Chaos Key

[Ena Marklund]

Andmost importantly: The fancier your title, the more you must avoid causing chaos. If Bob the Intern frantically flip-flops on the plan for his summer project, people will patiently help him towards a good path and perhaps make sure he has less caffeine tomorrow. If Bob the CEO flip-flops on his strategy, people will capsize the ship trying to enact his will. Remove chaos and you and your team will be happier and more successful.

I went to the World Domination Summit in Portland earlier this month with few plans. I had a speech to give, a couple smaller sessions to hold, a bike tour scheduled, a plane ticket and a hotel room. But the large majority of the weekend I left open, with no plans.

The two planned events, even though they happened as planned, were totally unpredictable and uncontrollable. The more we embrace this chaos, the more we embrace the brilliant possibilities that might emerge. The more we try to control our day and actions with plans, the more we limit ourselves.

We try to hold onto the illusion of control, but what if we instead embraced the chaos? What if we left ourselves open to the changing, unfolding moment, and the possibilities we could never have planned for?

When we acknowledge the fluidity of our lives, we learn to use that fluidity to our advantage. We flow. We are open to changing currents. We see things with open eyes, instead of trying to make the world fit to our plans and goals.

The first chapter introduces the basics of one-dimensional iterated maps. Take a function y = ƒ(x). Substitute some number into it. Take the answer and run it through the function again. Keep doing this forever. This is called iteration. The numbers generated exhibit three types of behavior: steady-state, periodic, and chaotic. In the 1970s, a whole new branch of mathematics arose from the simple experiments described in this chapter.

The second chapter extends the idea of an iterated map into two dimensions, three dimensions, and complex numbers. This leads to the creation of mathematical monsters called fractals. A fractal is a geometric pattern exhibiting an infinite level of repeating, self-similar detail that can't be described with classical geometry. They are quite interesting to look at and have captured a lot of attention. This chapter describes the methods for constructing some of them.

The third chapter deals with some of the definitions and applications of the word dimension. A fractal is an object with a fractional dimension. Well, not exactly, but close enough for now. What does this mean? The answer lies in the many definitions of dimension.

The fourth chapter compares linear and non-linear dynamics. The harmonic oscillator is a continuous, first-order, differential equation used to model physical systems. The logistic equation is a discrete, second-order, difference equation used to model animal populations. So similar and yet so alike. The harmonic oscillator is quite well behaved. The paramenters of the system determine what it does. The logistic equation is unruly. It jumps from order to chaos without warning. A parameter that discriminates among these behaviors would enable us to measure chaos.

CHAOS challenge aims the segmentation of abdominal organs (liver, kidneys and spleen) from CT and MRI data. CHAOS was held in The IEEE International Symposium on Biomedical Imaging (ISBI) on April 11, 2019, Venice, ITALY. The results of the challenge were published: -challenge.org/Results_CHAOS/ Also detailed analyses of these results have been published in the challenge article. You may find further information on Publications and Citation page.

If you would like to attend to CHAOS but missed it, do not worry. It is possible to join CHAOS and submit results online. The up-to-date scores are being published on our Results page. You may read the information on this website and join our Google group. !forum/chaos-challenge

Understanding prerequisites of complicated medical procedures plays an important role in the success of the operations. To enrich the level of understanding, physicians use advanced tools such as three-dimensional visualization and printing, which require extraction of the object(s) of interest from DICOM images. Accordingly, the precise segmentation of abdominal organs (i.e. liver, kidney(s) and spleen) has critical importance for several clinical procedures including but not limited to pre-evaluation of liver for living donor-based transplantation surgery or detailed analysis of abdominal organs to determine the vessels arising from and entering them for correct positioning of a graft prior to abdominal aortic surgery. This motivates ongoing research to achieve better segmentation results and overcoming countless challenges originating from both highly flexible anatomical properties of abdomen and limitations of modalities reflected to image characteristics. In this context, the proposed challenge has two separate but related aims:

In addition to exhibiting sensitive dependence, chaotic systemspossess two other properties: they are deterministic and nonlinear(Smith 2007). This entry discusses systems exhibiting these threeproperties and what their philosophical implications might be fortheories and theoretical understanding, confirmation, explanation,realism, determinism, free will and consciousness, and human anddivine action.

A third example discussed by Poincar is of a man walking on astreet on his way to his business. He starts out at a particulartime. Meanwhile unknown to him, there is a tiler working on the roofof a building on the same street. The tiler accidentally drops a tile,killing the business man. Had the business man started out at aslightly earlier or later time, the outcome of his trajectory wouldhave been vastly different!

To begin, chaos is typically understood as a mathematical property ofa dynamical system. A dynamical system is a deterministicmathematical model, where time can be either a continuous or adiscrete variable. Such models may be studied as mathematical objectsor may be used to describe a target system (some kind of physical,biological or economic system, say). I will return to the question ofusing mathematical models to represent actual-world systems throughoutthis article.

A simple example of a dynamical system would be the equationsdescribing the motion of a pendulum. The equations of a dynamicalsystem are often referred to as dynamical or evolution equationsdescribing the change in time of variables taken to adequatelydescribe the target system (e.g., the velocity as a function of timefor a pendulum). A complete specification of the initial state of suchequations is referred to as the initial conditions for the model,while a characterization of the boundaries for the model domain areknown as the boundary conditions. An example of a dynamical systemwith a boundary condition would be the equation modeling the flight ofa rubber ball fired at a wall by a small cannon. The boundarycondition might be that the wall absorbs no kinetic energy (energy ofmotion) so that the ball is reflected off the wall with no loss ofenergy. The initial conditions would be the position and velocity ofthe ball as it left the mouth of the cannon. The dynamical systemwould then describe the flight of the ball to and from the wall.

This definition is both qualitative and restrictive. It isqualitative in that there are no mathematically precise criteria givenfor the unstable and aperiodic nature of the behavior in question,although there are some ways of characterizing these aspects (thenotions of dynamical system and nonlinearity have precise mathematicalmeanings). Of course can one add mathematically precise definitions ofinstability and aperiodicity, but this precision may not actually leadto useful improvements in the definition (see below).

The presence of such a mechanism in the dynamics, Battermanbelieves, is a necessary condition for chaos. As such, this definingcharacteristic could be applied to both mathematical models andactual-world systems, though the identification of such mechanisms intarget systems may be rather tricky.

To construct the Smale horseshoe map (Figure 2), start with the unitsquare (indicated in yellow). First, stretch it in the \(y\)direction by more than a factor of two. Then compress it inthe \(x\) direction by more than a factor of two. Now, fold theresulting rectangle and lay it back onto the square so that theconstruction overlaps and leaves the middle and vertical edges of theinitial unit square uncovered. Repeating these stretching and foldingoperations leads to the Samale attractor.

This definition has at least two virtues. First, it can be proventhat Chaos\(_h\) implies Chaos\(_d\). Second, it yieldsexponential divergence, so we get SD, which is what many people expectfor chaotic systems. However, it has a significant disadvantage inthat it cannot be applied to invertible maps, the kinds of mapscharacteristic of many systems exhibiting Hamiltonian chaos. AHamiltonian system is one where the total kinetic energy pluspotential energy is conserved; in contrast, dissipative systems loseenergy through some dissipative mechanism such as friction orviscosity. Hamiltonian chaos, then, is chaotic behavior in aHamiltonian system.

Of these three features, (c) is often taken to be crucial to definingSDIC and is often suspected as being related to the other two. Thatis to say, exponential growth in the separation of neighboringtrajectories characterized by \(\lambda\) is taken to be a property of aparticular kind of dynamics that can only exist in nonlinear systemsand models.

Though the favored approaches to defining chaos involve globalLyapunov exponents, there are problems with this way of defining SDIC(and, hence, characterizing chaos). First, the definition of globalLyapunov exponents involves the infinite time limit (seethe Appendix), so, strictly speaking, \(\lambda\) only characterizes growth inuncertainties as \(t\) increases without bounds, not for anyfinite \(t\). So the combination \(\exists \lambda\) and\(\exists t\gt 0\) in SD is inconsistent. At best, SD can onlyhold for the large time limit and this implies that chaos as aphenomenon can only arise in this limit, contrary to what we take tobe our best evidence. Furthermore, neither our models nor physicalsystems run for infinite time, but an infinitely long time is requiredto verify the presumed exponential divergence of trajectories issuingfrom infinitesimally close points in state space.

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