Applied Chaos: A Detailed Look At Fractal Training!

Learn what the chaos theory is and it can help you but first you must know your body and what it responds to, but by keeping detailed logs and planning specific programs, you can begin to maximize hypertrophy in your training!
In the 1960s, a new math was born. It fostered a paradigm shift that was subtle but far-reaching. The new math, known as chaos theory, began to describe systems and equations that formerly eluded classification. New models appeared that used simple mathematical equations to generate graphics and information that was infinitely complex. Around the time that tie-die shirts peaked in popularity, computerized tie-die printouts of Mandelbrot sets, Cantor dust, and Sierpinski gaskets began appearing in the emerging online community that would later become known simply as the Internet.

Like nutrition and weight training, bits and pieces of chaos theory have been known for thousands of years. In recent decades, this knowledge has coalesced to form a greater understanding of the universe around us. More importantly, however, chaos theory describes the systems of our body more precisely than could be known before. In 1987, for example, Richard J. Cohen and his colleagues at MIT found that systems described in chaos theory were associated with the onset of a heart attack. This led to new breakthroughs in prevention and recovery from cardiac arrest.

Chaos theory has a direct application to every day fitness and training as well. Chaos is not about the chaotic routines performed by fitness buffs ad naseum every time a particular piece of equipment is needed but some dude has decided that fast-paced, jerky reps for 20-minutes at a time is somehow going to improve his lat spread. This flavor of chaos can not only describe some seemingly sporadic results that are obtained through efforts in the gym and attention to nutrition, but can also help design the right approach to obtain hypertrophy-related goals.

The crux of chaos theory is that while systems may be extremely complex, they can be described in more simple ways. Understanding this can not only help set more realistic expectations of how the human organism responds to nutrition and training, but also how to best stimulate the body to produce the changes desired.

The Butterfly Effect

A few centuries ago, for want of a nail, an entire kingdom was lost!

"For want of a nail, the shoe was lost;
For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
For want of a rider, a message was lost;
For want of a message the battle was lost;
For want of a battle, the kingdom was lost!"

And so the popular folklore poem goes. Welcome to the Butterfly Effect. Edward Lorenz coined this phrase in 1961 to describe what was happening with his weather prediction models. He had built a fairly complex model, and was analyzing how parameters changed with time. After a particular run, he decided to re-run the system to find out how it would change after a longer period of time.

Using a printout, he faithfully keyed in the starting parameters. At first, the system began to change in predictable fashion. Then, much to his surprise, the model changed and became abruptly different - in fact, completely incomparable to the original model. Knowing that computers are not ambiguous machines, he began to search for the reason behind this apparent divergence.

What he found was a simple matter of precision. The computer was dealing with parameters accurate to several decimal places, but his printouts only listed the three most significant digits. What was incredible, however, was that such a small difference could make such a tremendous change over time. The Butterfly Effect describes this phenomenon - that weather is too complex to be accurately modeled or predicted, because even the smallest eddy created by the flapping of a butterfly's wings can affect the tropical storm that brews over the ocean a month later.

This concept is very important to understand. The human body is an extremely complex system, and to be able to predict exactly how the equations will proceed with so many variables to input is, in short, chaos. This is presumably the reason why so many people struggle with the concept of what the ideal diet is. One person says that a calorie is a calorie, while the other emphatically denies this, and swears that the type of food is the most important parameter. As will soon be discovered, both are correct.

A calorie is calorie. When one consumes 9 calories from fat or 4 calories from carbohydrate, that precise amount of energy is what enters the body's system - the human set of equations, so to speak. Where the logic breaks down is assuming that if one uses some fancy set of numbers to compute metabolic rate - the amount of calories expended throughout the day - one can apply this to the food one eats and create the right balance to gain fat or lose muscle. In reality, the equation is more complex.

Using protein as the example, when the body ingests protein, many things can happen. Protein is composed of amino acids and these are used for a variety of processes in the human body. Ultimately, bodybuilders strive to shuttle amino acids to their muscle tissue with hopes that protein synthesis - the building of new muscle tissue - will take place. This process requires energy - if the amino acid is the brick, someone must exert the effort to lay the brick, apply mortar, and append it to the existing piece of wall.

Even when the amino acid is ultimately used for energy, things are not so simple - the amino acid has a nitrogen molecule which is not needed for energy, so the body must first strip this molecule from the amino acid to convert it into sugar and thereby utilize it for energy (these processes are described by words like deaminate, gluconeogenesis, and oxidation). This process requires, not surprisingly, more energy.

The influence that food has on metabolism is known as the thermic effect. It turns out that 100 calories of protein require, on average, 30 calories to process, as opposed to around 10 for carbohydrate and 4 or less for fat! The calorie still is the calorie going in, but what changes is the equation of the body... suddenly one is forced to burn more calories, and so the foods one eats changes one's metabolism. If one is trying to be strict about increasing lean mass and losing fat, changes one's demands for food!

The butterfly effect is at work in everyone's bodies, and being aware of this phenomenon can help remove some of the mystery behind calories. One may not be able to predict with precision the exact calories that are going in or being expended, but one can certainly understand the general model or pattern that they conform to. Lorenz is known for more than one butterfly - beyond the Butterfly Effect, he introduced concept of strange attractors, and the beautiful mathematical butterfly known as the Lorenz Attractor.

Strange Attractors

The Lorenz Attractor is a beautiful rendering of a mathematical equation. It is based on a simplified model of convection in the Earth's atmosphere. It is a stunning example of what is known as deterministic chaos. What this means is that, while you cannot predict the exact path that subsequent points will be plotted along, the path is still constrained within a set of parameters. Therefore, you see the butterfly - a distinct shape, finitely bounded but infinitely complex.

Training is no different. While the method of training can certainly determine the stimulus - strength, power, hypertrophy, endurance, etc. - the exact response to that stimulus is complex. While genetics may determine the bounds of our own butterfly, we must take on and master the process necessary to explode through that infinite complexity and transform to those limits - by breaking down perceived limits to shine through to the reality made possible through exertion and determination.

This is important to understand for anyone who has hit a plateau, become frustrated because they failed to drop a certain amount of weight in a given week, or allowed their inability to break a personal record affect their performance. One cannot expect the body to evolve in a completely linear fashion. While there are definite bounds to what can be achieved, these bounds are complex, interrelated components that have sensitive dependence on initial conditions.

In short, the human body is subject to the Butterfly Effect, and as such, is but a strange attractor. One cannot predict the next muscle fiber to grow in size nor the next fat cell to empty its contents, but one can certainly figure out how those changes are bounded and make certain one is plugging away at the right equation.

This is the beauty of the strange attractor, and why it is called such. Gone is the pinpoint accuracy of landing the lunar module on a very specific plot of ground. Instead, parameters are input with the same mind-boggling precision and the result is a strange attractor.

There is a point of height, weight, density, muscle mass, and fat mass that one strives to obtain, but the body may hover tenaciously close, overshoot the mark, and then return to just below it.

Real world examples of this can be seen every day. When the bodybuilder wants to come in lean and muscular at 190 pounds for a show, the goal is typically 200 or 210, followed by a cutting phase. All of these parameters are designed to pummel the body through the "strange attractor" so that it eventually materializes into that wonderful moment on stage, when the athlete "peaks". This is no different than the distance runner doing bracket training (shorter distance, faster speed, and longer distance, slower speed) or the seasonal changes in training focus in the sport of football.

So, the question is, how does one master this strange attractor? Believe it or not, the answer, which many people are familiar with - periodization - has it's own part to play in our application of chaos. A well-conceived periodization program has macrocycles and mesocycles and minicycles - in short, there are many levels of phases at play. This makes the periodization model self-similar. And that lends it to a name not often associated with periodization, but believe it or not, a periodized training model is a fractal.

Fractals & Self-similarity

The Sierpinski triangle is a work of art. The rules are simple. Take a triangle then remove the middle by cutting out an inverted triangle whose vertices are at the midsection of the original triangles sides. Then repeat this process through infinity to each side of the triangle. The shape is interesting to look at, but what is even more fun is to try and determine what its actual surface area is. It turns out it is no longer a two-dimensional shape, but exists between dimensions, or in fractional space.

This is how Benoit Mandelbrot came to coin the term, fractal. One need not be so scrupulous in one's endeavors to apply chaos to training. Begin with the next event. Call the time between now and then a "season," or macrocycle. Train in higher rep ranges some of the time to give joints, tendons, and ligaments a break, engage in some hypertrophy training then train that muscle to be used effectively with strength and power training.

Each of these phases will be called a mesocycle. Within the phases, there may be different types of workouts - the so-called splits - called these minicycles. The actual workout is, well, heck, a workout. Within a workout, various exercises are performed. One might perform several sets of each exercise. For each set, reps are completed.

Individual reps are done to a particular tempo, so they have a specific time under tension. The rep itself is divided into a concentric or explosive phase and an eccentric or negative phase. Some people like to work through what they call "sticking points" so even the phases of the repetition can be further subdivided.

As can be seen by this example, there are many levels or dimensions to training, like it or not. All levels of training are, to some extent, "self-similar". Consider the person who spends a few weeks doing sets with 12 reps, a few weeks doing sets with 10 reps, and a few weeks doing sets with 8 reps. Now consider someone else who spends several weeks using a pyramid that goes from 12 to 10 to 8 repetitions. Here it can be seen how the same concept - in this case, the training continuum (from endurance down to power) can be applied anywhere from the set level to the mesocycle, yet possibly with different results.

Period 3 Implies Chaos

The Yi-Lorke theorem states that if "f" is a continuous function, and "f" has a periodic point order of 3, then "f" has a periodic point of all orders. More simply stated, any equation with a period of 3 eventually leads to chaos.

The same equation that supplies the model for the growth of bacterial cultures may provide a clue to the inner workings of hypertrophy. The "period 3" phenomenon can be illustrated with a simple recursive equation. The equation is recursive because the result of one "iteration" or computation is then fed into yet another. For example, consider the equation:

F(x) = x + 1

If I feed in a value of 3, the function F(3) = 3 + 1 evaluates to F(3) = 4. If I then take the result (4) and feed it back into the equation, I get F(4) = 5. This is recursion, and be expressed like this:

x[n+1] = x[n] + 1

In biology, a similar equation was used to describe the behavior of bacteria. Consider a dish that contains a biological culture. x describes the percentage of maximum population that exists - a 1.0 indicates that no more bacteria could possibly fit in the culture, and 0 means that the bacteria are extinct. The equation used to describe how that population may grow is:

x[n+1] = r ( x[n] ) * (1.0 - x[n])

R is a parameter that is used to manipulate the environment, and can account for the availability of food and the space of the environment. Taking an r value of 1.0 and feeding 0.5 into x, the equation looks like this:

x[0] = 0.5 x[1] = 1.0 ( 0.5 ) * ( 1.0 - 0.5 ) = 0.25 x[2] = 1.0 ( 0.25 ) * ( 1.0 - 0.25 ) = 0.1875

Carrying this forward, the number rapidly converges to a single value, as demonstrated by the following graph:

This indicates that the higher population may use of all of the resources, and therefore the population ultimately declines.

Taking a higher value for r, or 3.0, the following graph is produced:

In this case, the population fluctuates between two values. This is again indicative of natural phenomenon: a population consumes available resources, and declines. The decline results in an overabundance of resources, and the population surges.

Increasing r results in several of these "periodic" results (periodicity meaning the numbers periodically fluctuate). In fact, if the equation is allowed to run for several iterations so that the final, fluctuating values are evident, the equation can be plotted across time. This produces what is known as a "bifurcation diagram" because of the way the population splits over time. From left to right, increasing values of r are used. Then, from bottom to top, the resulting values are plotted. A single dot indicates the population has stabilized at a fixed point, while two or more dots in a particular column demonstrates a fluctuation between various levels. The entire bifurcation diagram looks like this:

The equation seems to be fairly stable to a point, splitting in cycles of 2 and then 3 and so on. However, at a certain point, suddenly there does not seem to be any pattern at all - the results have become chaotic. Using a value of r = 3.599, for example, the following graph is produced:

The Li-Yorke theorem states that any time a continuous, or recursive, function results in a fluctuation between three discrete values, there is an implication that a certain phase of that equation will result in a fluctuation between all values. This is chaos - all orders of magnitude are visited, none are repeated. Yet even with higher values of r, the stripes in the diagram can be seen. When these are magnified, it turns out that the equation has settled back into order and repeated the bifurcation diagram at a lower level - self-similarity once again rears its head.

Periodization & The Training Effect

When hypertrophy is the ultimate goal, the training effect is what helps achieve that goal. Muscle is not built while training. Instead, training is the stimulus. Training applies the blueprint and creates the need for the body to respond. Given sufficient recovery, and an ample supply of nutrients, the training effect describes how the body will respond through a combination of physiological adaptation, such as increasing the cross-sectional diameter of muscle fibers, and neurological adaptation, such as becoming more efficient with recruitment of motor units within the muscle.

While the term chaos implies disorder, chaotic equations are actually very stable and therefore important for biological systems. To understand how chaos and recursion can apply to the body, consider homeostasis, a word that describes the body's attempts to "remain the same". Going back to the recursive equations, using the bifurcation diagram as an example, consider the example where r = 3.0.

The equation ultimately fluctuates between two points that are close to 0.6 and 0.7. In the example below, the value fed into the equation was suddenly switched to 0.2 - well outside of the range. As can be seen, despite this sudden deviation from the sequence, the equation rapidly returns to normal, first overcompensating above 0.7 and then returning to the original sequence:

This behavior is seen in many systems within the human body. When arrhythmia sets in, or irregular heart beat, in most cases the body responds and manages to bring the heart rate back to a normal, steady rhythm. When blood sugar levels are too low, the liver releases stored glycogen to raise these levels. The so-called "sugar crash" refers to how the liver sometimes overcompensates when blood sugar is too high by driving blood sugar levels below normal - the resulting graph is very similar to the one depicted above.

While these systems can be protective of the body, by rapidly returning to the original state, the converse is also true. Chaotic systems, by nature, can enter a state of true chaos, whereby the next state cannot be predicted and may lie anywhere on the spectrum of possibly outcomes.

It is hypothesized that heart attack may be characterized by the body's chaotic system that drives heart rate entered a state of chaos - it is still operating using the same function or equation, but the "r" parameter was driven high by some influence and the result was chaos. It is, in fact, chaos theory that led in part to the popularity of defibrillators to treat heart attack. The application of the electric pulse drives the "system" back into a periodic phase, so the regular heart rate can return.

Since the training effect is the sum of multiple systems at work in the body - specifically the central nervous system (CNS), the cardiovascular system, and multiple other biological systems that are suspected to be chaotic in nature, it stands to reason that an understanding of chaos can help manipulate these systems to maximize the effect. In fact, hypertrophy is directly related to the example above - the body's "system" is influenced by an external parameter (resistance training) and then rapidly tries to return to the normal state.

Instead of labeling 0 "no population" and 1 "total population," consider setting 0 as "catabolic" and 1 as "anabolic". While the body, through what is known as the diurnal cycle, is constantly moving between anabolic and catabolic states, hypertrophy implies that the anabolic phase was more pronounced than the catabolic phase, resulting in net growth. The bodybuilder constantly strives to achieve this "diurnal asymmetry" by influencing the body a number of ways, including but not limited to:

  • Resistance training, to provide stimulus for increased anabolism (ironically, resistance training itself is slightly catabolic, but that is far outweighed by the anabolic response)

  • Nutritive manipulation, by ingesting nutrients that encourage anabolism and suppress catabolism

  • Supplementation of substances that influence the anabolic and catabolic phases

The training effect can also utilize this concept. There is a scientific principle that explains why progressive overload, or increasing the load of a particular exercise (by lifting a heavier weight or slowing the tempo to keep the muscle under tension for a longer period of time) is so effective. It is known as the General Adaptation Syndrome, or the GAS principle. Essentially, organisms, when subjected to stress, undergo three distinct phases.

The alarm phase is the initial response to the stress - for example, the incredible muscle soreness you experience when performing a new exercise. The adaptation phase is the body's attempt to handle the stress. This can be manifested in callouses when the stress is something rubbing against our skin, or in larger muscles when the stress is excessive contraction created by moving a muscle through resistance. If the stress continues, however, the organism eventually responds through exhaustion, fatigue, and possibly death.

The human body is very clever when it comes to training. The adaptation phase involves not only a physiological response - increasing the muscle size - but a neurological response as well. The central nervous system becomes more efficient at coordinating motor units to move the load. If the center of our graph is optimal efficiency, then the initial introduction of a new exercise would represent our deviation. Each subsequent workout would eventually converge to the center of the graph - this is known as "dampening". In order to continue the desired result, muscle growth as opposed to neurological efficiency, we must continue to force the system away from that "comfortable" or "periodic" state. How can this be achieved?

The initial goal might be to induce chaos by training completely random, and trying a new exercise every workout. While this may work, it is far from efficient. It takes time for the body to optimally adapt, so there is no reason why a particular exercise or protocol cannot be used two or three times before moving to a new system. There are many ways that this protocol can be changed. Some common methods used to induce this type of change, and to keep the chaotic system from settling include:

  • Shock training, i.e. German Volume Training
  • Progression of reps - For example, moving from 15 rep sets to 10 rep sets and so on
  • Variation in tempo - Training explosively, then training slowly
  • Manipulation of rest periods

These are just a few examples. These traditional methods, however, may also fall short of the goal to gain optimal mass in a short period of time. Remember, the goal is to minimize the ability of the central nervous system to become more efficient or to "settle" and force the body to respond physiologically through hypertrophy. For this reason, it makes since to approach the complexity of the exercise rather than simply time under tension, recovery, or other aspects.

Consider a plan to increase the size of your chest. How can we build a program that will optimize growth in a short period of time? You might design something like this:

This is just an example. As you can see, the chest is the primary focus throughout the cycle. However, we move from the most simple version, involving the least amount of stability (the Smith machine enforces the plane of movement for the bar) to the most complex exercise involving core stabilization along with unilateral coordination.


Chaos theory is a valid science and is very prevalent in nature. It is believed to be the mechanism behind many biological functions such as homeostasis, where a complex system is able to "settle" despite external influences. By recognizing our diurnal cycles as a chaotic system subject to these influences, we can design an effective routine to constantly "challenge" the system to produce prolonged results over time. This, of course, requires a good knowledge of your own body and what it responds to, but by keeping detailed logs and planning specific programs, you can begin to maximize hypertrophy in your training!

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