Is Small Actually Big? The Chaos of Technological Change

(A full paper submitted to The Fourth Organization Studies Summer Workshop:

Embracing Complexity; 5-7 June 2008, Pissouri, Cyprus)

by

Shih-Chang Hung1,2, Min-Fen Tu1,2, and Richard Whittington2

1 Institute of Technology Management, National Tsing Hua University 2 Saïd Business School, University of Oxford

Biographic Notes:

Shih-Chang Hung is Distinguished Professor in the Institute of Technology Management, National Tsing Hua University, Taiwan. He received his PhD in strategy from the University of Warwick Business School. He has held visiting positions in the School of Economics and Management, Tsinghua University, Beijing and the Saïd Business School, University of Oxford. His works have appeared in Human Relations, Organization Studies, IEEE Transactions on Engineering Management, R&D Management, Long Range Planning, Management and Organization Research, and various Taiwanese management journals. His current research focuses on institutional entrepreneurship and action-structure linkages. Email: schung@mx.nthu.edu.tw.

Min-Fen Tu is a Ph.D. student in the Institute of Technology Management, National Tsing Hua University. During 2007~2008, she is a visiting student in the Saïd Business School, University of Oxford. Her current research interests focus on technological change and the application of chaos and complexity theories to management studies. Email: d949401@oz.nthu.edu.tw.

Richard Whittington is Professor of Strategic Management and Millman Fellow at New College, University of Oxford. His current interest is in the historical and contemporary evolution of strategy work in Europe and North America. He has authored or co-authored eight books, including Strategy as Practice: Research Directions and Resources (Cambridge University Press, 2007), Exploring Corporate Strategy (8th edn, Pearson, 2008), the Handbook of Strategy and Management (Sage, 2002) and The European Corporation (Oxford University Press, 2000). He has published in journals such as Industrial and Corporate Change, Organization Science, the Strategic Management Journal and Organization Science, as well as Organization Studies. Email: Richard.Whittington@sbs.ox.ac.uk.

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Is Small Actually Big? The Chaos of Technological Change

Abstract:

This article examines the utility and applicability of chaos theory in technological

change. Using data collected from the USPTO, we develop and find a

theoretically-based chaotic model of the change process in panel display technology.

We draw from our findings a set of propositions, which are tested through a

retrospective case study. Based upon this pattern-matching approach, we conclude

that chaos theory can be useful in describing change processes in technology.

Keywords: chaos theory, technology change process, panel display

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I. INTRDOCUTION

The past decade or so has seen the increasing importance of chaos theory in

management studies (Levy, 1994; Thietart & Forgues, 1995; Cheng & Van de Ven,

1996; Eve, Horsfall, & Lee, 1997; Polley, 1997). Chaos theory initially developed

from Lorenz’s work on weather systems, in which he found that even a change as

small as a butterfly wing-beat can lead to large-scale and radical consequences

(Lorenz, 1963). This butterfly effect of sensitivity to initial conditions is non-linear in

that complex patterns of behavior are not proportional to their original causes. A small

cause can have a big impact.

In this paper, we examine the utility and applicability of chaos theory in

accounting for technological change process. To do so, we rely on a mathematical tool

of chaos theory – Local Lyapunov Exponents (LLEs) – to define and measure degrees

of change. The empirical context is panel display technology between 1976~2003.

Using data collected from the United States Patent and Trademark Office (USPTO),

we develop and find a theoretically-based chaotic model of panel technology change

process. We draw from our findings a set of propositions regarding possible change

patterns of the technology. Through a case study, these patterns are matched against

our historical analysis of change forms of panel display technology in reality. Based

upon this “pattern-matching” approach (Yin, 1994: 106-110), we conclude that chaos

theory can be valid in describing technological change processes. Sometimes, small is

actually big, as chaos can amplify small technological changes, causing the instability

necessary to stimulate large-scale and path-breaking changes. First, however, we

begin in the next section with a brief review of chaos theory, focusing particularly on

its relevance to technology change processes.

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II. THEORY

What is Chaos Theory?

Chaos theory was a new science developed in the 1970s that seeks to understand and

explain disorderly conditions existing in the universe (Gleick, 1987). Considered by

some as a subset of complexity science, it has challenged the prevalent view of how

the world works (e.g., a mechanistic view with simple, linear cause-effect

relationships) and discovered new kinds of laws, associated with non-linear and

complex dynamics (Loye & Eisler, 1987). Accounts of the theory vary, but generally

the notion of butterfly effect is seen to define patterns of interaction and behavior

within particular trajectories or systems of change. The butterfly effect signifies

sensitive dependence on initial or starting conditions, emphasizing that even a

microscopic fluctuation can send a complex, chaotic system off in a new direction.

Small changes matter, and the potential for chaos can reside within every dynamic

system of activity – ranging from weather conditions (Lorenz, 1963), population

growth (May, 1974), volatility in financial markets (Hsieh, 1991), diplomatic crisis

(Thietart & Forgues, 1997), to innovation processes (Cheng & Van de Ven, 1996) and

the dynamic evolution of industries (Levy, 1994).

A Chaotic View of Technological Change

In this paper, we apply chaos theory to describe the technological change process.

This application should be workable, because chaos is concerned with how order, or

sensitive dependence on initial conditions, emerged from disorderly conditions and

fluctuation (Hansson, 1991; Levy, 1994; Cheng & Van de Ven, 1996; Mathews, White,

& Long, 1999). This concern broadly echoes the literature on technological change

and management, indicating the significance of path dependence in characterizing

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innovation activity. In general, path dependence depicts technologies as evolving

through relatively long periods of stability and continuity, punctuated by relatively

short bursts of fundamental, radical change (Abernathy & Utterback, 1975; Nelson &

Winter, 1977; Dosi, 1982; Sahal, 1985; Wade, 1996). While sensitivity to initial

conditions is generative of path dependence, the possibility of path-breaking activity

or sudden shifts is likely to occur because small variations can lead to major changes

in a complex, dynamic system.

III. DATA AND METHODS

In this paper, we propose chaos theory has much to inform our thinking about

technological change. In order to test our ideas, we chose to study panel display

technology in 1976~2003, mainly because of its rich history of expansion and

dynamics. Empirically, the study is designed in two stages. In the first stage of

analysis, we employ patent statistics as a proxy (Basberg, 1987; Jaffe & Trajtenberg,

2002) to measure degrees of technological change. Consistent with the literature

(Basberg 1987; Anderson, 2001; Jaffe & Trajtenberg, 2002), we draw on the USPTO

to generate a times series of data, including 336 absolute counts of monthly patent

application from 1976 to 2003 (see Figure 1). This process involved four steps. First,

data are extracted by keywords searching on titles (field code in USPTO as TTL),

abstracts (as ABST), claims (as ACLM) and/or specification (as SPEC). Second, a

monthly interval is used as the appropriate unit of application date, in order to

increase the sample size. Third, longitudinal data is captured between the applicant

dates (as APD) of January 1976 to December 2003. Fourth, only utility-type patents

(as APT) are included. The searching syntax is specified as below.

5

(TTL or ACLM or ABST or SPEC/"Display Panel" or "Flat Panel" or "Video Display" or "Video Monitor" or "Visual Display" or "Visual Monitor") and APD/Month/Day/Year-> Month/Day/Year and APT/1

(Source: the USPTO, updated in Nov 2007; spikes due to end-of-year effects)

Figure 1. Panel display patents, 1976~2003

In the second stage of using case study to test propositions, we draw on two different

classes of data to analyze the real-time change processes of panel display technology.

The first was a broad database of archival materials including industrial journals,

periodicals, magazines, newspapers, company annual reports and scholarly studies of

the technology (e.g. Murtha, Lenway, & Hart, 2001; Kawamoto, 2002; Polgar, 2003;

Castellano, 2004; Mathews, 2005; Weber, 2006). The second type of information

source came from over thirty personal, unstructured interviews, combined with

numerous occasional talks with industrial experts and engineers. Data collection from

all these sources was continued until we were able to establish a concise chronology

showing the sequence and flow of key technological events in panel display over

time.

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IV. MATHEMATICAL ANALYSIS: CHAOTIC FORMS OF CHANGE

To detect chaotic forms of technological change (that is, to decide if a system exhibits

sensitive dependence on initial conditions), we rely on a mathematical tool of chaos

theory – Local Lyapunov Exponents (LLEs) – for analysis. Many studies indicate that

this method provides more than metaphorical or analogical benefits to the

deconstructing of chaotic dynamics and processes in a visual way (Wolff, 1992; Kiel

& Elliot, 1996; Cheng & Van de Ven, 1996).

Local Lyapunov Exponents (LLEs)

Appendix 1 details a step-by-step analysis of LLEs. The LLEs can be used to quantify

the effect of perturbations at different time points on a trajectory, for one-dimensional

dynamical system, especially for small and noisy data sets such as ours (Datta &

Ramaswamy, 2003). The LLEs are based on Lyapunov exponents, which are used to

measure the rate at which evolving motions in the whole phase space of exponential

convergence or divergence along an orbit and to provide a classification of the nature

of the dynamical system (Wolf, Swift, Swinney, & Vastano, 1985). Attaching the

epithet “local” before Lyapunov exponents (Wolff, 1992), LLEs present the local

characteristic and provide an indication of the effect a perturbation would have on a

particular effect over finite times.

Results of LLEs Analysis for Panel Displays

Through analysis of LLEs, we obtain Table 1 using data summarized in Figure 1. This

table contains both positive and negative values.

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Table 1. Results of LLEs Analysis

Xi LLE Xi LLE Xi LLE Aug-77 (-) 0.14** Apr-92 (-) 0.20** Mar-95 (-) 0.18** Dec-80 (+) 0.24** Jul-92 (+) 0.09*** Jul-95 (-) 0.12** Dec-83 (-) 0.13** Sep-92 (+) 0.13** Jan-96 (-) 0.24** May-84 (+) 0.27** Oct-92 (+) 0.11*** Jan-97 (+) 0.12**Aug-84 (+) 0.14** Nov-92 (+) 0.17** Sep-97 (+) 0.21**Mar-86 (-) 0.12** Jan-93 (+) 0.10** Jun-99 (+) 0.18**Aug-86 (-) 0.13** Feb-93 (+) 0.14** Apr-00 (+) 0.29***Mar-88 (-) 0.16** Jul-93 (+) 0.11** May-00 (+) 0.20**Feb-91 (+) 0.15** Aug-93 (+) 0.12***

Note: p<0.05**, p<0.01*** (Please refer to Step 7 in Appendix 1).

Consider a symbolic encoding of the dynamics in negative (-) and positive (+)

signs, with negative representing the stable state and positive the chaotic. Then a

finite-time segment of the trajectory is a negative-positive string of (-)’s and (+)’s. If

the segment includes chaotic bursts, then the signs have some (+)’s, corresponded

with the positive values of the LLE – with all being validated via the least-squares

straight-line fit for the exponential-divergence assumption, given by the LLE

definition (say, P<0.05 in practice). In Figure 2, each of the dots λ(xi) shown stands

for the positive sign.

Figure 2. Positive LLEs for panel displays, 1976~2003

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Overall, we find that these dots – positive LLEs – are visibly clustered into three

groups. Table 2 details some descriptive statistics for each of the clusters, including:

(1) duration of the positive LLE, (2) number of dots, (3) average value of λ (defined

as summation of each positive LLE, divided by number of dots), (4) length of the

positive-LLE duration, (5) height of the difference between maximum and minimum

positive LLE, (6) measure of area by length multiplying height, and (7) density of

cluster (defined as the number of dots divided by the measure of area). Based on

“length of the positive-LLE duration” and “height of the difference between

maximum and minimum positive LLE”, we draw a cross symbol to highlight the

scope of chaotic dynamics.

Table 2. Descriptive statistics for positive LLEs in panel displays, 1976~2003

Cluster 1 Cluster 2 Cluster 3 Duration of the positive LLE Dec. 1980 ~

Aug. 1984 Feb. 1991 ~ Aug. 1993

Jan. 1997 ~ May 2000

Number of dots 3 9 5 Average value of λ 0.22 0.12 0.20 Length of the positive-LLE

duration (unit: month) 45 31 40

Height of the difference between max. and min. positive LLE

0.1 0.08 0.17

Measure of area (length×height) 4.5 2.5 6.8 Density of cluster 0.07 3.6 0.7

As detailed in Table 2, there appeared to be three clusters of chaos which include

the periods of December 1980 ~ August 1984, February 1991 ~ August 1993, and

January 1997 ~ May 2000. Based on the characterization of chaos theory, we arrive at

our first proposition:

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Proposition 1: The periods of 1980~1984, 1991~1993, and 1997~2000 all exhibited initial conditions that cause chaotic dynamics. In other words, some small changes in the three periods can lead to dramatically new paradigms or patterns of technological activity.

With reference to the three clusters of chaos, we observe that the period of

1991~1993 is characterized by an obviously low average value of λ (0.12) when

compared with the periods of 1980~1984 (0.22) and 1997~2000 (0.20). Given the

limited number of our observation, it is reasonable to assume that the values 0.22 and

0.20 should have no statistical difference. In this sense, we suggest our second

proposition as follow:

Proposition 2: Small changes in 1980~1984 and 1997~2000 were more liable to trigger bigger subsequent path-breaking changes than in 1991~1993.

For each of the clusters, we go on to compare the potential of their capacity for

turning small changes into major revolution. Consider a symbolic rectangular space

(the cross symbol), constructed by the length (L) of positive-LLE duration and the

height (H) of minimum-maximum difference range. For the measure of area by length

multiplying height, the area of the first cluster equals 4.5 (L: 45, H: 0.1), the second

2.5 (L: 31, H: 0.08), and the third 6.8 (L: 40, H: 0.17). Within each of the clusters,

there are 3, 9, and 5 dots respectively, indicating that the density, or dispersion, of dots

in the period of 1991~1993 (D=3.6) is obviously more concentrated than in

1980~1984 (D=0.07) and 1997~2000 (D=0.7). The higher the concentration, the

greater the salience of chaotic dynamics and the more likely the conditions of

path-breaking activity occurring. We expect thus:

Proposition 3: The change force of chaos in 1991~1993 was more rigorous and

salient than in 1980~1984 and 1997~2000.

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V. CASE STUDY

To test the three propositions, we develop into a historical review of panel display

technology from 1976~2003. This development yielded a chronology of key events in

technological change, based upon which we identified a set of key technological

forces that characterize the evolution of panel display. These forces are matched

against our propositions, shown in Table 3.

Table 3. Matching between chaotic and case analyses

Time Propositions Case study: Key forces 1976~1979 Non-chaotic

CRT as a standardized design and stable, paradigmatic force.

1980~1984 Chaotic (more liable to trigger

bigger subsequent

path-breaking changes)

1. CRT continuing as a dominant, standardized design.

2. With the creation of a new PC market, some new technologies, notably LCDs, began to rise as a promising new design of panel display.

1985~1990 Non-chaotic On the one hand, CRT continued to dominate the

market. On the other hand, (STN-)LCDs began to govern the portable PCs and other pocket-size screens markets.

1991~1993 Chaotic (more rigorous and

salient)

1. CRTs remained a dominant design, but were severely threatened by TFT-LCDs.

2. High investment in TFT-LCDs by the Japanese.3. The emergence of PDPs in large-size screen

markets.

1994~1996 Non-chaotic The quickly growing TFT-LCDs attracted the entry of the Koreans firms that draw on the Japanese strategy and technology for competition.

1997~2000 Chaotic (more liable to trigger

bigger subsequent

path-breaking changes)

1. The Japanese and Korean firms were severely hit by the 1997 Asian financial crisis.

2. Radical changes in LCD competition, with the dramatic rise of the Taiwanese players.

3. Increased competition between LCDs and PDPs. 4. The creation of some new technologies in

micro-displays.

2000~2003 Non-chaotic Continually moving towards some major breakthroughs in the commercialization process.

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Test of Proposition 1:

In general, we find our case review is supportive of proposition 1. To explain in detail,

a chaotic form of technological change would consider the period of 1976~1979 as

characterized by non-chaotic or path-dependent activity, followed by a chaotic,

path-breaking period of change in 1980~1984. This consideration is realistic, as our

review shows that till 1979 panel display technology was still dominated by CRT

(Wisnieff & Ritsko, 2000), whereas the early 1980s began to see the introduction of

some new technologies – notably STN and a-Si TFT LCDs – to challenge the industry

de facto standard. Despite problems of body thickness, CRT as a dominant design was

in the twentieth century the key to the development of panel display markets,

including television sets, radar displays and oscilloscopes. Since 1980, the rise of PCs

had created a new, huge demand for panel displays which, in turn, promised further

alternative trajectories for new product development. These alternatives include LCDs,

PDPs, and LED; among which LCDs had draw the greatest attention, with core

technologies advancing rapidly from DSM, to TN and a-Si TFT bases in just four to

five years (Kawamoto, 2002).

While chaos theory would see the period of 1985~1990 as path-dependent, our

case study also reveal 1985~1990 as a period of immovability in which competition

was built on the technologies of CRT and (STN-)LCDs; the former dominant in

desktops and TVs, the latter solely used in portable computers. During these years,

competence-enhancing activity in LCDs was common, though their visual quality was

still considerably inferior to that of CRTs.

In addition, the theory would highlight 1991~1993 as a period of chaotic

dynamics. In line with this indication, our historical review shows that in the early

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1990s the LCD technology had reached maturity, leading to the rise of TFT-LCD as a

guidepost for further product and process change. Although not perfect for desktops

and TVs, TFT-LCDs were then commonly expected to replace CRT in the foreseeable

future. This expectation, together with a growing notebook market and the wide

application of PDPs in large-size screen markets, induced large-scale investment,

almost all undertaken in Japan. Yield competition was widespread, and the Japanese

were striving to make the TFT-LCD another hi-tech success. There was a clear sense

of chaos, as a sudden shift from CRTs to LCDs was taking a promising start. In turn, it

is also reasonable to expect the period 1994~1996 as non-chaotic, as the industry was

heavily underlined by the rash entry of Korean firms (Samsung, LG, and Hyundai)

that draw on the technology and strategy of the Japanese for competition (Linden,

Hart, Lenway, & Murtha, 1998).

While chaos theory points out another chaotic change in the period of

1997~2000, our case review also shows that this is a period that radical competition in

process innovation really took off. Beginning in 1997, the Asian economic crisis made

producers in Korea and Japan delay some 13 new LCD fabrication projects

(Arensman, 1998). This in turn created opportunities for the Taiwanese to break into

the flat panel markets. These new Taiwanese firms, together with the Korean giants,

had been aggressively pursuing scale economy and low cost, resulting in years of

price and production competitions. The Japanese firms were forced out of the LCD

industry, which was increasingly shifting towards more investment in scale and

equipment. Competition between LCD and PDP in large-size screens accelerated

(O’Donovan, 2006). As LCD TV technology matured, this period had also seen the

dramatic rise of some new technologies – including OLED, SED, and CNT-FED – in

micro-display markets. There were some fundamental changes in ways of

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competition – hence a sign of chaos, and the period that followed (2000~2003) tended

to extend this newly-structured path to provoke also major breakthroughs in the

commercialization process.

Overall, despite some differences in degrees of matching, proposition 1 is

generally supported by our retrospective and real-time analyses of technological

change.

Tests of Proposition 2 and 3:

Our proposition 2 predicts the degrees of chaos in 1980~1984 and 1997~2000 was

obviously higher than in 1991~1993. In retrospect, while the period of 1980~1984 had

seen – for the first time – the feasible introduction of new panel display technologies

(STN and a-Si TFT LCD), the period of 1997~2000, too, had seen the rise of some

alternative technologies including OLED, SED, and CNT-FED. Both periods were

characterized by a relatively high degree of technology diversity when compared with

the period of 1991~1993, in which technological competition was based mainly on the

process innovation of TFT-LCDs by the Japanese. The higher the degree of

technology variety, the more likely that path-breaking activity will occur subsequently

(Klevorick, Levin, Nelson, & Winter, 1995). Hence it makes sense to expect some

small changes in 1980~1984 and 1997~2000 were more liable to trigger bigger

subsequent path-breaking changes than in 1991~1993. Proposition 2 is thus by and

large supported by our case review.

In comparison with the periods of 1980~1984 and 1997~2000, 1991~1993 may

show a relatively low degree of technology variety, but this in turn implies the

tendency of the Japanese firms in 1991~1993 to concentrate more on technology

process innovation, usually involving large-scale investment. In 1991~1993,

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TFT-LCD was rising as a promising technology standard for panel display, that

shaped new wealth and industry creation. Boulton and Furukawa (1995: 3) write:

If semiconductors are equivalent to Japanese rice, then flat panel displays such as LCDs are a new strain of rice that are indispensable to Japan’s sustained economic growth. LCDs were considered the second semiconductor industry and were expected to sustain Japan’s prominence in high-tech products into the 21st century.

With a series of heavy investment and process improvement, Japanese firms

began to successfully commercialize the LCDs, which together with the rise of PDPs

and FEDs, were likely to create a larger display market than ever (West & Kent, 1998).

The higher the technology consensus is, the more likely that a large-scale and

path-breaking is likely to occur (Utterback, 1994). Thus, our case analysis shows

supports for proposition 3 which predicts that the sensitivity of initial conditions in

1991~1993 was more rigorous and salient than in 1980~1984 and 1997~2000.

VI. IMPLICATIONS

The results of this research suggest that chaos theory can be useful for describing

change process in technology. In proposing a chaotic view of technological change,

our study may have important implications for research on technological change and

chaos theory.

First, a common suggestion in the technology change literature has been that,

change unfurls in a linear manner, and large-scale, or revolutionary, change occurs

rapidly (Dosi, 1982; Foster, 1986; Tushman & Anderson, 1986; Christensen, 1997).

Our study examines the nonlinearity of the change process. Under conditions of

chaotic dynamics, even a small, incremental change can generate disproportionate,

fundamental technological change, leading to the rise of new paradigms or patterns of

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behavior. Thus, while the simple patent statistics seem to show a burst of activity in

1994-96, the LLEs indicated that the really significant burst came earlier, as

corroborated by our case account of the rise of TFT-LCDs.

Second, although ideas of chaos theory are simple, there seem to be no

straightforward techniques for developing a model from a chaotic time series. Using

the LLEs to measure degrees and forms of technological change, our study should

contribute to a chaos theoretic program of research by extending the confines of a

metaphor or analogy to an instrument of analysis in management studies.

Finally, a quasi-natural study of this kind - mixing case study with mathematical

tools - should have implications for organizational research methods that often rely on

either quantitative approaches or qualitative explanations. Combining a mathematical

analysis with a case narrative should also help us to better explain social phenomena

of interest.

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APPENDIX 1: COMPUTATION OF LOCAL LYAPUNOV EXPONENTS

In this appendix, we provide a step-by-step analysis of LLEs from a given time series

as shown in Figure 1.

Step 1: The time series (shown in Figure 1) is labeled x(t0), x(t1), x(t2),… as t0＜t1

＜t2＜… Within this series, the time interval between observations is equal to month,

and then written

tn - t0 = nτ, (equation 1)

where τ is the time interval of month between observations. We treat the whole time

series as a system and transform the sum of each count into 1. Thus, every positive

integral number of patent application counts is changed to a decimal as the truly

analyzed value xi.

Step 2: We regard every point xi (happening at some specific time) as an initial

point - then from each initial point to the after would constitute a sub-sequence or

sub-system. That is, within the whole time series, there exist many sub-systems,

starting with the corresponding initial point. If small changes of the initial point could

cause huge influences in the future, we say, this has the characteristic of sensitivity to

initial conditions; otherwise, no.

Step 3: We select xi as an initial point, followed by its sub-sequence, and locate

the nearest neighbor point xj, i≠j, to the point xi, then the difference between the two

sub-sequences over time becomes

d0 = | xj – xi |

d1 = | xj+1 – xi+1 |

d2 = | xj+2 – xi+2 |

17

………

dn = | xj+n – xi+n |, (equation 2)

There is a restriction that we should not choose j that is too close to i. If the two

values are close in time, we expect their behaviors to remain similar, leading in turn to

an anomalous result. To avoid this problem, we determined that the minimum time

delay between i and j should be at least greater than 24 lags in practice. We also note a

practical limitation on how small d0 can be. For example, if the data is recorded with

only three decimal places, then it is meaningless to expect a difference smaller than

0.001.

Step 4: We move on to use the Lyapunov exponent definition that the sequence of

distances change exponentially after n time iterations. More formally, we assume that

dn = d0eλn (equation 3)

or, after taking equation 4 as the definition of the Lyapunov exponent λ,

λ=0

n

ddln

n1× . (equation 4)

The exponential growth of distance within the bounded system cannot go on forever,

so the number of time steps n to determine of λ should not be too large. In practice, we

limit n to 12 iterations.

Step 5: After substituting the sequence of distances into equation 4, we obtain

λ(xi) (namely LLEs) for the corresponding initial point xi. λ(xi) may have a negative,

positive, or ineffective value. The results of ineffective values are due to step 3 in

finding xj closest to xi. The “closest” principal may possibly permit us to find two

identical values (say, xi equal to xj) and then facilitate d0 into zero. A zero value could

not have been taken in the natural logarithm in equation 4.

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Step 6: We repeat steps 3, 4, and 5 until the data sequence is exhausted.

Step 7: The exponential-divergence assumption in the Lyapunov exponent

definition should be tested and validated. This can be done by plotting the natural

logarithm of the difference dm as a function of the index m. If the divergence is

exponential, the point will fall on, or close to, a straight line, the slope of which is the

LLE. A least-squares straight-line fit (say, P<0.05 in practice) to that data will give a

measure of the goodness of that fit. If the data do not approximate a straight line on a

semi-log plot, then the quoted LLE should be considered meaningless and be

discarded.

That done, we obtain LLEs for a time series (detailed in Table 1). Figure 2

exhibits the positive values of LLEs (shown in dots – 17 points in total, roughly

clustered into 3 groups), referring to the distribution of chaotic dynamics for panel

displays.

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APPENDIX 2: ABBREVIATION

a-Si amorphous Silicon CNT Carbon Nano Tube CRT Cathode Ray Tube DSM Dynamic Scattering Model FED Field Emission Display LCD Liquid Crystal Display LED Light-Emitting Diode OLED Organic Light Emitting Diode Display PDP Plasma Display Panel SED Surface-conduction-electron Emitter Display STN Super Twisted Nematic TFT Thin-Film Transistor

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