Routes to Chaos: The Non-Linear Dynamical System Approach – Part I

Give me the velocity and position of every molecule, and I will predict your future… Pierre-Simon Laplace – Mécanique Céleste

Introduction

This next post might be a little less approachable to some, especially those mostly attracted to the physical phenomenon standpoint, yet absolutely rejecting and keeping themself as safe as they can from mathematical valuable approaches. As for me I found myself always attracted to the secrets mathematics may expose about physical phenomena, even sometimes could imagine the mathematical evolution as if it wasn’t just a tool, but as a view of a gifted artist’s mind at the moment of sliding he’s paintbrush through the canvas after lit by a creative thought.
The field of non-linear dynamical systems and the study how solutions from non-linear deterministic equations, such as the Navier–Stokes equations, make a transition from a stable and ordered regime into an unstable and chaotic regime has very much that quality I love so much about mathematics.

  • As far as non-linear dynamical systems are concerned I’ve based my knowledge mostly of the book “Chaos: An Introduction to Dynamical Systems” by  Kathleen (T. Alligood, Tim D. Sauer, James A. Yorke)

The Logistic Map

Perhaps the easiest way for me to make an acquaintance with non-linear systems is the logistic map.
The logistic map is defined as follows:

Xi+1=μ(1-Xi)
Xi ∈ [0, 1], μ ∈ [0, 4]

Then substitute:


Xi=(ui+1)/2
as: ui ∈ [−1, 1]

Now it can be shown that the Logistic Map can be written in such a form that has an equivalent to a non-linear term, a forcing term, and a viscous term which is linear, terms we know from Navier-Stokes Equations:

This equation is often referred to as “Poor Man’s Navier-Stokes Equations”, and a special notable fact about the equation is that it has no spatial character. The poor man’s Navier–Stokes could be used to demonstrate several aspects of non-linear dynamical systems as a view to the gifted artist’s mind at moment he thought and began sliding his paintbrush through the canvas to paint some of the phenomenological counterparts that we find in turbulence. Especially transition to turbulence, fractal structures, intermittency, and in the end such as when you take a look at every artist’s completion of his creation – the finite range of predictability in meaning.

A Logistic Map – cobweb graph
in a cobweb graph subsequent iterations are visualized.

We may draw the cobweb graph for the “transients” Xi for all initial values X0 ∈ [0, 1], then check how stability behaviour comes about. A solution X=0 is considered stable (or fixed point). In other words, a solution which has reached this fixed point shall remain there forever.
When μ>1 there is still a fixed point to converge to, yet it is not at X=0 anymore (it is easy to see that it is now at = 1 − μ−1).


Bifurcations

We can also keep following the logistic map while enlarging μ and we shall see that the fixed point keeps changing until μ=3, then, the solution becomes periodic and starts alternating between two values. This is what is called a bifurcation. the periodicity itself is stable, but the original fixed point has a different character, and it is now an unstable node (sometimes referred to as repler). This kind of bifurcation should be a beloved one by most for it’s the stage where the artist gets’s sudden spark of creativity his right hemisphere lit up by a super-critical value of μ, and the paintbrush starts its initial divergence to geniousity.

We keep enlarging the values of μ, still finding a doubled value periodic orbit till lo and behold when μ=(1+61/2) the solution starts alternating between four values. In other words, there’s yet another period-doubling – a second bifurcation occurs.

Such a process of period-doubling (another bifurcation) continues and becomes more and more frantic in rapid succession. At a value of μ=μc (where μc is about 3.58 – i.e.very rapidly) the dynamic system undergoes what seems to be an infinite number of bifurcations such that also an orbit period becomes infinitely long. At this point, periodicity is distinguishable anymore.


Fractalization and The “Strange Attractor”

When magnifying the state reached beyond μc=3.58 it is revealed that each part of that “infinite orbit” is a band with a finite width, in each such width three narrower bands, in these ones three narrower once… Ad infinitum (there’s a nagging song to describe such a process: if such and such and such and such passed in front of you to where you were going so how many sis actually pass?… well, just you!).
Anyhow, such an orbit is not periodic anymore, but this structure is found for different initial values X0. Such a non-linear dynamical system has some degree of stability, however, it doesn’t reach a state that is made up of a finite number of fixed points, but a state that has a fractal nature that has a dimension related to him. In non-linear dynamical system language, such a solution is called a “Strange Attractor”.

Examples:

  • the formation azimuthal rolls in Taylor–Couette flow (at specific flow conditions):

    (a) Even period doubling is observed when these rolls become modulated in the axial direction.

    (b) Subsequent bifurcations (again, at specific flow conditions) suggest a possible transition to reach
    a turbulent state after an infinite number of bifurcations characterized by a strange attractor. This
    transition scenario is known as Landau’s route to chaos (it is interesting to mention that
    experimentally Landau’s route has never been observed, notwithstanding the fact that
    mathematical advancement in the field of non-linear dynamical systems suggest different insights
    in the emergence of turbulence…).
Possible Transition
scenarios (Just 2005)
  • Logistic Map: here it is shown graphically that for values μ>μc the logistic map shows chaotic behavior, apart from some small regions of somewhat of a with laminar motion.
    But this is not the most interesting point. actually it is far from it. The bifurcation diagram shown below shows clearly that once an initial value X0 is given, the transient process for all i>0 is fully deterministic – as should be… The essential aspect of Newton’s laws, and thus of the Navier–Stokes equations, is that they are deterministic. This means that in principle, given the equations of motion together with the initial and boundary conditions, the evolution of the flflow fifield can be computed as a function of time; hence, the solution to the equations and conditions that describe the flow is completely determined. In other words, the deterministic character of Newton’s mechanics implies full predictability of the fluid motion.
0<μ<3.55
3.4<μ>4

Highly recommended for deepening the understanding in non-linear dynamical systems:

  • Nonlinear Dynamics and Chaos (Professor Steven Strogatz, Cornell University

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