Small Wars Journal

Nonlinearity and the Proper Use of Buzzwords

Fri, 08/21/2015 - 1:09pm

Nonlinearity and the Proper Use of Buzzwords

Justin Lynch

War is nonlinear. The implications of this statement should shape our understanding of war’s nature, and the role of its unpredictability in our planning process. Unfortunately, the word nonlinear has lost much of its definition. It is an increasingly popular term used to describe modern warfare, one with a wide variety of uses and common definitions. A Google search of nonlinear war returns articles about Russian strategy in the Ukraine, also often referred to as hybrid warfare. Joint Publication 3-03 labels operations nonlinear when “forces orient on objectives without geographic reference to adjacent forces.” Cognoscenti also use nonlinear as a synonym for asymmetric, irregular, and unconventional. But nonlinear has a specific definition, one whose implications are lost when nonlinear is used as a vague buzzword.

Nonlinearity achieved its buzzword status after New Science theories gained popularity outside of math and science circles. The study of nonlinearity, chaos theory, fractals, and complexity began in earnest in the late nineteenth century. The concepts seized their fifteen minutes of fame in the late 1980s and early 1990s through the publication of several accessible books and articles, particularly the pop-science book Chaos. The ideas influenced several military theorists, resulting in an explosion of articles about the subjects’ effects on military matters, notably influencing John Boyd’s theories. Of the subjects commonly explored in New Science, nonlinearity is the most militarily relevant concept.

What Does Nonlinear Mean?

Linear systems have two characteristics, superposition and homogeneity. Superposition means the output of the system is equal to the sum of its parts when the parts are isolated. Homogeneity means that when the system multiplies, its output multiplies proportionately.  Because of these characteristics, analysts can use data from one point in a linear system to predict the behavior of the system at any other point. Observers can also reduce linear systems to their constituent parts and analyze them without damaging the accuracy of the results.

In nonlinear systems, this is not the case.  Their lack of superposition makes them difficult to analyze using reductive methods. Their lack of homogeneity means that instead of small changes in the input resulting in small changes in the output, as in linear systems, in nonlinear systems small changes in the input may result in large changes to the output, or vice versa. Nonlinear systems are also inherently less predictable than linear systems. Newton was able to solve the linear two body problem relatively easily (once he developed calculus, which was somewhat more challenging). Centuries later, it is still impossible to make accurate predictions about nonlinear three body systems. Due to the properties listed above, linear systems are far easier to understand and predict than nonlinear systems. Unfortunately for those in the business of predicting things, nonlinear systems are far more common.

How is War Nonlinear?

Warfare violates both superposition and homogeneity. Wars’ outcomes are the result of the interaction between two or more parties, each of whose actions are shaped by many interacting internal factors. The powerful influence of the interactions means that we cannot analyze the different parts of a society, military, or conflict in isolation, then understand how the system as a whole functions, violating superposition. The relationships between factors heavily influence the factors’ effects. Doubling the number of infantry available for a particular engagement does not directly equate to doubled combat power. A huge number of other factors like tactical decisions and sustainment also prevent a directly proportionate output, violating homogeneity.

Also true to nonlinear systems, unpredictability plays a large role in warfare. Clausewitz said that “War is the realm of chance. No other human activity gives it greater scope: no other has such incessant and varied dealings with this intruder. Chance makes everything uncertain and interferes with the whole course of events.” Clausewitz also highlighted the impossibility of creating a math-based, useful theory of war, insulting Bulow’s attempts to do so. Even before the popularization of nonlinear mathematics, military theorists could identify warfare’s nonlinearity.

The Creation of Models

Though discussing superposition and homogeneity is exciting, we need to understand how to apply nonlinear principles to operations. To do so, we need to incorporate the characteristics of nonlinear systems into our models. We understand the world through the creation of models that represent a simplified, easier to comprehend version of our environment, which we then use to make predictions. Any physical, computational, or mental simulation of a structure, relationship, or series of events is a model. Wherever we think ‘if I do this, then that will happen,’ we are using a model to understand and make predictions about our environment. Models provide a pre-existing structure or context in which their users can place facts, creating much faster and more efficient comprehension and processing.

It is extremely important to build models using the correct underlying concept. Models that use incorrect concepts will mislead their users about the role of information and causal chains in a system, leading to inaccurate or overambitious predictions. Models that use the correct concepts, can help make some predictions, help link tactical, operational, and strategic objectives, and help us gain an understanding of the role of unpredictability and information in a particular system.

When we create models of nonlinear systems, we need to incorporate unpredictability, non-superposition, and non-homogeneity. As explained above, we can forecast some events, but confident, long-term predictions are unlikely to be accurate. While that may seem like common sense, the military’s culture discourages words like ‘probably’ and ‘should,’ instead rewarding the subordinates that speak confidently about the outcome of inherently unpredictable events. We need to holistically assess systems that do not have superposition. Instead of using one measure of performance or success, we need to use as many relevant factors as possible and appreciate the way they influence each other. Systems that do not display homogeneity will sometimes have disproportionate cause and effect. This makes attempts to use correlation to determine causality dangerous. Instead, we need to use more deliberate intelligence gathering and holistic analysis to develop a specific causal chain. If we incorporate these three concepts into our models, we can more effectively understand real world events.

It is tempting to build linear models for nonlinear real-world system. Linear models are easier to create  and understand, and more readily generate the confident predictions leaders want to hear. Mathematicians and scientists sometimes attempt to resolve nonlinear systems by using linear approximations. Unfortunately, this oversimplifies the system, reducing its accuracy and ability to portray real-world phenomenon. Military planners often use the same reductive method, either wittingly or unwittingly understanding complex problems that defy prediction by creating simplified approximations. Unfortunately, these approximations, whether they are in the guise of the principles of war or concepts like information dominance, also fail to accurately portray real world phenomena.

The Problem with Reductive Models

Even if they use the correct underlying principles, models of nonlinear systems will run into problems. Unfortunately, models are always reductive, diminishing their accuracy. For a model to perfectly represent a real world system, it has to be as large and complex as the modeled system, losing site of the greater efficiency and ease for which models are created. Given that there are no truly closed systems, an absolutely complete model would have to be the size of the universe to be completely accurate, so we’ll have to accept reduction as a necessary evil. Reduction is not a problem for linear systems, which allow for reduction without greatly reducing the accuracy of their models’ predictions.

That same cannot be said for nonlinear systems. Reductive models oversimplify or eliminate interactions between parts, a key element of the output of nonlinear systems. Because of this, models of real world nonlinear systems will always eventually fail to predict their system’s outcome. Even if their initial predictions are accurate, the real system’s results will deviate from those of the model. Unfortunately, the future of most real world systems cannot be reliably predicted from observations of the past and present.


Richard Feynman said that “physicists like to think that all you have to do is say these are the conditions, now what happens next.” We need to avoid the physicists’ tempting mentality and instead appreciate the limitations of information and the importance of holistically understanding our environment. But the challenges inherent in the creation of nonlinear models do not mean that we should somehow attempt to avoid creating models, which would be impossible, or rely on much easier but overly simple linear models. Instead, we should incorporate nonlinear concepts and reject methods that rely on an ability to forecast long-term events or prioritize precise planning over flexible responses. 

Nonlinear is a buzzword with relevance for military operations. But it is often misused, and its importance for the understanding the nature of war is often understated. Accepting that modern, and perhaps all war is nonlinear in nature requires us to place more effort on understanding the many interacting factors that influence our environment, deliberately exploring causality instead of trusting correlation, and to except that we live in an inherently unpredictable world.

About the Author(s)

Justin Lynch is an infantry officer in the U.S. Army with a B.S. in history. He has served in both Afghanistan and Iraq, and is a member of the Military Writers Guild. The views expressed are his own and not those of the U.S. Army, Department of Defense, or the U.S. Government.


Robert C. Jones

Sat, 08/22/2015 - 3:39pm

Any term that implies a commonality of characteristics, but does not group in ways that demand a common solution approach is a dangerous exercise of meaningless intellectualism.

War is non-linear. Noted, and so what?

More important for the US military to learn is that not all political conflict is war at all. Applying war-theory to all such conflicts and then dreaming up new terms to rationalize why that fails to yield desired strategic results is something best left to think tank SMEs and 20-something DASDs

Outlaw 09

Sat, 08/22/2015 - 8:09am

Here is the inherent problem with newly coined words in this case "hybrid warfare" or the Russian version "non linear warfare".

I would argue in the 2014 timeframe up to the Russian active military invasion of eastern Ukraine in August 2014 there was in fact a non linear warfare model underway.

But with each day after August 2014 it has been no longer a non linear war but more to the point--- an actual interstate war with an identified aggressor nation state blatantly attacking another nation state.

Actually if one takes the time to read the definition of invasion and war in Websters--there is an ongoing war in Central Europe and there is nothing non linear about it.

Even the current use of Spetsnaz as a SOF force is not being used in the traditional non linear way but more as a recon and attack leading element more along the concept of Ranger attacks trying to find a break through for follow on armored forces.

So in fact we need to finally shift gears and reassess what we are currently seeing in the daily shellings and ground attacks.

Yes there are still several non linear elements in play ie weaponization of information and cyber--but the actual fighting is now interstate.


Fri, 08/21/2015 - 2:36pm


Good timing on this piece. I'm looking forward to the rebuttals defending the linearity of war.

Additionally, Ochagovyy makes some good points about the "buzzword". Below are a few new and old examples from a western perspective involving "non-linear" and "non-linearity" when it comes to conflict:

1999: Dr. Linda P. Beckerman's piece "The Non-Linear Dynamics of War" outlines in great detail the application of non-linearity, chaos and complexity theory to warfare. I'll re-read that piece also and provide some additional comments if I can find it again...

2015: General Stan McChrystal's book "Team of Teams" discusses nonlinearity in Chapter 3: From Complicated to Complex in the Comets and Cold Fronts section.

"Because of these dense interactions, complex systems exhibit nonlinear change." p 58.

"The same technologies that provided organizations like the Task Force with enhanced transportation, communication and data abilties simultaneously imbue our operating environment with escalating nonlinearity, complexity and unpredictability." p 61.

The Tunisia catalyst for the Arab Spring, pool and chess are examples of nonlinear escalation.

Good work.



Fri, 08/21/2015 - 1:44pm


Appreciate the article. I would agree that warfare has always been characterized by complexity and thus - is nonlinear. It is becoming increasingly common to read of the interrelated, intersubjective, and multifactorial aspects of political, social, and economic features of the conflict environment nested within the more traditionally understood insurgent and terrorist activity of a specific country’s adversaries - thus a nonlinear conflict.

Yet --- Current and future complex environments would benefit from additional theoretic models that, arguably, are oriented on the complex. The positivist worldview, attributed to Auguste Comte from the 18th century, holds that an objective truth can be identified through a rigorous application of scientific methodologies. It has almost been ingrained in analysts to think this way - looking at the relationship of how to test and investigate (think) --that is, testing, in a linear and reductionist manner, variables that would yield knowledge and provide for some explanation of behavior. The much-championed empirical, repeatable methods of analysis, while applicable in areas of study where mechanistic principles and behaviors are present, require new frameworks for complex and dynamic conflict issues. An example of reductionist approaches (positivism) would be when the analyst focuses on governments, political structures, and adversarial groups without regard for the existing social systems within the conflict space. Complexity theory, systems thinking, op design and nonlinear methodologies of analyst are very much needed.

Understanding the complexity (the nonlinear character) of conflict boundaries must include not just physical borders, political groups, and balance-of-power boundaries; it must expand to social systems, belief systems that reach beyond traditional borders, cultural systems, weapons systems, communications systems, governance systems, cyber systems, and an ever-expanding set of systems yet to be imagined (e.g., the emergent field of robotics). (The emphasis here is the term 'systems' which is directly applicable to the term nonlinear.)

To your term 'buzzword' -- I would add that this is not so much a buzzword as it is a term that has been used in warfare doctrine for some time. When a country (or group) uses the term nonlinear as a word that defines how they view the future of war - namely a nonlinear war -- it becomes very important that we understand the use of the term - and use it (meaning that we don't change it to a word like Hybrid war). In the case of the Soviets (Russians) the term nonlinear has been used for several decades. When we staring translating a Russian word into something it is not -- 'ochagovyy' does not translate to the word hybrid or any other buzzword ---- we might be not be well served.