Introduction to fractional linear systems. Part 1 : continuous-time case

IEE Proceedings - Vision, Image, and Signal Processing, Vol. 147, nº 1

Constraint reasoning for differential models

The basic motivation of this work was the integration of biophysical models within the interval constraints framework for decision support. Comparing the major features of biophysical models with the expressive power of the existing interval constraints framework, it was clear that the most important inadequacy was related with the representation of differential equations. System dynamics is often modelled through differential equations but there was no way of expressing a differential equation as a constraint and integrate it within the constraints framework. Consequently, the goal of this work is focussed on the integration of ordinary differential equations within the interval constraints framework, which for this purpose is extended with the new formalism of Constraint Satisfaction Differential Problems. Such framework allows the specification of ordinary differential equations, together with related information, by means of constraints, and provides efficient propagation techniques for pruning the domains of their variables. This enabled the integration of all such information in a single constraint whose variables may subsequently be used in other constraints of the model. The specific method used for pruning its variable domains can then be combined with the pruning methods associated with the other constraints in an overall propagation algorithm for reducing the bounds of all model variables. The application of the constraint propagation algorithm for pruning the variable domains, that is, the enforcement of local-consistency, turned out to be insufficient to support decision in practical problems that include differential equations. The domain pruning achieved is not, in general, sufficient to allow safe decisions and the main reason derives from the non-linearity of the differential equations. Consequently, a complementary goal of this work proposes a new strong consistency criterion, Global Hull-consistency, particularly suited to decision support with differential models, by presenting an adequate trade-of between domain pruning and computational effort. Several alternative algorithms are proposed for enforcing Global Hull-consistency and, due to their complexity, an effort was made to provide implementations able to supply any-time pruning results. Since the consistency criterion is dependent on the existence of canonical solutions, it is proposed a local search approach that can be integrated with constraint propagation in continuous domains and, in particular, with the enforcing algorithms for anticipating the finding of canonical solutions. The last goal of this work is the validation of the approach as an important contribution for the integration of biophysical models within decision support. Consequently, a prototype application that integrated all the proposed extensions to the interval constraints framework is developed and used for solving problems in different biophysical domains.

Playing Newtonian Games with Modellus

This article is a short introduction on how to use Modellus (a computer package that is freely available on the Internet and used in the IOP Advancing Physics course) to build physics games using Newton’s laws, expressed as differential equations. Solving systems of differential equations is beyond most secondary-school or first-year college students. However, with Modellus, the solution is simply the output of the usual physical reasoning: define the force law, compute its magnitude and components, use it to obtain the acceleration components, then the velocity components and, finally, use the velocity components to find the coordinates.

Comments on ‘‘Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions

Applied Mathematical Modelling, Vol.33

R&D spillovers in an endogenous growth model with physical capital, human capital and varieties

There is a family of models with Physical, Human capital and R&D for which convergence properties have been discussed (Arnold, 2000a; Gómez, 2005). However, spillovers in R&D have been ignored in this context. We introduce spillovers in this model and derive its steady-state and stability properties. This new feature implies that the model is characterized by a system of four differential equations. A unique Balanced Growth Path along with a two dimensional stable manifold are obtained under simple and reasonable conditions. Transition is oscillatory toward the steady-state for plausible values of parameters.

Transitional dynamics of an endogenous growth model with an erosion effect

The convergence features of an Endogenous Growth model with Physical capital, Human Capital and R&D have been studied. We add an erosion effect (supported by empirical evidence) to this model, and fully characterize its convergence properties. The dynamics is described by a fourth-order system of differential equations. We show that the model converges along a one-dimensional stable manifold and that its equilibrium is saddle-path stable. We also argue that one of the implications of considering this “erosion effect” is the increase in the adherence of the model to data.