Period doubling is chaos?

Question

I've already read something about chaos and it's origins but I am not sure that this affirmative statement is true for all the cases. Can anyone help me? Thanks, Bruno

Answer 1

You're confusing two related concept: "chaos" (in quotes because there are several distinct definitions) is a property of an attractor, in other words of a specific dynamical system (though note, nonlinear dynamics can support multiple attractors), for example:

Rössler attractor (ODE)
Lorenz system (ODE)
Hénon map (discrete map)

On the other hand, period doubling cascade to chaos refers to bifurcation scenarios, which are observed when one or more parameters are varied - so refers to behavior across a parametrized family of dynamical systems.

Although period doubling cascades are common mechanisms in continuous and some discrete dynamics, it's not true in general: border-collision and corner-collision bifurcations in hybrid systems (dynamics whose state space is a product of continuous and discrete states - so they can model switching and jump systems) can move a system from, say, periodic motion to chaos instantaneously.

Also important to keep in mind that different classes of dynamics support very different types of chaos - eg, contrast turbulence in Navier Stokes PDEs, versus avalanche models in cellular automata as developed by late physicist Per Bak (see "How Nature Works" - it's not about the butterfly effect).

Answer 2

Bifurcation theory refers to the branching of solutions at some critical value parameter, which results in a loss of structural stability.

Bifurcation is one of the routes to chaos.

A few main ruotes to chaos are period doubling, intermittency, horseshoes and homoclinic orbits.

You can look at the Logistic Map for an example of where period doublings can lead to chaos.

Regards