover we prove a kind of variational principle for this
complexity.Comment: 29 page
Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials
We study initial boundary value problems for the convective Cahn-Hilliard
equation \Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0. It is well-known that without
the convective term, the solutions of this equation may blow up in finite time
for any . In contrast to that, we show that the presence of the convective
term u\px u in the Cahn-Hilliard equation prevents blow up at least for
. We also show that the blowing up solutions still exist if is
large enough (). The related equations like
Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard
equation, are also considered
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