**ABSTRACT**

**THE METHOD OF CRITICAL CURVES FOR DISCRETE **

**DYNAMICAL SYSTEMS IN TWO DIMENSIONS**

**by**

**Ronald Joe Record**

The study of discrete dynamical systems has been primarily limited to maps of an interval of the line and/or diffeomorphisms. A rigorous theory of the dynamics of noninvertible maps of the plane has only recently developed. Many of the analytical techniques and theory applied to diffeomorphisms is either unavailable or limited in the context of two-dimensional endomorphisms of the plane. The method of critical curves provides researchers in this field with analytical and experimental tools. This work presents research in the study of noninvertible maps of the plane carried out by developing the method of critical curves and incorporating this theory in experimental digital simulations.

**DEDICATION AND ACKNOWLEDGMENTS**

I would like to dedicate this Doctoral dissertation to my father, Dr. Joe Neil Record. There is no doubt in my mind that without his continued support and counsel I could not have completed this process.

I would like to acknowledge the inspirational instruction and guidance of Dr. Ralph Abraham and the initial impetus to study Dynamical Systems Theory given me by Dr. John Guckenheimer. Both of these men have given me a deep appreciation and love for the beauty and detail of this subject.

I would also like to acknowledge the support and assistance given me by The Santa Cruz Operation and my co-workers there. SCO has been very generous in their support of my academic pursuits and many of my co-workers have contributed ideas, feedback and advice. Hiram Clawson, in particular, has assisted this research with his photographic and X11 expertise. Finally, I would like to thank my wife, Jodi, for her support and encouragement. I could not have completed this effort without her assistance, tolerance, and enthusiasm.

The text of this dissertation includes reprints of the following previously published material :

Gardini, L., R. H. Abraham, R. J. Record, D. Fournier-Prunaret,
[1994] "A Double Logistic Map", __International Journal
of Bifurcation and Chaos__, Vol. 4, No. 1, 145-176.

Abraham, R. H., G. Chichilnisky, R. J. Record, [1994] "Dynamics
of North-South Trade and the Environment", __Environmental
Economics__, Graciela Chichilnisky, ed., Mattei Foundation,
Milan, in press.

The co-author Dr. Ralph Abraham listed in these publications directed and supervised the research which forms the basis for the dissertation.

**INTRODUCTION**

**Overview**

The method of *critical curves* has recently been employed in the analysis of discrete dynamical
systems in the plane defined by noninvertible endomorphisms. This work attempts to detail and illustrate the method of
analysis developed and used by the author and colleagues by presenting
the following :

This introduction to the context and history of iterated maps.

A published research paper analyzing an iterated coupled logistic map in the plane.

A published chapter of a book analyzing an economic model incorporating an iterated map of the plane.

The software developed by the author and used in the analyses above.

A User's Guide to the software, **endo**, with chapters detailing
the analytical and software methods.

**Endo** is a software laboratory designed for the analysis
of discrete dynamical systems in two dimensions. The user is provided
with several graphical representations of the dynamics including
the point trajectories, basins of attraction,
Lyapunov exponents, critical and precritical curves, and bifurcation diagrams. **Endo
** is designed to be highly interactive. Using the keyboard
and mouse, the user can select from a menu containing a wide variety
of maps, toggle analytical features, select specific areas for
closer examination and interactively explore the dynamical properties
of the planar endomorphism.

The name "endo" is short for "endomorphism"
, a noninvertible smooth map. The
study of the dynamics of these maps has generally been restricted
to one dimension. This software attempts to utilize current developments
in the theory of planar endomorphisms, especially the theory of critical curves and the role they can play in the dynamical analysis of such
maps. **Endo** is an **X11** graphical client written in
the **C** programming language. It has been compiled and run
on a wide variety of platforms including Intel, Mips, Motorola,
Sparc and RS6000 architectures.

**Context**

Dynamical systems theory has concerned itself with both continuous
and discrete systems. In the case of continuous systems, the theory
of *flows*** **is the study of actions of the
real line on a manifold, generated by a vector field. The study
of *cascades*** **arose naturally from the
study of flows when, at the turn of the century, Henri Poincaré
used the *return map* of a cross-section of a flow. Cascades are actions
of the integers on a manifold, generated by a diffeomorphism. The theories of flows and cascades, initiated
by Poincaré, has enjoyed a century
of rich exploration and development.

The third area of dynamical systems theory, that of *semi-cascades
*,* *are actions of the non-negative
integers generated by a noninvertible endomorphism. The research presented in this paper concerns itself with
the iteration of planar endomorphisms which are noninvertible.
Many of these noninvertible endomorphisms arose in the study of
population biology, ecology, economic systems and other natural
phenomena. An example from the field of population dynamics is
the density dependent Leslie matrix of Guckenheimer, Oster and
Ipaktchi who explored the map . These efforts were an attempt
to model predator-prey and host-parasitoid interaction. In these
simple population biology models the iteration usually represented
a strobed census of the population, typically at annual or generation
length intervals.

Iterated maps of the Complex plane as investigated by the pioneering French school of Poincaré, Picard, Montel, Fatou and Julia can also be thought of as iterated maps of the Real plane by a suitable transformation of coordinates. For instance, the function used in the calculation of the Mandelbrot set is where z is a Complex variable and c is a Complex parameter. Letting and , the map is transformed into a map of the Real plane into itself defined by , a function of two Real variables, x and y, and two Real parameters, a and b.

Often a physical or biological system will be modelled by a system of partial differential equations. The process of numerically investigating theses systems usually involves transforming the partial differential equations into ordinary differential equations and, in the nonlinear case, numerically integrating these using one of several popular techniques, all of which involve a discrete time increment. Thus, many mathematical models comprised of partial differential equations eventually are transformed into the discrete time and space domain of numerically integrated ordinary nonlinear differential equations. An excellent treatment of this phenomena is presented by Lorenz in 1989 in an investigation of chaotic behavior induced by using difference equations as approximations to ODE's solved numerically with a large time increment .

**History**

Perhaps the most widely known use of iterated maps in the analysis of dynamical systems is the Poincaré return map. Henri Poincaré reduced the study of an autonomous differential equation of the 2nd order to the study of a transformation of the line into itself by interpreting the second order differential equation as a 1st order dynamical system in the plane and drawing a section across its trajectories. He also considered a transformation of a circle into itself defined by trajectories on a torus and their section. Poincaré generalized this method to dynamical systems in three dimensions where the reduced section is a surface and the return map a transformation of the plane into itself. The restricted problem of three bodies, reduced to three dimensions, was further reduced to a planar endomorphism by Poincaré. Investigating the invariant curves passing through a saddle point of a planar endomorphism, Henri Poincaré discovered the existence of homoclinic and heteroclinic (doubly-asymptotic) points. He described the figure obtained as an inextricable tangle of curves.

Examples of the use of iterated maps (or recurrence relations) throughout the history of mathematics abound. The Fibonacci sequence, Newton's method and the Naperian logarithm can or have been defined as recurrence relations. Fibonacci's sequence can be defined by the recurrence relation with the initial conditions . Viewed as a map of the plane, it becomes which has a unique fixed point at the origin. Calculating the eigenvalues, we see that implies that or , the Golden Mean and the opposite of its inverse.

Newton's method for finding the roots of involves the iteration of the recurrence relation . Hurwitz defined the Naperian logarithm of a number by means of the recurrence relation with solution . Thus, .

The theory of semi-cascades is more recent than that of flows and cascades and has mainly been studied for maps of the line. A rigorous study in two dimensions dates only from the 1960s. In 1964 Christian Mira introduced the method of critical curves. One of the distinguishing features of a noninvertible endomorphism is the existence of these curves. The geometric theory of semi-cascades, as developed by Mira and his school, relies heavily on the analysis of the critical curves.

Back to Dr. Record's Resume or proceed to Chapter 1 of Dr. Record's Ph.D. Thesis