Can chaos Theory be used to manage companies given that Chaos, justifiably, has confused managers? For them chaos means disorder. Indeed, managers’ perception is that chaos is a state of turmoil, confusion, and lack of order. Chaos, moreover, is considered as the amorphous lifeless void from which earth emerged at creation (Hesiod, 8^th c. B.C.), depriving God from one of his most important works: The creation of the Universe.

The contemporary opinion is, however, different: chaos has structure, high complexity, and many patterns, like the weather and the waterfalls. It is deterministic, with laws of organizing, it appears random, but it is not. The only weak point is that chaos’s ability to forecast long periods is diminishing with time (Lyapunov exponent). Chaos is expressed by fractal (non-integer) mathematics, which dominates par excellence in Nature [1] and in the structure of the human body. The word fractal comes from the Latin verb “frangere”, meaning “to break”. Fractal means a fraction of a whole. Readers unfamiliar with chaos theory may read the scientific book of a journalist [2].

The Greek mathematician Euclid (3^rd c. B.C.) disorientated his colleagues from fractal geometry, as he dealt exclusively with solid bodies and shapes (e.g. the circle = a harmonic divine shape) expressed by integers. Moreover, all have been brought-up in linear sciences by education. Linearity is based on the principle that a 10% cause creates exactly an equal result. Real life has shown that this is not true. Life is mostly, some argue 95%, nonlinear.

Apart from the puzzle of mathematicians, bankers and investors were, and still are, perhaps in a lesser degree, disappointed by the inaccurate predictions provided by using traditional tools embedded in linear philosophy (i.e. normal distribution) [3].

In addition, four at least severe economic crises (depressions) repeated themselves undisturbed in the last 120 years or so (Figure 1). This confirmed the inability of economists to pre-act to a forthcoming economic disaster, even if they saw it coming, as in 2007 [4].

As shown, the first major economic crisis was the one in 1929 (04/09)-1933-1936, known also as the Great Depression, with the USA stock market crash (29/10/1929; called “black Tuesday”). International trade fell by more than 50%. Unemployment in USA rose to 25% and 33% elsewhere. Heavy industry declined sharply, construction stopped and crop prices fell by 60%, while mining and logging suffered.

In Black Monday (19/10/1987) stock markets round the globe crashed starting in Hong Kong (−45.5%), Europe and USA (−23%) and elsewhere. DJIA fell by 508 points to ~1739 (−23%). Interesting is that the DJIA departed from its mean 22σ! It is recognized as a sudden drop in stock prices, which turned into a rout by a wall of insurance options crashing down on the market.

The dot-com bubble or crash (2001) attributed to speculation (mainly in USA) for a rather long period (1994-2000) using as a platform the internet. NASDAQ after March 2000 crashed, while many companies failed and closed down. Some companies’ stocks fell by 86%.

The Global financial crisis in end-2008 is due to various market and regulatory failures and macro-economic environment of cheap credit with new models of securitization¹ and relaxed international capital flows. It started in USA,

where plethora of loans was given on house mortgages (called sub-prime) on the basis of an income from employment, which suddenly lost due to global recession. The system of securitization helped the crisis to spread to the rest of the world till 2019.

Chaos Theory caused the establishment² of the Santa Fe Institute (SFI) (1984) for the study of Complexity Science. The City bank provided the required financial support (apropos 1987 is known for Black Monday).

The aims of the paper are: 1) to show how to manage a (tramp³ shipping) company using a tool from Chaos Theory. 2) To clarify such terms like mission, vision, goals and objectives. 3) To apply limit cycle [5] and 4) to show how to find-out companies’ order of chaos.

The paper is organized in five parts. After literature review, Part I dealt with methodology and terminology. Part II, dealt with how Chaos has evolved over time (since 1886). Part III, dealt with Linear Management (since 1911). Part IV, dealt with the basic terms in management of mission and vision. Part V, dealt with a case-study from shipping industry. Finally, we conclude.

We believe that the paper has a practical significance because for example promotes the use of computers, which have cancelled hierarchy, as chaos theory does not work with the persons in company’s hierarchy, but with the persons called attractors (those having a solution to any problem). If hierarchy is not useful, then simpler organizations structures can be adopted with a view to decide fast and in good timing. The difficulty is to be able to spot the attractors in office and among crew. Moreover, if real world is nonlinear how a manager can approach business world using linear tools?

Morgan [6] argued that the word “organization” derives from Greek “organon”, meaning instrument; so, the concept of a company is usually loaded with a mechanical significance (p. 343). Morgan coined the word imaginization to break free of any mechanical meaning, and more important to symbolize the close link between images and actions. We organize as we imaginize, he wrote. This is a creative process where the new images and ideas can create new actions (Architecture).

The literature on chaos theory as applied to management of business companies goes back to early 1990s. Stacey (1991), [7] argued that in models we show the importance of chaos theory in the practice of management. Before Stacey only two publications are recorded.

Priesmeyer [5] wrote an excellent book (p. 254) titled: “Organizations and Chaos”. He argued: organizations are nonlinear systems and they should be managed using the tools provided by chaos theory (p. xiii). It is a pity that he turned his attention to other subjects thereafter.

Kellert (1993), [8] defined the theory of chaos as the qualitative study of the unstable non-periodic behavior of the nonlinear determinants of dynamic systems. Levy (1994), [9] wrote that chaos tends to be seen in almost all business systems. He used a nonlinear simulation of the chain of the international supplies, where managers underestimated their cost. Managers had to control this process and to relocate the whole system back to a stable condition.

Phelan (1995), [10] wrote that Chaos Theory provides the ability of a substantial insight in complex systems in the world of businesses. He argued that the increased ups and downs in business world and the global speeding-up of changes observed, introduce chaotic influences into the system.

Battram [11] wrote a book (p. 267) titled “navigating complexity” with sub-title: “the essential guide to complexity theory in Business and management” for Industrial Society (UK). He wrote that (p. vi) unlike traditional science, which studies simple “ideal” phenomena based on “perfect laws”, which can only be applied to a very narrow range of conditions, complexity theory studies the other 95%.

Chaos theory was not known in management of shipping companies till 2002, when I wrote a relevant paper (Goulielmos in 2002)⁴, which attracted the congratulations of my colleagues. In 2015, a further analysis has appeared (Goulielmos in 2015)⁵. Chaos theory then applied in many maritime subjects in 2004 (Goulielmos in 2004)⁶ like Marine Accidents (4 papers between 2002 and 2009 have appeared⁷).

Ramnswamy, K. and Youngdahl, W. [12] argued that 44% of employees state that their leaders do not clearly communicate firm’s purpose and direction.

Goulielmos [13] presented the two faces of management, i.e. linear and nonlinear, and argued that linear management provides just a list of strategies with 22 versions, which do not help much shipping managers.

Goulielmos [14] argued that turbulence is an extreme economic phenomenon caused by forces, which should have a certain degree of intensity/energy, but they exhibit always chaotic behavior.

A Limit Cycle-LC emerges when points in a phase space are attracted by a circle. Moreover, a LC has also attractor (Figure 2).

The circle has a period T (not specified), which is the number of repetitions for a full round of motion. This is the 2nd more complicated attractor after point one attractor. Limit cycles are distinguished by their period; in management we study those with 1 to 8 periods [5]. The 1^st limit cycle is with period⁸ 1; let one variable be sales (the independent) and the other be profits (the dependent). From now on we work with changes in variables which we plot. The plot of the 1^st limit cycle trajectory is a single dot.

If the two variables are not constant in their changes, but one is constant and the other changes, then the limit cycle is in period 2; if plotted will give a straight line either horizontal or vertical. There is a possibility to plot a straight line rising from left down to right up if profits and sales oscillate proportionately. For period 4 and 8 limit cycle we abandon the one-quadrant Cartesian diagrams we mentioned so far and adopt the 4-quadrants one. Figure 3 is an 8-period high order chaos.

A more complex limit cycle is the one belonging to USA’s Company “Thiokol Corporation”—(TKC) (Figure 3) in NYSE.

The new element in the above figure is that we plot in axes changes only in the variables. Also we have to plot the trajectories from each pair of variables to next pair, starting from past to present; for a simpler exposition points are not indicated by their distances from each axis but their common point is connected by

trajectory to the next chronological one. Within each quadrant variables either increase or fall or one increases and the other falls, as shown. The interesting part is to see what quadrants trajectories have visited⁹ over time. The company that its sales and its profits both increase over time are marked in quadrant Q1. This is the first best.

In horizontal axis are changes in sales, and in vertical changes in earnings (per share). This departs from tradition in which values are plotted only in Q1 quadrant. This pattern indicates an 8-period trajectory, or in other words, a chaotic attractor, (called apropos strange), and a high order of chaos. TKC’s trajectories visited: 1^st, 2^nd, 3^rd and 4^th quadrants (Appendix 1).

TKC had a turbulent (external) environment: in 1986, (28/01), it faulted¹⁰ (together with NASA and Marshall SF center), as a contractor, in the explosion of the shuttle orbiter Challenger, with 7 dead. TKC manufactured and assembled the “booster motors”. TKC’s management apparently paid no attention to the changing structural patterns bound TKC. This picture is also applicable to shipping companies after a sock from a marine accident with dead.

TKC had to reduce its order of chaos from an 8-period, where it was, to a 4-period; or in other words, TKC’s trajectories had to visit: 1, 4, 2, 3 quadrants (one option out of 16, or 6% of all combinations), and with this order. This means a rise in both sales and earnings (visit at Q1); a rise in sales, despite a fall in earnings (visit at Q4); a rise in earnings, despite a reduction in sales (visit at Q2), ending at Q3 (i.e. 1423 = period-4; a medium order of chaos).

In general, trajectories which visit frequently quadrants 2¹¹ and 3¹², head at high order chaos (period 8). In a lower order of chaos (period 4) head trajectories, which visit frequently 1¹³ and 4¹⁴. Given that the greater majority (61%) of cases falls in high order chaos, we presume, this led Priesmeyer [5] to state that a high order chaos is common in businesses! Hallelujah!

Chaos theory [17] begun in 1889 emerging from the attempted solution of the problem of the “three bodies”, or in other words, of the stability of Solar system. The French mathematician Poincare A (1854-1912) worked-out a partial solution, when he first glanced at the exceptional complexity involved (=chaos) (known as the “homoclinic tangle”, meaning a tangle having same bend) [14].

People’s attitude, however, was against the above endeavor: people avoided all along complexity, and preferred simple structures, which reduced complex things to simple ones (=reductionism). Beauty surely can be found in simplicity. This does not mean that complex systems do not exist, or are inferior in beauty [1]. Managing firms and industries, as well economies are considered as equivalent to managing complex nonlinear systems (CNS)! The concept of the attractor was not unknown in the past, but it was known as the ellipticorbit (trajectory) of a planet, where disturbances are kept within specific limits. This concept passed as

a metaphor to humans and thus we have “management by attractors” and not by hierarchy!

Chaos is clearly shown in mathematics: e.g. in the simple logistic (difference) equation of the second order: z = λz (1 − z) {1} by giving, in a computer, various values (0 - 4) to the (unique) parameter λ (e.g. λ = 0.8, 1.5, 3 and 3.55) and z = 0.5. This equation neither departs, nor approaches one single value, but most important is that, at some point, becomes chaotic!

On the other hand, Randomness, which is the antipode of Chaos, has no structure. Important is that many physical dynamic systems (PDS) behave chaotically. Thus a whole theory emerged i.e. Complexity Theory to study how stable structures emerge.

The main point of Chaos Theory is that certain physical systems (PS) have the same characteristics of social systems (SS), with similar mathematical behavior, like the complex adaptive systems (CAS). These are subsets of nonlinear dynamic systems¹⁸ (NDS). Important is that these systems have negative or positive feedback; the second term is by now used in nonlinear Management. The characteristics of these systems are: self-similarity¹⁹, complexity, emerging ability and a self-organizing behavior [19]. Worth noting is also that the DNA structure, invented in 1953, fits to the analysis of Von Neumann²⁰ J (1903-1957) of the mathematical properties of the self-replicating systems.

In 1984, Wolfram S (1959-) remarked that the automata have intense similarities with the ND. He classed them in four universal classes: I, II, III and IV (Figure 6). Worth noting is that class IV comes before class III, as argued by Langton C (1949-), meaning that chaos comes after complexity! This is important for management, as companies have a strategy to get out of chaos if they are in and not to get into it if they are not in.

As shown, class I (=stasis) and class II (=order) offer static solutions, after a small number of (life) cycles; Class I ends at one steady and fixed structure, while class II ends at a stable periodic structure. Given that the world of businesses is dynamic, then these classes and especially I, must be considered as transitory. In Class III we encounter chaos, without a visible structure (a metaphor = the dangerous area in surfing is chaotic)!

The game of life is played in Class IV, as well in other systems of emerging order and for innovation. Their companies must try to be and stay. Chaos Theory thus provides managers a business strategy: get one’s company across classes I and II in their early stages and then companies must try to be and stay in class IV. IBM e.g. was in class II in 1980s and Microsoft was also in class II in end-1970s (Figure 7).

As shown, Microsoft, over six years in 1970s, performed a sustained rise in both assets and profits; also, company showed a stability; company’s trajectories progressively advanced into Q1 [5] indicating a balanced course between assets and profits. It is, however, recommended that company pursued the blue trajectory where profits rise faster than assets.

We cannot say that humans discovered management for the first time in 1911. Management is an everyday science practiced all along with mankind’s history, even with no theory, by necessity and common sense. It is the science of action, than inaction. In many occasions, men had to: hire staff, to organize, to direct and to span management (Moses in Bible applied delegation on suggestion from his father in law; Pyramids; Chinese Great Wall; Acropolis (Athens)), by instinct using their brain.

Only Noah, the first global shipbuilder, and ship owner of passengers, and livestock in bulk, commanded by God, was the one to need help for ark’s plans. Robbins and Coulter [20] distinguished management approaches in five, which due to overlapping we reduced them to three: 1) the early: from 3000 B.C. to 1776, including a behavioral one from 1700 to 1950; 2) the classical: from 1911 to 1947, including a quantitative one: from 1940 to 1950 (or 1960?) and contemporary: from 1960 till this day. In this last during, we have to include the Digital Management, which appeared in 1980s and followed the electronics and information technology in mid-1970s. This will occupy us in another paper.

Most probably EL was not pre-aware of the market conditions, at the time when its newly-built ships delivered. These deliveries succeeded so that EL to transport, in 1999, 3.2 m additional metric tons. EL showed a technology syndrome by ordering ships 3 years before their technology became obligatory by law.

This, we believe, led EL to have higher operating costs [24]. This differentiation is considered as providing a competitive advantage, but not for shipping, which is cost-based! Differentiation is the ability to provide unique and superior value to buyers(charterers) in terms of service quality, as well as other special features (= double hull; dedicated ballast water tanks; new-buildings). This policy is inherently more costly.

In shipping, a premium price cannot be asked for a superior quality, as in other industries! This justifies other Greek owners to buy larger than hitherto second-hand ships at a rather lower age and at rock bottom prices. Owners were stuck-in (locked-in) newly-buildings at the time when market was falling, offer their customers services of higher cost and perhaps were preferred by them, but they are paid only what market determined! Market decides who is paid and how much provided a ship is seaworthy, no matter new or old.

The four falls we saw (1989; 1995; 1997; 1998) in EL’s revenue had to be avoided. EL could either lay-up a number of ships (in 1989; 1995; 1996; 1997 and 1998) or resort to time-charters or plan better its new-buildings. This policy could have led EL to an 1-period limit cycle (low order of chaos) vis-à-vis 2-period it was in 1988-1996; while in the last period (1997-2000) company could be in period 4 (=medium order chaos), instead of being in period 8 (=high order chaos).

The cumulative revenue rises of EL between 1988 and 2000 was ~$146 m. If we add the amounts by which revenue fell during certain years, then potential revenue rise could reach $189 m (the forgone rise in revenue = $43 m). We believe that EL’s management did know EL’s limit cycle. EL, in chaos language, had to avoid visits to quadrants 3 and 4 (i.e. falls in revenue, Y)! This recommends as one option a switch to a time-charter policy.

EL had also a “contest syndrome”, i.e. to be better than the other owners though everyone was worse. Business is not a contest, but a profit-making enterprise! EL won the race by getting the gold medal, but it did not break the world record.

Given that three deaths occurred among the original owners, the young generation, which took over, we hope to change EL’s mission from “supplying newly-built ships with latest technology independently from profits”, to supply relatively young ships according to the intensity of demand and the level of freight rate. There is no free meal in businesses of a high order of chaos!

Paper’s deficiency is that Chaos Theory is not well-known or even taught in business schools, with few exceptions. Matters change of course. People also dislike chaos as they consider chaos as a synonym of disorder. All we have been brought-up with the linear paradigm: the straight lines, which we call them curves; the normal distribution, which misleads investors; the variance as risk bell, which is not. A paradigm shift is needed. We never imagine a small effect to have a proportional greater result—life we believe is equally proportional. The % of action = the % of the result. We never imagined that a small crack in the ballast tank of a diesel tanker will result in cutting the ship in two under severe weather conditions outside Italy!

The authors declare no conflicts of interest regarding the publication of this paper.

Goulielmos, A.M. (2019) Management by Chaos Theory with a Case-Study from Shipping Industry. Modern Economy, 10, 2004-2029. https://doi.org/10.4236/me.2019.108127

Chaos period-1: when limit cycle trajectories visit the following quadrants, and in this order: 1111, 2222, 3333 and 4444. Chaos Period-2: when trajectories visit: 1112, 1222,2122, 2332, 3232, 3434, 4344, 1113, 1311, 2211,2333, 3233, 3443, 4411, 1114, 1313, 2212, 2422, 3311, 3444, 4422, 1121, 1331, 2221, 2424, 3313,4111,4424, 1122, 1333, 2223, 2442, 3322, 4114, 4433, 1131, 1411, 2224, 2444, 3323, 4141, 4434, 1133, 1414, 2232, 3111, 3331, 4222, 4441,1141, 1441, 2233, 3113, 3332, 4224, 4242, 1144, 1444, 2242, 3131, 3334, 4242, 4443, 1211, 2111, 2244, 3133, 3343, 4244, 1212, 2112, 2322, 3222, 3344, 4333, 1221, 2121, 2323, 3223, 3433, 4334. Chaos Period-4: when trajectories visit: 1423, 1243, 1342, 1324, 4123, 4321, 4231, 4213, 3241, 3421, 3142, 3124, 2134, 2314, 2413, and 2431. Chaos Period-8: trajectories visit the remaining 155 quadrants sequences (not shown here for their number).

EL is a major Greek tramp shipping company, established by four related sea captains, 53 years ago (1966). It eventually specialized in product tankers. EL applied specialization and economies of scale by going to every increasing individual sizes. Moreover, EL’s company’s policy is that of maximizing ownership, and almost exclusively, trade in spot market.