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Quiz. Logistic map - populations … a<a_1 - 1-cycle a_1< a<a_2 - 2 cycle etc a=a_c=3.59665… chaos! Parameter a_1=3.0 For a< a_1 motion regular trajectories converge L<0 For a>a_c motion chaotic trajectories diverge L>0. Quiz 2. Bifurcation diagram - plot a along x, values on cycle on y
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Quiz ... • Logistic map - populations … • a<a_1 - 1-cycle • a_1< a<a_2 - 2 cycle etc • a=a_c=3.59665… chaos! • Parameter a_1=3.0 • For a< a_1 motion regular • trajectories converge • L<0 • For a>a_c motion chaotic • trajectories diverge • L>0
Quiz 2 • Bifurcation diagram - plot a along x, values on cycle on y • Feigenbaum delta D • D=(a_2-a_1)/(a_4-a_2) • universal - many maps have same D • chaos - dynamical variables lie on strange attractor - fractal • self-similar • non-integer dimension!
2D Maps and chaos • Regular fractals like Sierpinski are somewhat atypical • Need an example of a 2D map which possesses a chaotic regime with fractal structure • Henon map xn+1=a-xn2+byn yn+1=xn • a,b parameters
Henon • Michel Henon - French astronomer • studies of globular clusters • dense collections of stars • dynamics controlled by gravity • many-body problem - very hard! • Assumed that cluster was self-similar • Used computer simulation to study a simplified model
Orbits in globular clusters • Look at a 2D slice of a globular cluster • Focus on the orbit followed by just 1 star. • Plot the (x,y) position of the star each time passes thru slice • Low energies - regular and predictable ...
Chaos in the Henon Map • Fix b=0.3. As we increase the parameter a (energy) we see transition from cyclic motion to chaos • We can see period doubling • In chaotic regime see (x,y) values not random but lie on a fractal - Henon attractor • Self-similar and possesses a fractional dimension.
Lorenz model • Very simplified model of fluid flow (weather is fluid model also) • Consider fluid between 2 flat surfaces held at different temps • 3D map - describes how temperature, density and velocity of fluid depend on time • 1 important parameter - temp difference
Lorenz map dx/dt=s(y-x) dy/dt=-xz+rx-y dz/dt=xy-bz • Just a map again ! • For small r (temp) fixed pt observed. • As r is increased approached more slowly … • For r>24 see chaotic motion • motion lives on fractal !
Where is the map ? • Replace dx/dt by (xn+1-xn)/Dt • eqn. 1 becomes xn+1=xn+DT*s(yn-xn) ... • Plot say x against y to show motion ( 3 possible planes) • Since eqns for general fluid flow contain Lorenz’s model, we expect they too have chaotic regimes (the weather!)
Chaos in the Lorenz model • Notice that for r>28 the trajectory never intersects itself! • If it did so then it would begin to • repeat • But it is certinly confined to a finite region of 3D space • analogous to logistic map • infinite number of pts(curves) confined to a finite region but • unable to cover that region.
