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BACKGROUND Sir Isaac Newton hrought to the world the idea of modeling the motion of physical systems with equations. It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, of position. His greatest single success was his discovery that which are derivatives the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the hodies. He demonstrated that the ohserved motion of the planets could he explained hy assuming that there is a gravitational attraction he tween any two ohjects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them. The circular, elliptical, and parabolic orhits of astronomy were v INTRODUCTION no longer fundamental determinants of motion, but were approximations of laws specified with differential equations. His methods are now used in modeling motion and change in all areas of science. Subsequent generations of scientists extended the method of using differ­ ential equations to describe how physical systems evolve. But the method had a limitation. While the differential equations were sufficient to determine the behavior-in the sense that solutions of the equations did exist-it was frequently difficult to figure out what that behavior would be. It was often impossible to write down solutions in relatively simple algebraic expressions using a finite number of terms. Series solutions involving infinite sums often would not converge beyond some finite time.

Inhaltsverzeichnis

Frontmatter

Chapter One. One-Dimensional Maps

Abstract
THE FUNCTION f(x) = 2x is a rule that assigns to each number x a number twice as large. This is a simple mathematical model. We might imagine that x denotes the population of bacteria in a laboratory culture and that f(x) denotes the population one hour later. Then the rule expresses the fact that the population doubles every hour. If the culture has an initial population of 10,000 bacteria, then after one hour there will be f(10,000) = 20,000 bacteria, after two hours there will be f(f(l0,000)) = 40,000 bacteria, and so on.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Two. Two-Dimensional Maps

Abstract
IN CHAPTER 1 we developed the fundamental properties of one-dimensional dynamics. The concepts of periodic orbits, stability, and sensitive dependence of orbits are most easily understood in that context.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Three. Chaos

Abstract
THE CONCEPT of an unstable steady state is familiar in science. It is not possible in practice to balance a ball on the peak of a mountain, even though the configuration of the ball perfectly balanced on the peak is a steady state. The problem is that the trajectory of any initial position of the ball near, but not exactly at, the steady state, will evolve away from the steady state. We investigated sources and saddles, which are unstable fixed points of maps, in Chapters 1 and 2.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Four. Fractals

Abstract
A FRACTAL is a complicated geometric figure that, unlike a conventional complicated figure, does not simplify when it is magnified. In the way that Euclidean geometry has served as a descriptive language for the classical mechanics of motion, fractal geometry is being used for the patterns produced by chaos. Trajectories of the two-body problem, for example, consist of conic sections: ellipses, parabolas, and hyperbolas. Chaotic attractors, on the other hand, often have features repeated on many length or time scales.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Five. Chaos in Two-Dimensional Maps

Abstract
THE CONCEPTS of Lyapunov numbers and Lyapunov exponents can be extended to maps on ℝ m for m ≥ 1. In the one-dimensional case, the idea is to measure separation rates of nearby points along the real line. In higher dimensions, the local behavior of the dynamics may vary with the direction. Nearby points may be moving apart along one direction, and moving together along another.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Six. Chaotic Attractors

Abstract
AN IMPORTANT aspect of explaining dynamical phenomena is the description of attractors. Newton knew of two types of attracting motion that systems settle into: the apple sitting on the ground is in equilibrium, and the planets in the solar system are undergoing periodic, or more properly quasiperiodic motion, at least to good approximation. For the next 300 years, these were the only kinds of motion known for simple dynamical systems. Maxwell and Poincaré were among a small number of scientists who were not content with this view. The small number grew, but it was not until the widespread availability of desktop computers in the last quarter of the 20th century that the third type of motion, chaos, became generally recognized.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Seven. Differential Equations

Abstract
IN THE FIRST six chapters, we modeled physical processes with maps. One of the most important uses of maps in scientific applications is to assist in the study of a differential equation model. We found in Chapter 2 that the time-T map of a differential equation may capture the interesting dynamics of the process while affording substantial simplification from the original differential equation.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Eight. Periodic Orbits and Limit Sets

Abstract
THE BEGINNING of this book was devoted to an understanding of the asymptotic behavior of orbits of maps. Here we begin a similar study for solution orbits of differential equations. Chapter 7 contains examples of solutions that converge to equilibria and solutions that converge to periodic orbits called limit cycles. We will find that the dimension and shape of the phase space put serious constraints on the possible forms that asymptotic behavior can take.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Nine. Chaos in Differential Equations

Abstract
In the late 1950s, a meteorologist at MIT named Edward Lorenz acquired a Royal-McBee LGP-30 computer. It was the size of a refrigerator carton and contained 16KB of internal memory in a thicket of vacuum tubes. It could calculate at the rate of 60 multiplications per second. For the time, it was a staggering cache of computational power to be assigned to a single scientist.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Ten. Stable Manifolds and Crises

Abstract
WE INTRODUCED the subject of stable and unstable manifolds for saddles of planar maps in Chapter 2. There we emphasized that Poincaré used properties of these sets to predict when systems would contain complicated dynamics. He showed that if the stable and unstable manifolds crossed, there was behavior that we now call chaos. For a saddle fixed point in the plane, these “manifolds” are curves that can be highly convoluted. In general, we cannot hope to describe the manifolds with simple formulas, and we need to investigate properties that do not depend on this knowledge. Recall that for an invertible map of the plane and a fixed point saddle p, the stable manifold of p is the set of initial points whose forward orbits (under iteration by the map) converge to p, and the unstable manifold of p is the set whose backward orbits (under iteration by the inverse of the map) converge to p.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Eleven. Bifurcations

Abstract
OF ALL POSSIBLE motions in dynamical systems, a fixed point or equilibrium is the simplest. If the fixed point is stable, the system is likely to persist in that state, even in the face of minor disturbances.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Twelve. Cascades

Abstract
CASCADES OF period-doubling bifurcations have been seen in the great majority of low-dimensional systems that exhibit chaotic behavior. A “cascade” appears as an infinite sequence of period-doubling bifurcations. A stable periodic orbit is seen to become unstable as a parameter is increased or decreased and is replaced by a stable periodic orbit of twice its period. This orbit in turn becomes unstable and is replaced by a new stable orbit with its period again doubled, and the process continues through an infinity of such period-doubling bifurcations.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Chapter Thirteen. State Reconstruction from Data

Abstract
THE STATE of a system is a primitive concept that unifies the approach to many sciences. In this chapter we will describe the way states can be inferred from experimental measurements. In so doing we will revisit the Belousov-Zhabotinskii chemistry experiment from Lab Visit 3, the Couette-Taylor physics experiment from Lab Visit 4, and an example from insect physiology.
Kathleen T. Alligood, Tim D. Sauer, James A. Yorke

Backmatter

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