From: AAAI-91 Proceedings. Copyright ©1991, AAAI (www.aaai.org). All rights reserved. Toyoaki Nishida and Kenji Department of Information Science Kyoto University Sakyo-ku, Kyoto 606, Japan email: [email protected] Abstract intelligent mathematical reasoner. We have chosen two-dimensional nonlinear difFeren- It has been widely believed that qualitative analysis tial equations as a domain and analysis of long-term guides quantitative analysis, while sufficient study has behavior including asymptotic behavior as a task. The not been made from technical viewpoints. In this pa- domain is not too trivial, for it contains wide varieties per, we present a case study with PSX2NL, a pro- of phenomena; while it is not too hard for the first step, gram which autonomously analyzes the behavior of for we can step aside from several hard representational two-dimensional nonlinear differential equations, by in- issues which constitute an independent research sub- tegrating knowledge-based methods and numerical al- ject by themselves. The task is novel in qualitative gorithms. PSX2NL focuses on geometric properties of physics, for nobody except a few researchers has ever solution curves of ordinary differential equations in the addressed it. phase space. PSX2NL is designed based on a couple of novel ideas: (a) a set of flow mappings which is an abstract description of the behavior of solution curves, Analysis of Two-dimensional Nonlinear and (b) a flow grammar which specifies all possible pat- Ordinary ifferentid Equations terns of solution curves, enabling PSX2NL to derive the The following is a typical ordinary differential equation most plausible interpretation when complete informa- (ODE) we investigate in this paper: tion is not available. We describe the algorithms for dx deriving flow mappings. ifz = : - x2 + 0 . 2y + 0 .3xy . 0) dt = Introduction This is two-dimensional in the sense that it is specified In spite of tremendous research efforts, most technical by two independent state variables x(t) and y(t), and it developments obtained in qualitative physics are still is nonlinear in the sense that the right-hand sides con- quite naive from the standards of other disciplines and t ain nonlinear terms such as xy and x2. Unlike linear real applications. An obvious way of escaping from the differential equations, no general algorithm is known trap would be to avoid building the theory from the for solving nonlinear ODES analytically. In addition, scratch. It would deserve a serious effort to develop a the behavior of nonlinear differential equations may computational theory on top of existing theories with become fairly complex in certain situations. keeping the spirit of qualitative physics in mind. A strategy taken by applied mathematicians is to In this paper, we present how such a schema is in- study geometric properties of solution curves in the stantiated in PSXlLNL, a program which autonomously phase space spanned bY analyzes the behavior of two-dimensional ordinary dif- independent state variables [Hirsch and Smale, 1974, ferential equations, by integrating knowledge-based Guckenheimer and Holmes, 19831. That approach has methods and numerical algorithms. PSX2NL focuses made it possible to understand the qualitative behav- on geometric properties of solution curves of ordinary ior of nonlinear differential equations, even when the differential equations in the phase space. PSX2NL is explicit form of solution is not available. Figure 1 illus- designed based on a couple of novel ideas: (a) a set of trates a portion of the phase portrait of (l), a collection flow mappings which is an abstract description of the of solution curves contained in the phase space. (a) is behavior of solution curves, and (b) a flow grammar drawn in an rather ad-hoc manner by manually invok- which specifies all possible patterns of solution curves, ing a numerical integrator and (b) is one that PSX2NL enabling PSX2 N L to derive the most plausible inter- has produced to understand the behavior in adjacent pretation when complete information is not available, regions ABGM and MGCD. Notice that only cru- Taken together, these two techniques provide a well- cial orbits are drawn there. Even from (a) it would founded computational framework for an autonomous be much easier to grasp the global characteristics of NISHIDA, ET AL. 811 (a) manuzdly drawn using a numerical integrator (b) produced by PSX’LNL for local analysis .---. __-_----. _---__------ --.9-L.. --- - -.-- -- - ----_ --- ____ _-_. -* -2 -1 3 2 35 - I -2; ~~ Arrow-heads and some symbols are added by hand. Figure 1: The Phase Portrait for (1) behavior such as: y decreases as time passes, there ex- an orbit by using a numerical integration algorithm ists some solutions which approaches to an equilibrium such as the Runge-Kutta algorithm, as was done for statex=l,y=Oast-+--oo,andsoon. examples in this paper. However, numerical methods The collection of solution curves is also called a flow, support only a small portion of the entire process of un- for it introduces a mapping which maps a given point in derstanding the behavior, as pointed out in [Yip, 19881. the phase space into another as a function of t. Each It is necessary to plan numerical simulation and inter- solution curve, called an orbit, represents a solution pret the result. In order for a program to carry out under some initial condition. the entire process without much external assistance, Theoretically, an orbit is a directed curve such that the program has to possess sufficient knowledge about its tangent vector conforms to the vector field specified nonlinear differential equations. In order to integrate by a given ODE at each point in the phase space. A numerical and knowledge-based methods, one must ad- fized point is a special orbit consisting of a single point dress several questions concerning (a) representation of orbits, (b) algorithm for generating the representation, at which g = 3 = 0. A fixed point corresponds and (c) algorithm for reasoning about global behav- to an equilibrium state which will not evolve forever. ior based on the representation. The central concern Fixed points are further classified into sinks, sources, of this paper is the second and to demonstrate how saddles, and some other peculiar subcategories accord- qualitative and quantitative analysis are integrated to ing to how orbits behave in their neighborhood. If the achieve the goal. Before describing that, we present an solution of a given ODE is unique as is often the case, overview of our solution to these three questions in the orbits never cross with themselves or other orbits. An next section. orbit must be cyclic if it intersects with itself. Under many circumstances, it is crucial to under- stand the long-term behavior of a given ODE. In struc- Outline of our Solution turally stable two-dimensional ODES, orbits may di- verge or approach a fixed point or a cyclic orbit when PSX2NL is a program which autonomously analyzes t -+ foe. An orbit is called an attracting orbit when the behavior of two-dimensional ODES. PSX2NL takes all orbits in some neighborhood approach it arbitrar- as input the specification of an ODE such as (1) and ily after a long run. Similarly, a repelling orbit is an a region of the phase space to analyze. As output orbit which all orbits in some neighborhood approach PSX’LNL produces (a) a list of possible asymptotic ori- as t + -co. Conversely, when an orbit 01 approaches gins and destinations and (b) a set of flow mappings, an orbit 02 as t --) 00 (-00)~ we call 02 the asymptotic des- abstract description of the behavior of the given ODE tination (origin) of 01. In this paper, we study the be- in the given region. In this section, we briefly describe havior of qualitatively coherent bundles of orbits rather flow mappings and their use in reasoning about long- than individual orbits. term behavior, and then we present the overall picture It is possible to obtain an approximate picture of of the algorithms for deriving flow mappings. 812 QUALITATIVE ANALYSIS Flow Mappings as Abstract and hence, Representation of the Phase Portrait 41 0 42(FQ) c FQ. (8) We represent a flow as a set of flow mappings which (8) entails that there exists an attracting bundle of specifies how the flow maps points in the phase space. orbits containing at least one limit cycle transverse to For example, consider the flow in the region MGCD 41 0 h(FQ) = h(Q) b42(F). Seefigure 2(b)- shown in figure l(b). Orbits transverse to segment T Q Notice that qualitative and quantitative analysis are continuously map points on the segment to polyline integrated in the lines of reasoning described above. 27 G H. We represent the fact as 42,2 : T Q + U G H, The main thread is concerned with topological aspects where 42,2 is a label attached to the bundle of orbit of bundle of orbit intervals and hence it is qualitative, intervals between T Q and U G H. We call U G H the while the basic facts are obtained by quantitative anal- destination of T Q, and T Q the or&n of U G H (with ysis. respect to the given region). The whole flow 42 in The global analyzer incorporated in PSX2NL rec- region MGCD is represented as a sum of sub-flows ognizes attracting or repelling bundles of orbits con- corresponding to bundles of orbit intervals in MGCD, taining a limit cycle by chaining flow mappings in as follows: turn and looking for such patterns of argumentation as described above. The domains of attraction and 42 = 42,1 : MT + LMB~~,~ :TQ-+UGH (2) 7 - repelling are also determined in this process.