The Impact of Chaos on Science and Society (UNU, 1997, 415 pages): 4. The impact of chaos on mathematics: II. The Lorenz attractor

The Impact of Chaos on Science and Society (UNU, 1997, 415 pages)

4. The impact of chaos on mathematics

		(introduction...)
		Abstract
		I. Introduction: A historical view
		II. The Lorenz attractor
		III. The Feigenbaum bifurcation
		IV. Hydrodynamic turbulence
		V. Ergodic theory of differentiable dynamical systems: Axiom A systems
		VI. Ergodic theory of differentiable dynamical systems: General systems
		VII. Quadratic maps of the interval and the Hénon attractor
		VIII. Zeta functions
		IX. Conclusion
		References

II. The Lorenz attractor

The mathematical studies of Smale [5] have shown that the orbit of a dynamical system is in some cases asymptotic to a complicated set called an Axiom A attractor. The behaviour of the system is then chaotic and, since Axiom A attractors can be analysed in great detail, they provide very important examples of chaos.

Another example of chaos was obtained in an early computer study by Lorenz [3] when he analysed a (rather brutally) simplified model of convection described by the following equations

with s=10, b=8/3, r=28. The Lorenz attractor is chaotic, but different from the Axiom A attractors of Smale in that it contains an (unstable) fixed point for the time evolution. Interestingly we have, at this time, no mathematical proof that the solutions of the Lorenz system behave (chaotically) as we think they do. We have however a model inspired by the equations, and called geometric Lorenz attractor, for which a detailed mathematical study has been given by Guckenheimer [6] and Williams [7]. It is known in particular that the geometric Lorenz attractor has some properties of persistence when the equations are slightly changed (technically, what is proved is co-dimension 2 structural stability).