Routes to Chaos: The Non-Linear Dynamical System Approach and Lorenz Equations – Part II

If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of the same universe at a succeeding moment – Henri Poincaré

In “Routes to Chaos: The Non-Linear Dynamical System Approach – Part I” we gave a short introduction of the subject, then introduced the Logistic Map and Poor Man’s Navier Stokes Equations:

Poor Man’s Navier-Stokes Equations

We’ve enlarged the value of  μ and examined the behavior of the logistic map (the cobweb map) and saw how for small values of  μ the solution converges to a fixed point dependent upon the initial conditions. Then when  μ=3 the solution becomes periodic and starts alternating between two values – the first bifurcation – an unstable node (sometimes referred to as repler).
We kept enlarging μ and saw that more and more period doubling (e.g. more bifurcations) come about with smaller and smaller enlargements the second bifurcation (i.e. alternation of the solution between four values instead of two).
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 with an orbit period infinitely long and no distinguishable periodic cycle.
Magnifying our look at on the evolution in this stage we saw the fractal nature with an attached dimension, a solution called a “Strange Attractor”.

In part II we shall introduce a few non-linear dynamical systems procedures to aid us in introducing a very special and famous non-linear dynamical system – The Lorenz Equation.


Stretching and Folding

In a publication1982 (e.g. “The Evolution of a Turbulent Vortex”) Alexandre Chorin described at length a field of vortex tubes which tend to stretch and fold while bringing energy to dissipation scales. Such natural adaptivity is a real physical effect in turbulence, but it is one which must be controlled lest vast numbers of vortex elements appear to describe the dynamics of the dissipation process in excruciating detail.
In 1993 Chorin added yet another publication (e.g. “Hairpin removal in vortex interactions II”) and pioneered a solution to this problem by the removal of small scale hairpin shaped vortices that form in the course of the folding process as shown in the figure below.

Hairpin removal via Chorin’s (meshless) algorithm

The rationale for hairpin removal is that the velocity field associated with such hairpins is mostly local by nature. Moreover, further calculations about these hairpins will only track the movement of the localized energy to the dissipation scales. Thus, rather to invest in calculation of the small scale folding process, a simple estimate for the energy loss can be achieved by cutting away the small scale hairpin and reattaching the ends of the tube. The smallest resolvable scale would be determined by the minimum length of the vortex segments composing a filament (this is the basis of avoidance uncontrollable calculation proportions in meshless turbulence simulation methodologies – the smaller the allowed resolution, the more accurate the meshless model).

Top view of a developing and a fully developed mixing layer with meshless method following
Alexandre Chorin conceptual ideas

To explain the dynamics of vortex tubes we return to the 3D vorticity equation:

where Re is the Reynolds number. The non-linear dynamics is achieved by convection of an object, vortex stretching and reorientation are accommodated by moving the endpoints of the object. As these objects lengthen, they are divided to smaller segments. A decay model is added to account for long time evolution effects.
The point to be taken from the above description of the non-linear dynamical system is to stress the ease by which a turbulent evolution may be described by manipulating 3D vortical structures (sometimes referred to as Vortons) along with some additives to bring into effect dissipation and diffusion in the flow.
Chorin proposed a rationale for hairpin, or more generally, loop removal, as physically consistent means of simplifying the representation of turbulent structures without, possibly, altering the essential physics of the energy cascade. In fact, the vortex stretching process is accompanied by folding that brings energy to small dissipative scales. Direct elimination of folded vortices in the form of loops removes primarily local energy that is likely destined for subsequent dissipation at smaller scales. In this way there is justification for believing that the dynamics of the remaining vortices will not be unduly harmed if vortex loops are removed where and when they form.

Vortex stretching process is accompanied by folding that brings energy to small dissipative scales
(Nonlinear Dynamics and Chaos – Steven Strogatz – Cornell University)


Determinism and unpredictability:
Hadamard well-posed problem and Poincaré’s “three-body problem”

Let us now return to our main delight (e.g. folding and stretching). We began “Routes to Chaos: The Non-Linear Dynamical System Approach – Part I” with the quote by the French mathematician Pierre-Simon Laplace:

Give me the velocity and position of every molecule, and I will predict your future.

Following that, in the 19th century the mathematician Hadamard formulated the notion of a well posed problem. This means that a problem is well-posed when the solution of a set of differential equations obeys the following conditions:

  1. Existence: i.e. a solution exists.
  2. Uniqueness: i.e. there is only a single solution.
  3. Stability: i.e. small disturbances in the initial or boundary conditions lead only to small variations of the solution.

The first two condition imply determinism (given initial and boundary conditions, the solution is known). The third condition on the other hand, imposes a harsh restriction. Another way of describing this condition is to imply that a deterministic solution is in actuality only possible when it is not susceptible to small disturbances in the initial and boundary conditions. This is not a mathematical issue but purely a physical one since in real life initial and boundary conditions are only known with finite accuracy.
If the third Hadamard condition is not satisfied (i.e. initial and boundary conditions are not exactly known) it will result of what we perceive as unpredictability, and the problem shall mathematically be considered as ill-posed. Let the CHAOS begin…

These simple definitions by Hadamard were put into test by the iconic French mathematician Henri Poincaré in his famous work Méthodes Nouvelles de la Mécanique Céleste (1892) where he tried to solve the famous “three-body problem“. the two-body problem was solved by Newton of course, where the elliptic Kepler-trajectories as the solution, were considered a completely predictable solution. in other words, Newton proved that the “two-body” problem to be “Hadamard well-posed”.
On the other hand, Poincaré found that the “three-body problem” has no simple solution in terms of a smooth or differentiable function. moreover, he found that the solution had irregular and chaotic characteristics – intrinsically and fundamentally unpredictable, or according to Hadamard: ill-posed. This ended the view of Laplace’s Part I opening quote.

In modern dynamical systems theory words, dynamical systems can be imagined best as systems of coupled differential equations, describing the behavior of so-called system variables time evolution.
Sometimes, as is in a two coupled pendulums, the number of variables (or degrees of freedom) remains small, but in other cases such as in biological, economical and of course fluid dynamics it may certainly be very high. Saying all that, dynamical systems are deterministic by definition, and therefore full solutions time evolution can in principle be computed.
For such systems, consisting of sets of regular differential equations, it was proven that, given an initial condition, a single and unique solution exists. This implies confirmation of Hadamard’s first and second conditions. The third condition, however, is not always satisfied. When it does it can only be proven for a limited number of systems, often only the linear ones.
for many nonlinear dynamical systems it has been found that the solution is extremely sensitive to small variations of the initial conditions (such as forecasting of the weather as we shall shortly exemplify). As such solutions evolve in time it starts to fluctuate, and it is not possible to forecast a priori the amount or orientation of these fluctuations. This is what is called deterministic chaos.
One of the insights as the theory of chaos in mechanics had evolved much in the past decades is that such chaotic behavior is only anticipated for nonlinear dynamical systems.

Even the above findings has only been demonstrated for dynamical systems where the number of degrees of freedom is small. Nonetheless, we expect similar findings for systems that have many degrees of freedom, although this has only been proven in a few cases.
In fluid dynamics we interpret the equations of motion for the flow as a system with many degrees of freedom. Now consider a solution of the Navier–Stokes equations for a given flow problem where realistic initial and boundary conditions are given with finite accuracy. Suppose that all conditions for a well-posed problem are be satisfied, so that the solution for the flow is completely predictable. We define this as laminar flow.
However, the Navier–Stokes equations are nonlinear, hence we have to expect that only under very special circumstances it is possible to comply with the Hadamard’s conditions for a well-posed problem, especially the third one. In all other cases, the equations of motion and initial and boundary conditions for the flow shall not satisfy Hadamard’s third condition and thus shall be ill posed. Such solutions are susceptible to small variations in initial or boundary conditions, and as explained above in this case the solution eventually becomes completely unpredictable. For us fluid dynamicists these are what we call turbulent flows.
Turbulent flows are associated with the concept of deterministic chaos, and as such are affected by small variations in the initial and boundary conditions.


The Lorenz Equations

In “Routes to Chaos: The Non-Linear Dynamical System Approach – Part I” I’ve ended the post with the example of “Landau’s route to chaos”. I’ve also mentioned that although it’s predicted from a non-linear dynamical systems mathematical standpoint it has never been observed experimentally. Instead, it is observed that a flow can undergo only two bifurcations. after the second bifurcation the flow is not periodic anymore, but shows chaotic behavior (characterized by a strange attractor), which we identify with turbulence.
In other words, the flow reaches a turbulent state in only two steps. This transition scenario is known as the “Ruelle–Takens-Newhouse route to chaos”.

It is possible to illustrate a similar and much famous transition scenario using another simple dynamic system: Lorenz equations (Lorenz 1963). The actual background for the problem is the flow due to free convection between two flat parallel plates. The fluid circulation is known as Rayleigh–Bénard convection. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions
The complete derivation of the system of equation may be found the book by H.G. Shuster and W. Just (2005), but favorite written piece is downable here.
the equations after the derivation reads:

Lorenz Dynamical System

The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. σ represents the Prandtl number (the ratio between momentum diffusivity or kinematic viscosity to the thermal diffusivity), b and r functions of the aspect ratio of the convection problem and Rayleigh number, respectively. r is related to the temperature difference between the plates, in other words it is the Rayleigh number, and is the main forcing term in the problem.
Because of the last sentence above we consider the solution as a function of r, and the other parameters are taken as

The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. σ represents the Prandtl number (the ratio between momentum diffusivity or kinematic viscosity to the thermal diffusivity), b and r functions of the aspect ratio of the convection problem and Rayleigh number, respectively. r is related to the temperature difference between the plates, in other words it is the Rayleigh number, and is the main forcing term in the problem.
Because of the last sentence above we consider the solution as a function of r, and the other parameters are taken as σ=10 and b=8/3.

The next (visualization) procedure is one most often used in the description of the evolution in time of solution (for X, Y, and Z here) – a trajectory in phase space.

Phase Space
In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space (for example: a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane). For every possible state of the system or allowed combination of values of the system’s parameters, a point is included in the multidimensional space. The system’s evolving state over time traces a path (a phase space trajectory for the system) through the high-dimensional space. The phase space trajectory represents the set of states compatible with starting from one particular initial condition, located in the full phase space that represents the set of states compatible with starting from any initial condition. 

Not surprisingly the phase space has a similar function as the cobweb graph for the logistic map.

  • For r < 1 all trajectories approach the origin in phase space at X = Y = Z = 0, which is a fixed point. We can interpret this solution as a complete damping of the flow between the two plates ( same as the case where the heat transport occurs solely through molecular conduction).
  • For r= r1 = 1 the first bifurcation occurs as the fixed point at the origin becomes unstable.


The two new fixed points (the cylindrical flow patterns between the two plates called convection-
rolls) are at:

All trajectories for for r>1 reach one of the fixed points above
As we’ve shown in prior (transition process) related posts a good way to show the nature of stability
is to apply linear stability analysis to the Lorenz equations for r>1

Eigenvalues from stability analysis to Lorenz dynamical system of equations


For r=1, the three eigenvalues are real and negative. When r>1.35 two of the eigenvalues become
complex and are each others complex conjugate, but their complex part remains negative. The
result is an oscillating solution yet approaching the fixed points.

Phasse space trajectories for r=18 approach the still stable two alternating fixed points yet with long transient as r gets closer to18

  • The second bifurcation occurs for:

At this point both complex conjugates become positive, meaning the fixed points become unstable.
When r is just below yet almost r2 the trajectories show very long transients and the solution
somewhat starts to orbit the fixed points. After r>r2 no new fixed points emerge but the trajectory
(as these fixed points are no longer approachable as stable) forms the well known “butterfly” surface
around these two unstable fixed points.

Eigenvalues from stability analysis for the different r values

Conclusions

The Lorenz equations represent a very simplified of convection-derived flow, which serves as a model for weather forecasts. It was clearly demonstrated that as much as one tries to be accurate in measuring the initial conditions (a difficulty not to say impossible with nowadays technology in of itself in the wide sense of say weather forecasting) the solution at some point loses its predictive value.
This implies that a minuscule difference in initial conditions, for example the fluttering of a butterfly 🦋 in Brazil, could eventually make a critical difference in the trajectory of a hurricane 🌀 in Beijing.


(I love this phrase although the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat wrong…)

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