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The Lorenz equations : bifurcations, chaos, and strange attractors

Author: Colin Sparrow
Publisher: New York : Springer-Verlag, ©1982.
Series: Applied mathematical sciences (Springer-Verlag New York Inc.), v. 41.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
The equations which we are going to study in these notes were first presented in 1963 by E.N. Lorenz. They define a three-dimensional system of ordinary differential equations that depends on three real positive parameters. As we vary the parameters, we change the behaviour of the flow determined by the equations. For some parameter values, numerically computed solutions of the equations oscillate, apparently  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Sparrow, Colin.
Lorenz equations.
New York : Springer-Verlag, ©1982
(DLC) 82019435
(OCoLC)8974749
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Colin Sparrow
ISBN: 1461257670 9781461257677
OCLC Number: 727628446
Language Note: English.
Reproduction Notes: Electronic reproduction. [Place of publication not identified] : HathiTrust Digital Library, 2011. MiAaHDL
Description: 1 online resource (XII, 269 pages) : illustrations
Details: Master and use copy. Digital master created according to Benchmark for Faithful Digital Reproductions of Monographs and Serials, Version 1. Digital Library Federation, December 2002.
Contents: 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 --
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.
Series Title: Applied mathematical sciences (Springer-Verlag New York Inc.), v. 41.
Responsibility: Colin Sparrow.

Abstract:

The equations which we are going to study in these notes were first presented in 1963 by E.N. Lorenz. They define a three-dimensional system of ordinary differential equations that depends on three real positive parameters. As we vary the parameters, we change the behaviour of the flow determined by the equations. For some parameter values, numerically computed solutions of the equations oscillate, apparently forever, in the pseudo-random way we now call "chaotic"; this is the main reason for the immense amount of interest generated by the equations in the eighteen years since Lorenz first presented them. In addition, there are some parameter values for which we see "preturbulence", a phenomenon in which trajectories oscillate chaotically for long periods of time before finally settling down to stable stationary or stable periodic behaviour, others in which we see "intermittent chaos", where trajectories alternate beƯ tween chaotic and apparently stable periodic behaviours, and yet others in which we see "noisy periodicity", where trajectories appear chaotic though they stay very close to a non-stable periodic orbit. Though the Lorenz equations were not much studied in the years beƯ tween 1963 and 1975, the number of man, woman, and computer hours spent on them in recent years - since they came to the general attention of mathematicians and other researchers - must be truly immense.

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