9,056 research outputs found
Analysis of Reaction Network Systems Using Tropical Geometry
We discuss a novel analysis method for reaction network systems with
polynomial or rational rate functions. This method is based on computing
tropical equilibrations defined by the equality of at least two dominant
monomials of opposite signs in the differential equations of each dynamic
variable. In algebraic geometry, the tropical equilibration problem is
tantamount to finding tropical prevarieties, that are finite intersections of
tropical hypersurfaces. Tropical equilibrations with the same set of dominant
monomials define a branch or equivalence class. Minimal branches are
particularly interesting as they describe the simplest states of the reaction
network. We provide a method to compute the number of minimal branches and to
find representative tropical equilibrations for each branch.Comment: Proceedings Computer Algebra in Scientific Computing CASC 201
Transverse Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle
This paper derives a differential contraction condition for the existence of
an orbitally-stable limit cycle in an autonomous system. This transverse
contraction condition can be represented as a pointwise linear matrix
inequality (LMI), thus allowing convex optimization tools such as
sum-of-squares programming to be used to search for certificates of the
existence of a stable limit cycle. Many desirable properties of contracting
dynamics are extended to this context, including preservation of contraction
under a broad class of interconnections. In addition, by introducing the
concepts of differential dissipativity and transverse differential
dissipativity, contraction and transverse contraction can be established for
large scale systems via LMI conditions on component subsystems.Comment: 6 pages, 1 figure. Conference submissio
Structural identifiability of viscoelastic mechanical systems
We solve the local and global structural identifiability problems for
viscoelastic mechanical models represented by networks of springs and dashpots.
We propose a very simple characterization of both local and global structural
identifiability based on identifiability tables, with the purpose of providing
a guideline for constructing arbitrarily complex, identifiable spring-dashpot
networks. We illustrate how to use our results in a number of examples and
point to some applications in cardiovascular modeling.Comment: 3 figure
A Framework for Worst-Case and Stochastic Safety Verification Using Barrier Certificates
This paper presents a methodology for safety verification of continuous and hybrid systems in the worst-case and stochastic settings. In the worst-case setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method
Reduction of dynamical biochemical reaction networks in computational biology
Biochemical networks are used in computational biology, to model the static
and dynamical details of systems involved in cell signaling, metabolism, and
regulation of gene expression. Parametric and structural uncertainty, as well
as combinatorial explosion are strong obstacles against analyzing the dynamics
of large models of this type. Multi-scaleness is another property of these
networks, that can be used to get past some of these obstacles. Networks with
many well separated time scales, can be reduced to simpler networks, in a way
that depends only on the orders of magnitude and not on the exact values of the
kinetic parameters. The main idea used for such robust simplifications of
networks is the concept of dominance among model elements, allowing
hierarchical organization of these elements according to their effects on the
network dynamics. This concept finds a natural formulation in tropical
geometry. We revisit, in the light of these new ideas, the main approaches to
model reduction of reaction networks, such as quasi-steady state and
quasi-equilibrium approximations, and provide practical recipes for model
reduction of linear and nonlinear networks. We also discuss the application of
model reduction to backward pruning machine learning techniques
Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure
We generalize the idea of Vassiliev invariants to the spin network context,
with the aim of using these invariants as a kinematical arena for a canonical
quantization of gravity. This paper presents a detailed construction of these
invariants (both ambient and regular isotopic) requiring a significant
elaboration based on the use of Chern-Simons perturbation theory which extends
the work of Kauffman, Martin and Witten to four-valent networks. We show that
this space of knot invariants has the crucial property -from the point of view
of the quantization of gravity- of being loop differentiable in the sense of
distributions. This allows the definition of diffeomorphism and Hamiltonian
constraints. We show that the invariants are annihilated by the diffeomorphism
constraint. In a companion paper we elaborate on the definition of a
Hamiltonian constraint, discuss the constraint algebra, and show that the
construction leads to a consistent theory of canonical quantum gravity.Comment: 21 Pages, RevTex, many figures included with psfi
The Synthesis of Arbitrary Stable Dynamics in Non-linear Neural Networks II: Feedback and Universality
We wish to construct a realization theory of stable neural networks and use this theory to model the variety of stable dynamics apparent in natural data. Such a theory should have numerous applications to constructing specific artificial neural networks with desired dynamical behavior. The networks used in this theory should have well understood dynamics yet be as diverse as possible to capture natural diversity.
In this article, I describe a parameterized family of higher order, gradient-like neural networks which have known arbitrary equilibria with unstable manifolds of known specified dimension. Moreover, any system with hyperbolic dynamics is conjugate to one of these systems in a neighborhood of the equilibrium points. Prior work on how to synthesize attractors using dynamical systems theory, optimization, or direct parametric. fits to known stable systems, is either non-constructive, lacks generality, or has unspecified attracting equilibria.
More specifically, We construct a parameterized family of gradient-like neural networks with a simple feedback rule which will generate equilibrium points with a set of unstable manifolds of specified dimension. Strict Lyapunov functions and nested periodic orbits are obtained for these systems and used as a method of synthesis to generate a large family of systems with the same local dynamics. This work is applied to show how one can interpolate finite sets of data, on nested periodic orbits.Air Force Office of Scientific Research (90-0128
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