Practical Numerical Algorithms for Chaotic Systems

1. Front Matter

   Pages i-xiv

   PDF

2. Steady-State Solutions and Limit Sets

   • Thomas S. Parker, Leon O. Chua

   Pages 1-29

3. Poincaré Maps

   • Thomas S. Parker, Leon O. Chua

   Pages 31-56

4. Stability of Limit Sets

   • Thomas S. Parker, Leon O. Chua

   Pages 57-82

5. Integration of Trajectories

   • Thomas S. Parker, Leon O. Chua

   Pages 83-114

6. Locating Limit Sets

   • Thomas S. Parker, Leon O. Chua

   Pages 115-138

7. Stable and Unstable Manifolds

   • Thomas S. Parker, Leon O. Chua

   Pages 139-166

8. Dimension

   • Thomas S. Parker, Leon O. Chua

   Pages 167-199

9. Bifurcation Diagrams

   • Thomas S. Parker, Leon O. Chua

   Pages 201-235

10. Programming

    • Thomas S. Parker, Leon O. Chua

    Pages 237-267

11. Phase Portraits

    • Thomas S. Parker, Leon O. Chua

    Pages 269-300

12. Back Matter

    Pages 301-348

    PDF

One of the basic tenets of science is that deterministic systems are completely predictable-given the initial condition and the equations describing a system, the behavior of the system can be predicted 1 for all time. The discovery of chaotic systems has eliminated this viewpoint. Simply put, a chaotic system is a deterministic system that exhibits random behavior. Though identified as a robust phenomenon only twenty years ago, chaos has almost certainly been encountered by scientists and engi­ neers many times during the last century only to be dismissed as physical noise. Chaos is such a wide-spread phenomenon that it has now been reported in virtually every scientific discipline: astronomy, biology, biophysics, chemistry, engineering, geology, mathematics, medicine, meteorology, plasmas, physics, and even the social sci­ ences. It is no coincidence that during the same two decades in which chaos has grown into an independent field of research, computers have permeated society. It is, in fact, the wide availability of inex­ pensive computing power that has spurred much of the research in chaotic dynamics. The reason is simple: the computer can calculate a solution of a nonlinear system. This is no small feat. Unlike lin­ ear systems, where closed-form solutions can be written in terms of the system's eigenvalues and eigenvectors, few nonlinear systems and virtually no chaotic systems possess closed-form solutions.

• Nonlinear system
• algorithms
• bifurcation
• stability
• system

• Hewlett Packard, Santa Rosa, USA

  Thomas S. Parker

• Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, USA

  Leon O. Chua