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. His methods are now used in modeling motion and change in all areas of science.

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Alligood Tim D. Sauer James A. Used by permission of ARS. Differentiable dynamical systems. Chaotic behavior in systems. Sauer, Tim. Yorke, James A. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer-Verlag New York, Inc.

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It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, which are derivatives of position.

His greatest single success was his discovery that the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the bodies. He demonstrated that the observed motion of the planets could be explained by assuming that there is a gravitational attraction between any two objects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them.

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. When solutions could be found, they described very regular motion. If the solutions remained in a bounded region of space, they settled down to either A a steady state, often due to energy loss by friction, or B an oscillation that was either periodic or quasiperiodic, akin to the clocklike motion of the moon and planets.

In the solar system, there were obviously many different periods. The moon traveled around the earth in a month, the earth around the sun in about a year, and Jupiter around the sun in about Such systems with multiple incommensurable periods came to be called quasiperiodic.

Scientists knew of systems which had more complicated behavior, such as a pot of boiling water, or the molecules of air colliding in a room. However, since these systems were composed of an immense number of interacting particles, the complexity of their motions was not held to be surprising.

The new motion is erratic, but not simply quasiperiodic with a large number of periods, and not necessarily due to a large number of interacting particles. It is a type of behavior that is possible in very simple systems.

A small number of mathematicians and physicists were familiar with the existence of a third type of motion prior to this time. James Clerk Maxwell, who studied the motion of gas molecules in about , was probably aware that even a system composed of two colliding gas particles in a box would have neither motion type A nor B, and that the long term behavior of the motions would for all practical purposes be unpredictable.

He was aware that very small changes in the initial motion of the particles would result in immense changes in the trajectories of the molecules, even if they were thought of as hard spheres. Consider two atoms of equal mass, modeled as hard spheres. Give the atoms equal but opposite velocities, and assume that their positions are selected at random in a large three-dimensional region of space. Maxwell showed that if they collide, all directions of travel will be equally likely after the collision.

He recognized that small changes in initial positions can result in large changes in outcomes. In a discussion of free will, he suggested that it would be impossible to test whether a leopard has free will, because one could never compute from a study of its atoms what the leopard would do. See Chapter 2. His techniques were applicable to a wide variety of physical systems. Important further contributions were made by Birkhoff, Cartwright and Littlewood, Levinson, Kolmogorov and his students, among others.

By the s, there were groups of mathematicians, particularly in Berkeley and in Moscow, striving to understand this third kind of motion that we now call chaos. But only with the advent of personal computers, with screens capable of displaying graphics, have scientists and engineers been able to see that important equations in their own specialties had such solutions, at least for some ranges of parameters that appear in the equations.

The key requirement is that the system involve a nonlinearity. It is now common for experiments whose previous anomalous behavior was attributed to experiment error or noise to be reevaluated for an explanation in these new terms.

Taken together, these new terms form a set of unifying principles, often called dynamical systems theory, that cross many disciplinary boundaries. The theory of dynamical systems describes phenomena that are common to physical and biological systems throughout science. It is critical to the advancement of science that exacting standards are applied to what is meant by knowledge.

Beautiful theories can be appreciated for their own sake, but science is a severe taskmaster. Intriguing ideas are often rejected or ignored because they do not meet the standards of what is knowledge. The standards of mathematicians and scientists are rather different. Mathe- maticians prove theorems. Scientists look atrealistic models. The mathematicians feared that nothing was proved so nothing was learned. Scientists said that models without physical quantities like charge, mass, energy, or acceleration could not be relevant to physical studies.

Mathematicians found that these computer studies could lead to new ideas that slowly yielded new theorems. Scientists found that computer studies of much more complicated models yielded behaviors similar to those of the simplistic models, and that perhaps the simpler models captured the key phenomena. Finally, laboratory experiments began to be carried out that showed un- equivocal evidence of unusual nonlinear effects and chaotic behavior in very familiar settings.

In this sense, the chaotic revolution is quite different than that of relativity, which shows its effects at high energies and velocities, and quantum theory, whose effects are submicroscopic.

We will look at many pictures produced by computers and we try to make mathematical sense of them. For example, a computer study of the driven pendulum in Chapter 2 reveals irregular, persistent, complex behavior for ten million oscillations. Does this behavior persist for one billion oscillations? However, even if it continues its complex behavior throughout our computer study, we cannot guarantee it would persist forever.

Perhaps it stops abruptly after one trillion oscillations; we do not know for certain. We can prove that there exist initial positions and velocities of the pendulum that yield complex behavior forever, but these choices are conceivably quite atypical. There are even simpler models where we know that such chaotic behavior does persist forever. In this world, pictures with uncertain messages remain the medium of inspiration.

There is a philosophy of modeling in which we study idealized systems that have properties that can be closely approximated by physical systems. The experimentalist takes the view that only quantities that can be measured have meaning. For example, we will see immediately in Chapters 1 and 2 the way chaos develops as a physical parameter like friction is varied.

This is a mathematical reality that underlies what the experimentalist can see. It is there, but often hidden from view by the noise of the universe. All science is of course dependent on simplistic models. If we study a vibrating beam, we will generally not model the atoms of which it is made.

And we do not include in our model usually the tidal effects of the stars and the planets on our vibrating beam. We ignore all these effects so that we can isolate the implications of a very limited list of concepts. It is our goal to give an introduction to some of the most intriguing ideas in dynamics, the ideas we love most.

Just as chemistry has its elements and physics has its elementary particles, dynamics has its fundamental elements: with names like attractors, basins, saddles, homoclinic points, cascades, and horseshoes. As a reader, your ability to work with these ideas will come from your own effort.

We will consider our job to be accomplished if we can help you learn what to look for in your own studies of dynamical systems of the world and universe. The level is aimed at undergraduates and beginning graduate students. Typically, we have used parts of Chapters 1—9 as the core of such a course, spending roughly equal amounts of time on iterated maps Chapters 1—6 and differential equations Chapters 7—9. Some of the maps we use as examples in the early chapters come from differential equations, so that their importance in the subject is stressed.

The impetus for advances in dynamical systems has come from many sources: mathematics, theoretical science, computer simulation, and experimen- ix [ E Nusse and J.

A Yorke Springer-Verlag.

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