The Lorenz Equations

1. Front Matter

   Pages i-xii

   PDF

2. Introduction and Simple Properties

   • Colin Sparrow

   Pages 1-12

3. Homoclinic Explosions: The First Homoclinic Explosion

   • Colin Sparrow

   Pages 13-25

4. Preturbulence, Strange Attractors and Geometric Models

   • Colin Sparrow

   Pages 26-50

5. Period Doubling and Stable Orbits

   • Colin Sparrow

   Pages 51-75

6. From Strange Attractor to Period Doubling

   • Colin Sparrow

   Pages 76-114

7. Symbolic Description of Orbits: The Stable Manifolds of C1 and C2

   • Colin Sparrow

   Pages 115-131

8. Large r

   • Colin Sparrow

   Pages 132-150

9. Small b

   • Colin Sparrow

   Pages 151-178

10. Other Approaches, Other Systems, Summary and Afterword

    • Colin Sparrow

    Pages 179-191

11. Back Matter

    Pages 192-269

    PDF

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.

• Bifurcations
• Chaos
• Chaos (Math.)
• Equations
• Lorenzsche Gleichung
• Seltsamer Attraktor
• Verweigerung (Math.)
• ordinary differential equation

• King’s College, Cambridge, UK

  Colin Sparrow

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