Author | Sparrow, Colin. author |
---|---|
Title | The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors [electronic resource] / by Colin Sparrow |
Imprint | New York, NY : Springer New York, 1982 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-5767-7 |
Descript | XII, 270 p. online resource |
1. Introduction and Simple Properties -- 1.1. Introduction -- 1.2. Chaotic Ordinary Differential Equations -- 1.3. Our Approach to the Lorenz Equations -- 1.4. Simple Properties of the Lorenz Equations -- 2. Homoclinic Explosions: The First Homoclinic Explosion -- 2.1. Existence of a Homoclinic Orbit -- 2.2. The Bifurcation Associated with a Homoclinic Orbit -- 2.3. Summary and Some General Definitions -- 3. Preturbulence, Strange Attractors and Geometric Models -- 3.1. Periodic Orbits for the Hopf Bifurcation -- 3.2. Preturbulence and Return Maps -- 3.3. Strange Attractor and Homoclinic Explosions -- 3.4. Geometric Models of the Lorenz Equations -- 3.5. Summary -- 4. Period Doubling and Stable Orbits -- 4.1. Three Bifurcations Involving Periodic Orbits -- 4.2. 99.524 < r < 100.795. The x2y Period Doubling Window -- 4.3. 145 < r < 166. The x2y2 Period Doubling Window -- 4.4. Intermittent Chaos -- 4.5. 214.364 < r < ?. The Final xy Period Doubling Window -- 4.6. Noisy Periodicity -- 4.7. Summary -- 5. From Strange Attractor to Period Doubling -- 5.1. Hooked Return Maps -- 5.2. Numerical Experiments -- 5.3. Development of Return Maps as r Increases: Homoclinic Explosions and Period Doubling -- 5.4. Numerical Experiments on Periodic Orbits -- 5.5. Period Doubling and One-Dimensional Maps -- 5.6. Global Approach and Some Conjectures -- 5.7. Summary -- 6. Symbolic Description of Orbits: The Stable Manifolds of C1 and C2 -- 6.1. The Maxima-in-z Method -- 6.2. Symbolic Descriptions from the Stable Manifolds of C1 and C2 -- 6.3. Summary -- 7. Large r -- 7.1. The Averaged Equations -- 7.2. Analysis and Interpretation of the Averaged Equations -- 7.3. Anomalous Periodic Orbits for Small b and Large r -- 7.4. Summary -- 8. Small b -- 8.1. Twisting Around the z-Axis -- 8.2. Homoclinic Explosions with Extra Twists -- 8.3. Periodic Orbits Without Extra Twisting Around the z-Axis -- 8.4. Heteroclinic Orbits Between C1 and C2 -- 8.5. Heteroclinic Bifurcations -- 8.6. General Behaviour When b = 0.25 -- 8.7. Summary -- 9. Other Approaches, Other Systems, Summary and Afterword -- 9.1. Summary of Predicted Bifurcations for Varying Parameters ?, b and r -- 9.2. Other Approaches -- 9.3. Extensions of the Lorenz System -- 9.4. Afterword โ A Personal View -- Appendix A. Definitions -- Appendix B. Derivation of the Lorenz Equations from the Motion of a Laboratory Water Wheel -- Appendix C. Boundedness of the Lorenz Equations -- Appendix D. Homoclinic Explosions -- Appendix E. Numerical Methods for Studying Return Maps and for Locating Periodic Orbits -- Appendix F. Computational Difficulties Involved in Calculating Trajectories which Pass Close to the Origin -- Appendix G. Geometric Models of the Lorenz Equations -- Appendix H. One-Dimensional Maps from Successive Local Maxima in z -- Appendix I. Numerically Computed Values of k(r) for ? = 10 and b = 8/3 -- Appendix J. Sequences of Homoclinic Explosions -- Appendix K. Large r; the Formulae