786 research outputs found
Side conditions for ordinary differential equations
We specialize Olver's and Rosenau's side condition heuristics for the
determination of particular invariant sets of ordinary differential equations.
It turns out that side conditions of so-called LaSalle type are of special
interest. Moreover we put side condition properties of symmetric and partially
symmetric equations in a wider context. In the final section we present an
application to parameter-dependent systems, in particular to quasi-steady state
for chemical reactions.Comment: To appear in J. of Lie Theor
A geometric method for model reduction of biochemical networks with polynomial rate functions
Model reduction of biochemical networks relies on the knowledge of slow and
fast variables. We provide a geometric method, based on the Newton polytope, to
identify slow variables of a biochemical network with polynomial rate
functions. The gist of the method is the notion of tropical equilibration that
provides approximate descriptions of slow invariant manifolds. Compared to
extant numerical algorithms such as the intrinsic low dimensional manifold
method, our approach is symbolic and utilizes orders of magnitude instead of
precise values of the model parameters. Application of this method to a large
collection of biochemical network models supports the idea that the number of
dynamical variables in minimal models of cell physiology can be small, in spite
of the large number of molecular regulatory actors
Algorithmic Reduction of Biological Networks With Multiple Time Scales
We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting using a recent result by Cardin and Teixeira. The existence of invariant manifolds is subject to hyperbolicity conditions, which we test algorithmically using Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations
Comparing solution methods for dynamic equilibrium economies
This paper compares solution methods for dynamic equilibrium economies. The authors compute and simulate the stochastic neoclassical growth model with leisure choice using Undetermined Coefficients in levels and in logs, Finite Elements, Chebyshev Polynomials, Second and Fifth Order Perturbations and Value Function Iteration for several calibrations. The authors document the performance of the methods in terms of computing time, implementation complexity and accuracy and they present some conclusions about their preferred approaches based on the reported evidence.
A coordinate-independent version of Hoppensteadt's convergence theorem
The classical theorems about singular perturbation reduction (due to Tikhonov
and Fenichel) are concerned with convergence on a compact time interval (in
slow time) as a small parameter approaches zero. For unbounded time intervals
Hoppensteadt gave a convergence theorem, but his criteria are generally not
easy to apply to concrete given systems. We state and prove a convergence
result for autonomous systems on unbounded time intervals which relies on
criteria that are relatively easy to verify, in particular for the case of a
one-dimensional slow manifold. As for applications, we discuss several reaction
equations from biochemistry.Comment: 26 pages; small change
Computational reacting gas dynamics
In the study of high speed flows at high altitudes, such as that encountered by re-entry spacecrafts, the interaction of chemical reactions and other non-equilibrium processes in the flow field with the gas dynamics is crucial. Generally speaking, problems of this level of complexity must resort to numerical methods for solutions, using sophisticated computational fluid dynamics (CFD) codes. The difficulties introduced by reacting gas dynamics can be classified into three distinct headings: (1) the usually inadequate knowledge of the reaction rate coefficients in the non-equilibrium reaction system; (2) the vastly larger number of unknowns involved in the computation and the expected stiffness of the equations; and (3) the interpretation of the detailed reacting CFD numerical results. The research performed accepts the premise that reacting flows of practical interest in the future will in general be too complex or 'untractable' for traditional analytical developments. The power of modern computers must be exploited. However, instead of focusing solely on the construction of numerical solutions of full-model equations, attention is also directed to the 'derivation' of the simplified model from the given full-model. In other words, the present research aims to utilize computations to do tasks which have traditionally been done by skilled theoreticians: to reduce an originally complex full-model system into an approximate but otherwise equivalent simplified model system. The tacit assumption is that once the appropriate simplified model is derived, the interpretation of the detailed numerical reacting CFD numerical results will become much easier. The approach of the research is called computational singular perturbation (CSP)
The SYMBIONT Project: Symbolic Methods for Biological Networks
F1000Research 7:1341 (poster)SYMBIONT ranges from mathematics via computer science to systems biology, with a balanced team of researchers from those fields. At the present stage the project has a clear focus on fundamental research on mathematical methods and prototypes in software. Results are systematically bench-marked against models from computational biology databases. We summarize the motivation and aims for the project, and report on some existing results
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen
- …