Chaos in Nature

The following sections are included:

• Preface
• Foreword
• Acknowledgments
• Contents

The following sections are included:

• Kepler’s Empirical Laws
• The Law of Gravitation
• Theory of the Moon

The following sections are included:

• Imperfections in Newton’s Theory
• Challenges to the Law of Gravitation
• Problem of the Convergence of Series

The following sections are included:

• Simplification of the Geometry
• Simplification of the General Equations
• The First Exact Solutions

Despite the absence of exact general solutions, the magicians who practice celestial mechanics have obtained remarkable results using perturbation theory. In this setting, when the differential equations are expanded in series, a particular class of terms is singled out as particularly problematic: the secular terms. They have very little effect over the short term but as time evolves, their contributions grow continually and, in the long run, these terms make preponderant contributions. This property means it is impossible to make long term predictions of planetary trajectories. Worse, these trajectories are extremely sensitive to initial conditions. These are the two most important properties of chaotic motion.

The development of computational methods depending on perturbation theory improved the quality of predictions and the time intervals over which they were valid. Nevertheless, for very long times these predictions always contained large errors. The source of these errors was only understood through an entirely new approach, now known as the global analysis. The global nature comes from treating the evolution of an entire ensemble of trajectories, not a single isolated trajectory.

Urbain Jean-Joseph Le Verrier repeated Laplace’s calculations on planetary motions taking into account higher order terms. As far as Le Verrier was concerned, Laplace had shown that Newtonian mechanics is basically correct; he was not interested in testing whether the gravitation law was exact but rather was interested in constructing as precise a theory as possible. For this reason he took account of the perturbations due to all the planets and expanded perturbing functions between planet pairs up to sixth order with respect to inclinations, and up to seventh order with respect to eccentricities…

Chaotic behavior is intimately intertwined with the three-body problem. In celestial mechanics we treat the energy as a conserved quantity because dissipation is negligible since the planets move in a very high vacuum. This is very rarely observed for phenomena that occur on the earth (there is always friction somewhere): systems dissipate energy through heat. The trajectories of such systems are “attracted” to certain regions in their state spaces: these regions correspond to the existence of attractors of the type that we will encounter with the damped pendulum. In this case the attractor is a simple point in the state space that defines a stable state of rest (zero angle, zero angular velocity). There exist much more interesting situations in which the attractor is more complicated (chaotic, for example)…

Close behind celestial mechanics with its three-body problem, radio-engineering created a class of models leading directly to the study of chaotic behavior. These physical and mathematical problems appeared at the beginning of the 20th century in connection with the earliest vacuum tubes (diodes, triodes, …). The corresponding paradigm is designated under the name Theory of Oscillators, the title of the fundamental book on the subject by the founding members of the Russian school, namely Aleksandr Aleksandrovich Andronov, Aleksandr Aadol’fovich Witt and Semen Emmanuilovich Khaikin. The first edition appeared in 1937. The very practical context under which these studies were undertaken provided a central role for the differential equations that described these systems. The general problem is correctly and precisely presented in the introduction to the Theory of Oscillators…

The following sections are included:

• Prediction in Meteorology
• The Lorenz System
  • State Space
  • The Stability of Periodic Solutions
  • Numerical Integration and Application of Linear Theory
  • Topological Analysis
  • First-return Map to Maxima
• Sensitivity to Initial Conditions
• Turbulence, Aperiodic Solutions, and Chaos
• Hydrodynamics and the Lorenz Attractor
• Laser Dynamics and the Lorenz System
• Conclusion

With the studies of Lorenz, Ruelle and Ueda it began to be appreciated that a set of simple differential equations could produce chaotic behavior: at the same time this type of behavior began to be discovered in several very simple systems starting to the late 1970s. However, in order to become a real scientific discipline, it was necessary to identify the invariant structures underlying chaotic attractors. For dissipative systems there were several ideas that allowed a discussion of the architecture of chaotic behavior. To identify them, it was useful to study a system as simple as possible in order to identify in detail the ingredients that were necessary to produce chaotic behavior. This chapter is somewhat more technical than the previous chapters. Even read rapidly, it will give an idea of the concepts that are used to characterize chaos. The reader will note, however, that (s)he can read most of this book without mastering the contents of this chapter.

The following sections are included:

• The Earliest Experiments
• Chaos in an Experimental BZ-Reaction
• Chaotic Copper Electrodissolution
• Conclusion

The following sections are included:

• Theories of Malthus and Verhulst
• A Model with Two Species
• Models with Three Species
• Observational Evidence

Astrophysics has not escaped chaotic behavior: for example, stars like our sun or variable stars exhibit oscillatory behavior that may eventually turn out to be chaotic. Since the end of the 19th century, Assyriologists have come to the cuneiform texts of the Babylonians, dating from two or three thousand years ago, report the discovery of stars whose luminosity is variable. This hypothesis is supported by observations of a star in the constellation of Cetus the Whale in the eastern part of Aquarius. This star could become visible and very brilliant or go out like a torch; it could appear dark, yellow, or very red. This description applies accurately to the star Mira Ceti. However, there is no reference to stellar variability in the catalog of Hipparcus (-130) or the Almagest of Ptolemy (≃ 138). This absence extends through the star catalogue Zij-i Sultani of Ulugh Begh — his name means the great ruler — (1394–1449) dating from 1437. It seems that the main reason for the absence of these phenomena in the scientific tracts is that dogma blinded observers. One of the foundational principles of Aristotle’s physics was that change was not possible in the supralunary heavenly world, home of the celestial bodies and constituted of the quintessence, the divine, imperishable and inalterable matter. In the sublunary world, where nature consists of the four other elements (water, earth, air, and fire), that changes could occur. In fact, it seems that medieval scientists were unaware of a number of celestial events, for example the supernova in the Crab nebula, that occurred on 4 July 1054, was recorded by the Chinese but for which there is no European record. The Chinese called “guest stars” those appearing temporarily.

A dynamical system is a system that evolves as a function of time. Accordingly, all living systems are dynamical systems. Further, homeostasis is a fundamental property of the living and can be regarded as a propensity to maintain the living system within a particular dynamical domain. For example, a homeostatic dynamical system often works like a thermostat. Thus, a biological system remains within a certain dynamical range by self-regulating phenomena that can be considered in two ways. On the one hand, the system acts to modify the environment (like heating regulated by the thermostat modifies temperature) by controlling reactions that slow in order to stop some process. On the other hand, every change in the environment stimulates an appropriate adaption of the living system. Homeostatis is defined by the existence of specific regulatory mechanisms that maintain processes within a given dynamical regime…

While there is a firm foundation for the laws of chaos when a system has low dimension and is governed by highly dissipative dynamics, “chaoticians” do not always have a good understanding when behavior is governed by systems with low dissipation and/or high dimension (that is, dimension greater than 3). Very many difficulties are encountered for such systems, and so far it has not been possible to understand chaos in situations more complicated than those encountered in previous chapters.

The following sections are included:

• General Index
• Author Index