Renormalization and chaos in the logistic map

Logistic map

Many features of non-Hamiltonian chaos can be seen in this simple map(and other similar one dimensional maps).

Why? Universality.

Period doublingIntermittencyCrisisErgodicityStrange attractorsPeriodic/Aperiodic mixTopological Cantor set

Time series

Superstable points; convergence to attractor very rapidCritical slowing down near a bifurcation; convergence very slow

Time series

Superstable points; convergence to attractor very rapidCritical slowing down near a bifurcation; convergence very slow

Time series

Bifurcation to a two cycle attractor

Time series

A series of period doublings of the attractor occur for increasing valuesof r, the “biotic potential” as it is sometimes known for the logistic map

Time series

At a critical value of r the dynamics become aperiodic. However, note thatinitially close trajectories remain close. Dynamics are not ergodic.

Time series

At r = 4 one finds aperiodic motion, in which initially close trajectoriesexponentially diverge.

Self similarity, intermittency, “crisis”

Other one dimensional maps

Other one dimensional maps show a rather similar structure; LHSis the sine map and RHS one that I made up randomly.

In fact the structure of the periodic cycles with r is universal.

Experiment

Universality

2nd order phase transitions: details, i.e. interaction form, does not matternear the phase transition.

Depend only on (e.g.) symmetry of order parameter, dimensionality,range of interaction

Can be calculate from simple models that share these feature with(complicated) reality.

The period doubling regime

• Derivative identical for all points in cycle.• All points become unstable simultaneously; period doubling topology canonly be that shown above.

The period doubling regime

Superstable cycles so called due to accelerated form of convergence;in Taylor expansion first order term vanishes.

Use continuity

The period doubling regime

These constants have been measured in many experiments!

Renormalisation

Change coordinates such that superstablepoint is at origin

Renormalisation

Renormalisation

Renormalisation

Universal mapping functions

Each of these functions is an i cycle attractor.

Doubling operator for g’s

Doubling operator in function space g

Eigenvalues of the doubling operator

Linearize previousequation

Eigenvalues of the doubling operator

This was an assumption of Feigenbaum’s;proved later (1982)

Eigenvalues of the doubling operator

Liapunov exponents

Liapunov exponents

Fully developed chaos at r = 4

Bit shift map

Any number periodic in base 2 leadsto periodic dynamics of the logistic map

Irrational numbers lead to non-periodicdynamics.

Change variables, leads to exact solutionfor this case. (Other cases: r=-2, r=+2).

Ergodicity at r = 4

Strange attractor:

• Fractal• Nearby points diverge under map flow• All points visited: ergodic i.e., topologically mixing

Strange attractor of logistic map (and similarone dimensional maps) is a “topological Cantor set”Lorenz attractor

Ergodic: all points visited

Invarient density