Many interesting systems in physics consist of linked entities. Continuous system dynamics can be described by, possibly large, systems of differential equations. Here we consider the ordinary differential equations. Often a system does not have an exact analytical solution (non-linear differential equations) so a numerical solution is the best available. Two main phases are : discretization of time and then integration using, for example, some stepwise methods (iterative algorithms).

Notwithstanding simulation of pure continuous and pure discrete systems is well-understood, the building of initial models and their numerical resolution can become very complicated, especially when modeling of mixed continuous/discret behavior (hybrid phenomena) is needed.

The aim of the reactive approach is to propose a structured and flexible paradigm for programming dynamic reactive systems. A reactive system is composed of a set of synchronously executed parallel components communicating by instataneous broadcast events (communication is similar to the radio transmission where emitters send information that is instantaneously received by all listeners). The model proceeds in successive reactions, one at each logical instant. At each instant, the reaction is computed using the current tuple of input events and some internal information. The global behavior of such a system results from the cooperation of its components (subsystems).

A reactive object is an object containing the state variables plus associated reactive program that implements its behavior.

Using the synchronous execution, a complex system can be represented as a set of linked but independent entities with a common discret time (instants). Linkage translates into events (for example, forces between particles), thus the model is very modular : adding new receivers to the system is totally transparent and does not affect the functionality of the other components.

We can adapt many of the numerical algorithms of integration that are in standard use by mapping theirs steps onto the logical instants. Thus, one can simulate a complex physical system in very natural way by simply putting many reactive objects in parallel, that means that we have many parallel synchronized processes of integration communicating by broadcast events. Each process can be controled and preempted as well as new components can be dynamically added in the system without any global reinitialization.

Let's bring a simple example of the implementation of the principles mentioned above. The first image shows general block-function diagram of any component. Central block corresponds to a system of differential equations (for example, coupled equations for newtonian motion). The input entries are received events and the output events (g vector) are some functions depending on the value of the state vector (let's call them influences). Input and output entries of all the components are synchronized in a natural way.

The second image is a possible general schemes of reactive program of components corresponding to one step of calculation (the body of the main loop). It can be composed of three main parts : emission of influences, reception and collection of external influences and computation of the next state (or of a next estimation) using some of the simpler numerical methods (we use Euler, Verlet and Runge-Kutta4 methods). This program is self-starting.

Note that this scheme offers a natural way to build a behavior of an object just by putting appropriate elementary blocks (behaviors) in parallel. This facility can be employed, for example, in educational process

2002-07-18