Qualitative Reasoning and Bifurcations in Dynamic Systems Juan J. Flores juanf@zeus.umich.mx Division de Estudios de Postgrado Facultad de Ingenieria Electrica Universidad Michoacana Morelia, Mexico Andrzej Proskurowski andrzej@cs.uoregon.edu Computer Science Department University of Oregon Eugene, OR, USA Abstract A bifurcation occurs in a dynamic system when the structure of the system itself and therefore also its qual- itative behavior change as a result of changes in one of the system’s parameters. In most cases, an infinites- imal change in one of the parameters make the dy- namic system exhibit dramatic changes. In this paper, we present a framework (QRBD) for performing qual- itative analysis of dynamic systems exhibiting bifurca- tions. QRBD performs a simulation of the system with bifurcations, in the presence of perturbations, produc- ing accounts for all events in the system, given a quali- tative description of the changes it undergoes. In such a sequence of events, we include catastrophic changes due to perturbations and bifurcations, and hysteresis. QRBD currently works with first-order systems with only one varying parameter. We propose the qualita- tive representations and algorithm that enable us to reason about the changes a dynamic system undergoes when exhibiting bifurcations, in the presence of pertur- bations. Introduction When we think of dynamic systems, we think of Or- dinary (perhaps Partial) Differential Equations, and their solutions with time. They may be linear or non- linear. By system dynamics we understand the set of qualitative features a system exhibits when excited properly. Several works define qualitative descriptions and algorithms that solve dynamic systems, and pro- vide those solutions in qualitative terms (Kuipers 1986; Forbus 1984; de Kleer & Brown 1984). Those works consider a non-changing dynamic sys- tem and provide solutions to its transient response with time. But dynamic systems depend not only on state variables and their derivatives, but on parameters, and those parameters may be functions of time. For in- stance, the mass of a rocket changes as it burns fuel, the characteristics of an electrical machine change as it ages, the load of an electrical power system changes during the day, etc. Changes in the parameters, even if they are infinites- imal, may cause a dynamic system to exhibit totally Copyright c  2006, American Association for Artificial In- telligence (www.aaai.org). All rights reserved. different qualitative properties. For instance, a damped mass-spring system may stop oscillating if the damping increases. Those changes in the topology of the phase space representation of the dynamic system are known as bifurcations, and the values of the parameters for which a bifurcation occurs are called bifurcation points. The work presented in this paper concerns the deter- mination of the behavior of a dynamic system exhibit- ing bifurcations. The analysis will take into account changes in parameters and perturbations to the system and will derive a sequence of events the system exhibits under those circumstances. All the analysis is accom- plished at a qualitative level. The paper is organized as follows: The following sec- tion provides a gentle introduction to dynamic systems, ordinary differential equations, solutions, phase por- traits, and bifurcation diagrams; next, we define the problem to be solved; next section defines the quali- tative representation for the different components in- volved in the process; after that, we propose a repre- sentation for events and the dynamics exhibited by the system under analysis; once we have defined the repre- sentation, we present a simulation algorithm that allows us to reason about dynamic systems and bifurcation diagrams; following that, we show some results, where simulations include qualitative plots and accounts for the different events present in a given scenario; we then propose directions that need to be explored; the final section summarizes our findings. Dynamic Systems The main tool for modeling dynamic systems is the dif- ferential calculus. Differential equations can roughly be divided in two types: ordinary and partial differen- tial equations. This paper deals with dynamic systems that can be modeled by ordinary differential equations (ODEs). An ODE is an equation of the form F (t, x, dx dt , ··· , d n x d n t )=0 (1) Function f (t) is a solution to Equation (1) if F (t, f (t), df (t) dt , ··· , d n f (t) d n t )=0 (2)