103 research outputs found
Blooming in a non-local, coupled phytoplankton-nutrient model
Recently, it has been discovered that the dynamics of phytoplankton concentrations in an ocean exhibit a rich variety of patterns, ranging from trivial states to oscillating and even chaotic behavior [J. Huisman, N. N. Pham Thi, D. M. Karl, and B. P. Sommeijer, Nature, 439 (2006), pp. 322–325]. This paper is a first step towards understanding the bifurcational structure associated with nonlocal coupled phytoplankton-nutrient models as studied in that paper. Its main subject is the linear stability analysis that governs the occurrence of the first nontrivial stationary patterns, the deep chlorophyll maxima (DCMs) and the benthic layers (BLs). Since the model can be scaled into a system with a natural singularly perturbed nature, and since the associated eigenvalue problem decouples into a problem of Sturm–Liouville type, it is possible to obtain explicit (and rigorous) bounds on, and accurate approximations of, the eigenvalues. The analysis yields bifurcation-manifolds in parameter space, of which the existence, position, and nature are confirmed by numerical simulations. Moreover, it follows from the simulations and the results on the eigenvalue problem that the asymptotic linear analysis may also serve as a foundation for the secondary bifurcations, such as the oscillating DCMs, exhibited by the model
Analysis of the accuracy and convergence of equation-free projection to a slow manifold
In [C.W. Gear, T.J. Kaper, I.G. Kevrekidis, and A. Zagaris, Projecting to a
Slow Manifold: Singularly Perturbed Systems and Legacy Codes, SIAM J. Appl.
Dyn. Syst. 4 (2005) 711-732], we developed a class of iterative algorithms
within the context of equation-free methods to approximate low-dimensional,
attracting, slow manifolds in systems of differential equations with multiple
time scales. For user-specified values of a finite number of the observables,
the m-th member of the class of algorithms (m = 0, 1, ...) finds iteratively an
approximation of the appropriate zero of the (m+1)-st time derivative of the
remaining variables and uses this root to approximate the location of the point
on the slow manifold corresponding to these values of the observables. This
article is the first of two articles in which the accuracy and convergence of
the iterative algorithms are analyzed. Here, we work directly with explicit
fast--slow systems, in which there is an explicit small parameter, epsilon,
measuring the separation of time scales. We show that, for each m = 0, 1, ...,
the fixed point of the iterative algorithm approximates the slow manifold up to
and including terms of O(epsilon^m). Moreover, for each m, we identify
explicitly the conditions under which the m-th iterative algorithm converges to
this fixed point. Finally, we show that when the iteration is unstable (or
converges slowly) it may be stabilized (or its convergence may be accelerated)
by application of the Recursive Projection Method. Alternatively, the
Newton-Krylov Generalized Minimal Residual Method may be used. In the
subsequent article, we will consider the accuracy and convergence of the
iterative algorithms for a broader class of systems-in which there need not be
an explicit small parameter-to which the algorithms also apply
Emergence of steady and oscillatory localized structures in a phytoplankton-nutrient model
Co-limitation of marine phytoplankton growth by light and nutrient, both of
which are essential for phytoplankton, leads to complex dynamic behavior and a
wide array of coherent patterns. The building blocks of this array can be
considered to be deep chlorophyll maxima, or DCMs, which are structures
localized in a finite depth interior to the water column. From an ecological
point of view, DCMs are evocative of a balance between the inflow of light from
the water surface and of nutrients from the sediment. From a (linear)
bifurcational point of view, they appear through a transcritical bifurcation in
which the trivial, no-plankton steady state is destabilized. This article is
devoted to the analytic investigation of the weakly nonlinear dynamics of these
DCM patterns, and it has two overarching themes. The first of these concerns
the fate of the destabilizing stationary DCM mode beyond the center manifold
regime. Exploiting the natural singularly perturbed nature of the model, we
derive an explicit reduced model of asymptotically high dimension which fully
captures these dynamics. Our subsequent and fully detailed study of this model
- which involves a subtle asymptotic analysis necessarily transgressing the
boundaries of a local center manifold reduction - establishes that a stable DCM
pattern indeed appears from a transcritical bifurcation. However, we also
deduce that asymptotically close to the original destabilization, the DCM
looses its stability in a secondary bifurcation of Hopf type. This is in
agreement with indications from numerical simulations available in the
literature. Employing the same methods, we also identify a much larger DCM
pattern. The development of the method underpinning this work - which, we
expect, shall prove useful for a larger class of models - forms the second
theme of this article
A convex-programming-based guidance algorithm to capture a tumbling object on orbit using a spacecraft equipped with a robotic manipulator
An algorithm to guide the capture of a tumbling resident space object by a spacecraft equipped with a robotic manipulator is presented. A solution to the guidance problem is found by solving a collection of convex programming problems. As convex programming offers deterministic convergence properties, this algorithm is suitable for onboard implementation and real-time use. A set of hardware-in-the-loop experiments substantiates this claim. To cast the guidance problem as a collection of convex programming problems, the capture maneuver is divided into two simultaneously occurring sub-maneuvers: a system-wide translation and an internal re-configuration. These two sub-maneuvers are optimized in two consecutive steps. A sequential convex programming procedure, overcoming the presence of non-convex constraints and nonlinear dynamics, is used on both optimization steps. A proof of convergence is offered for the system-wide translation, while a set of structured heuristics—trust regions—is used for the optimization of the internal re-configuration sub-maneuver. Videos of the numerically simulated and experimentally demonstrated maneuvers are included as supplementary material
Blooming in a non-local, coupled phytoplankton–nutrient model
Recently, it has been discovered that the dynamics of phytoplankton concentrations in an ocean exhibit a rich variety of patterns, ranging from trivial states to oscillating and even chaotic behavior [J. Huisman, N.N. Pham Thi, D.M. Karl, and B.P. Sommeijer (2006), Reduced mixing generates oscillations and chaos in the oceanic deep chlorophyll maximum, Nature 439 322-325]. This paper is a first step towards understanding the bifurcational structure associated to non-local, coupled phytoplankton-nutrient models as studied in that paper. Its main subject is the linear stability analysis that governs the occurrence of the first nontrivial stationary patterns, the `deep chlorophyll maxima' (DCMs) and the `benthic layers' (BLs). Since the model can be scaled into a system with a natural singularly perturbed nature, and since the associated eigenvalue problem decouples into a problem of Sturm-Liouville type, it is possible to obtain explicit (and rigorous) bounds on, and accurate approximations of, the eigenvalues. The analysis yields bifurcation-manifolds in parameter space, of which the existence, position and nature are confirmed by numerical simulations. Moreover, it follows from the simulations and the results on the eigenvalue problem that the asymptotic linear analysis may also serve as a foundation for the secondary bifurcations, such as the oscillating DCMs, exhibited by the model
Equation-Free Analysis of Macroscopic Behavior in Traffic and Pedestrian Flow
Equation-free methods make possible an analysis of the evolution of a few
coarse-grained or macroscopic quantities for a detailed and realistic model
with a large number of fine-grained or microscopic variables, even though no
equations are explicitly given on the macroscopic level. This will facilitate a
study of how the model behavior depends on parameter values including an
understanding of transitions between different types of qualitative behavior.
These methods are introduced and explained for traffic jam formation and
emergence of oscillatory pedestrian counter flow in a corridor with a narrow
door
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