The Lorenz Equations: Bifurcations Chaos and Strange Attractors 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. 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 be­ 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 be­ 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.