Chaos: what it is; why it matters.

1. Chaos: what it is; why it matters. Chaos: we have no idea what will ever happen! (only statistics can save us)

2. What is chaos: first an example • Lorenz model of convection in atmosphere • three variables: • x, strength of convection • y, heat being transported • z, top to bottom temperature difference

3. What is chaos: first an example • Lorenz model of convection in atmosphere • three variables: • x, strength of convection • y, heat being transported • z, top to bottom temperature difference • And the parameters  and 

4. This model is not steady, for some parameters. Simplest model with "weather" • For some parameters, with weak temperature differences, it is "stable" • What does that mean? • No Weather! • What ever initial condition it has, it settles down to a steady state • stable_hasSteady.m; explain plot • explain evolution to steady states from very different ICs • For most initial conditions, two similar initial conditions end up in the same steady state • stable_noDiverge.m

5. For other parameters, model is chaotic • when larger temperature difference or bigger distance between top and bottom; also less stratification. • What is chaos? • Weather; does not settle down to steady state • show_weather.m • notice it alternates between two kinds of weather • like the stormy weeks and calm weeks we have in winter • They have "strange attractors" -- define. • Sensitive to initial conditions • even infinitesimal changes to initial conditions lead to totally different solutions • sensitive_to_ic.m

6. Is the weather like this? Yes! • The butterfly effect • Why only a 10 day forecast? • fundamentally limited by our knowledge of ICs!

7. It is worse than you might think • even if we know the IC perfectly, we cannot solve the problems for long times! • solve same problem with ode23 and ode45 • model_failure.m • why?

8. Is this problem limited to complex weather models? • No! • Simplest ecosystem model: • when population is low, grow at rate r Pn+1=rPn • but population growth decreases as increases towards Pmax (explain) Pn+1=r(Pmax-Pn)*Pn • observed in many ecosystems • in particular, in parasite populations and disease.

9. As growth rate r increases, solution becomes chaotic. • explain graph below, slowly! (Hastings et al. 1994) • Where is transition to chaos? Explain dynamics.

10. These dynamics are common in real life • a flag fluttering in the breeze • smoke rising from a cigarette • the path of a hurricane • the roll of dice • So are we doomed to a life of ignorance? • This question important on a very practical level • There are many who ask: if we cannot predict weather, how can we predict climate?

11. What do I mean by that? • rerun sensitive_to_ic.m • do they have the same strange attractors? • do they spend the same time around them? • overall, does it have the same behavior? • Just because we cannot predict the exact weather does not mean we cannot predict the climate! • This is not just a metaphorical point, but is relevant to current debate.

12. However, thinking about chaos does illuminate what we can know. • And more fundamentally, what we can predict. • In 1914, Gavrilo Princep:

13. Shot dead (after trying to blow up) • Archduke Franz Ferdinand of Austria

14. In order to, they said, to create a Greater Serbia or a Yugoslavia • This, in a complicated way, leads to the first world war

15. And • 9.71million military deaths • 6.8 million civilian deaths • try to put these #s to scale!

16. Did this single act "cause" WW1? • It was the IC that triggered it. • But was WW1 likely? • Inevitable? • We know enough about the future to have some ideas about what is likely • We might even know what is a bad idea • But reality is too sensitive to initial conditions to ever really predict • What if Hitler had died in the trenches? • Chaos is a good metaphor for many things • Beware of taking metaphors too far… • In all things humility & boldness.