Algebraically Generated Flows


Jerry Tutsch, Copyright © 2/13/01

Consider the flow in R2 defined by the solutions to the differential equations

which may also be written as

The local and global properties of the flow are traditionally arrived at by studying explicit solutions to the initial value problem and/or by studying local and/or global properties of the functions f and g.

In the simplest nontrivial case, where f and g are homogeneous linear functions of x and y, the system is generally written as

The topologically distinct flows defined by this simple system can be classified using the properties of the eigenvalues and eigenvectors of the (constant) coefficient matrix A. When the eigenvalues are real and distinct, the eigenvectors are in the directions of the separatrices of the system and the eigenvalues determine the rate of the flow on the separatrices. The separatrices themselves are straight lines through the origin.

In this document I propose a different approach to studying the properties of class of flows in R
2 based more on synthesis and less on analysis. The idea is to begin by defining a set of special solution curves, which will generally turn out to be separatrices, and rates on these special curves, and then study the flow generated by the set of curves.

I call a flow in R
2 that has a given algebraic curve, P(x,y) = 0, among its solutions, an algebraically generated flow. The curve can always be written as a product of its irreducible components in the complex field as

For the sake of simplicity let us assume that

The first order system of ordinary differential equations

has P(x,y) = c as its general solution. If we generalize this system to

where not all of the Ai's are equal, then it is easy to see that P(x,y) = 0 is a solution to the new system, that is, the n components of P are among the solution curves to equations (1) and (2). Equations (1) and (2) constitute the least complex (lowest degree) nontrivial system for which P(x,y) = 0 is a solution. I call the system of equations given by (1) and (2) the master equation. Flows given by the master equation are generated by P(x,y). The irreducible components of P(x,y) are called the generators of the flow.

The non zero constant coefficients, Ai, in the master equation are called the global rates. They are related to another set of parameters

called the local rates. The relationship between the global rates and the local rates, in the case that each Pi is a linear function, are given by

where ci and bi are real but ai and bi may be real or complex, is defined by

When the generators of the flow are linear, requiring dx/dt and dy/dt to be real forces Ai and Aj to be complex conjugates if Pi and Pj are complex conjugates. The local rates of the system are the eigenvalues of the linearized system at isolated critical points lying on exactly two generators and the two generators are in the directions of the corresponding eigenvectors.

To define a linearly generated flow in R
2 using the master equation given by equations (1) and (2), simply specify a set of real and complex lines, of the form given in equation (3), and the local rate on each line at its center, (ci, di).

The system of odes given by equations (1) and (2) is nonlinear when more than two linear generators are involved, consequently a computer is needed to study nontrivial AGFs. Go to for a free copy of OdeFactory which, in addition to being a general purpose ode development environment, allows you to create and study AGFs by entering and editing generators and local rates directly. If for some reason you want to see the right hand sides of equations (1) and (2) as polynomials in x and y, OdeFactory will show them to you. The program uses the form shown in (1) and (2) to compute solution curves internally.

It should be noted that in the more general case where P(x,y) is not algebraic, the components of P(x,y) are still solutions to equations (1) and (2). Furthermore, when all of the Ais equal 1, equations (1) and (2) become a Hamiltonian system.

Examples of AGFs are given in sample galleries: AGFExamples, TwoComplexLines, QuasiHamiltonian, AGFEMapExs, AGFFractals and AGFs&Mandelbrot.

In 1973 I wrote a book called "Algebraically Generated Flows." At that time access to computers that could do fast graphics was essentially nonexistent. Interesting AGFs are highly nonlinear so they are difficult to study without a computer. Consequently I never tried to publish the book. In 2000 I started to write OdeFactory to study AGFs. The program was expanded to include traditional forms of odes. I scanned the manuscript of the book to pdf files and put it on the web - in case anyone is interested in AGFs.

Here are the links to the pdf files:

The 273 line program was written in BASIC and was executed on a remote timesharing machine connected via a telephone line. The input/output device was a teletype with a paper tape for storing programs and a long roll of 8.5 inch paper. The right hand sides of odes were entered directly into the program. The graphics output consisted of letters and digits printed at rows and columns locations, acting like really big "pixels," on the long roll of paper.

A Sample run of the program is shown on page 281. It is the R2 view of figure 3.3 on page 89 of the book. Bring up gallery AGFExamples in OdeFactory and select the system. Switch to the R2+ view to see figure 3.3 on page 89 reproduced - about 40 years later.

The pdf files total 105 MB.The page numbers in the index are for the entire book so to find a page you need to find the correct chapter then scroll through pages until you find the desired page of the book.