Nonlinear science as a fluctuating research frontier

Chaos, Solitons and Fractals 41 (2009) 2533–2537

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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos

Nonlinear science as a fluctuating research frontier q Ji-Huan He Modern Textile Institute, Donghua University, Shanghai 200052, China

a r t i c l e

i n f o

Article history: Accepted 18 September 2008

a b s t r a c t Nonlinear science has had quite a triumph in all conceivable applications in science and technology, especially in high energy physics and nanotechnology. COBE, which was awarded the physics Nobel Prize in 2006, might be probably more related to nonlinear science than the Big Bang theory. Five categories of nonlinear subjects in research frontier are pointed out. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction ‘‘The most incomprehensible thing about the world is that it is at all comprehensible” (Albert Einstein), but how do we fully understand incomprehensible things? Nonlinear science provides some clues. The 2007 Impact Factors released by ISI brought the good news that most nonlinear journals are now being cited more than ever, reflecting the increasing awareness, visibility and importance of chaos, soliton, and fractals. A glance at Table 1 of the impact factors is sufficient to convince any objective scientists that the nonlinear science revolution did not wither way, but may have just started with profound applications in all ramifications of science and technology, especially in high energy physics and nanotechnology. 2. Triumph of nonlinear science In recent years we have been witnessing quite a triumph in all conceivable applications in engineering and applied physics. Some of the most fundamental theories can be explained with considerable ease and elegance using nonlinear science [1,2]. For instance COBE [3–5], which was awarded the physics Nobel Prize in 2006, might be probably more related to Nonlinear Sciencethan the Big Bang Theory [6,7], see Fig. 1. Take for example, the absolute zero temperature as derived by Einfinity[8,9]

T 0 ¼ ð4Þð10Þð1=/Þ4  1 K or the mass of an expectation proton is [10–12]

mp ¼ ð20Þð1=/Þ8 Mev: pffiffiffi where / ¼ ð 5  1Þ=2 is the golden mean. Then one notices that nonlinear dynamics is truly the way for standard nonlinear physics [12–19].

q

Part of this work was presented on 2007 International Symposium on Nonlinear Dynamics, Shanghai, China. E-mail address: [email protected]

0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.09.027

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Table 1 2004, 2005, 2006 and 2007 Impact factors of some nonlinear journals Journal

ISSN

Publisher

2004

2005

2006

2007

Int J Nonlin Sci Num Chaos Soliton Fract Chaos Physica D J Nonlinear Sci Phys Lett A Nonlinearity Int J Nonlinear Mech Nonlinear Dynam Int J Bifurcat Chaos Adv Nonlinear Stud J Nonlinear Math Phy

1565-1339 0960-0779 1054-1500 0167-2789 0938-8974 0375-9601 0951-7715 0020-7462 0924-090X 0218-1274 1536-1365 1402-9251

Freund Publishing House Ltd Elsevier Amer. Inst. Physics Elsevier Springer Elsevier IOP Publishing Ltd. Elsevier Springer World Scientific Publ. Advanced Nonlinear Studies, Inc. Atlantis Press

0.483 1.526 1.942 1.666 1.850 1.454 0.962 1.004 0.774 1.019 0.306 0.585

2.345 1.938 1.76 1.863 1.556 1.55 1.008 0.904 0.647 0.845 0.431 0.508

4.386 2.042 1.926 1.674 1.5 1.468 1.288 1.205 0.902 0.866 0.778 0.387

5.099 3.025 2.188 1.735 1.263 1.711 1.339 1.249 1.045 0.91 0.617 0.437

Brightness(10 -4ergs/sec/cm2/steradion/cm-1)

1.2

0

5

10

15

20

The smooth curve is the best fit blockbody spectrum

1.1 1

1.2 1.1 1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 0

5

10

15

20

Frequency(cycles/centimeter) Fig. 1. Blackbody spectrum with the temperature 2.735 ± 0.060 K. Only E-infinity can give a good explanation of the curve [3–5].

Some new ramifications arose recently, also in connection with E-infinity theory in Nanotechnology [20–24], Turbulence [25–27], biology [28–30], textile [31,32], and deterministic quantum mechanics [6]. 3. New research frontier in nonlinear science I introduce here five categories of nonlinear subjects which have been and will continue to be immensely important in science and technology [7]. 3.1. Complexity and nonlinear dynamics First I must mention the work of Mitchell Feigenbaum, who laid the foundations for studying universalities [13]. His fundamental work has proven to be incredibly significant in so many fields ranging from the stock exchange to fluid turbulence and lately, as pointed out by El Naschie in quantum field theory [33]. Very recently El Naschie gave some fundamental and interesting connections between Feigenbaum’s golden mean renormalization group and turbulence on the one side and high energy particle physics on the other [25].

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3.2. Fractals Second we have the work of Benoit Mandelbrot [34] who single handedly revived the theory of transfinite geometry which he called fractals. Without Mandelbrot the work of Cantor, von Koch and Menger would have remained unknown for many decades and would never have found their way to physics and changed our view of fundamental things like space and time as can be seen from the amazing Cantorian spacetime theory proposed by M.S. El Naschie [1,2]. 3.3. Turbulence The third category of nonlinear science is related to the modern work of D. Rouelle and F. Takeus on turbulence. Turbulence is one of the most difficult problems in science and engineering and suffice to know that the great Werner Heisenberg was quite capable of solving quantum mechanics but had great difficulties with fluid turbulence. E-infinity theory now sheds light on this problem [25–27]. 3.4. E-infinity theory The fourth group is the application of the new deterministically chaotic mechanics and its associate fractal geometry to fundamental problems in quantum physics [1,2]. As stated by Gerardus ‘tHooft [35] discrete space time may be the most radical and logical viewpoint of reality. Unfortunately most mainstream physicists are unwilling to adopt the picture that space and time consist of a collection of isolated points, where particles can be only on those points, but not in between. E-Infinity theory extends this notion to a transfinite setting where the collection of points can mimic the continuum. Such a collection of unaccountably infinite set of points is said to possess the cardinality of the continuum and as such it is a compromise between the discrete and the continuum. The main application of E-infinity theory shows miraculous exactness, especially in predicting the theoretically coupling constants and the mass spectrum of the standard model of elementary particles [36–42]. This theory is the only theory which explains why the COBE curve is the way it is[3–5]. These results were published some nine years ago but clearly they did not get the attention which they should have received and my guess is that this is because the papers were published in a nonlinear journal rather than the traditional mainstream journal of physics. In fact this theory makes many fundamental predictions and not only the one we have just mentioned. Furthermore, using this theory you can argue for the possible existence of a fractal quantum Hall effect [43]. The mathematical structure of El Naschie’s theory is based entirely on the infinite dimensional but hierarchal Cantor sets which are the backbone of nonlinear dynamics and the essence of fractals. Using this theory many scientists, for instance Tanaka [44,45] in Japan was able to predict the mass spectrum of the standard model with incredible accuracy in addition to predicting all the fundamental constants of nature as shown in the work of El Naschie and his associates [1,2]. 3.5. Analytical methods Last but not least I should mention some useful tools for nonlinear sciences. Discovery of Neptune and Pluto was made by the perturbation method. It is often useful to solve a nonlinear equation arising in physics, so that its physical understanding can be fully revealed. Among various methods, perturbation method is widely used for this purpose, but the obtained results often deteriorate quickly as the degree of nonlinearity increases. Consequently, if we are really determined to extract meaning from analytic formulations of physical processes, we must resort to amelioration of the classical perturbation methods using modern mathematical tools, such as calculus of variation, homotopy technology, and others. Recently some novel analytical methods have appeared, such as the homotopy perturbation method [46–53], the variational iteration method [54–65], the exp-function method [66–69], to mention a few. A complete review of recent development of analytical method is available in Ref. [70]. 4. Conclusions Scientists and engineers working in various ramifications in science and technology should realize now the usefulness of nonlinear science, especially chaos, solitions and fractals, to characterize complex systems. ‘‘The most incomprehensible thing about the world is that it is at all comprehensible”. Nature’s principle might be deceptively simple, and we might marvel that use of nonlinear science, e.g., E-infinity theory, can extremely simply and remarkably elegantly characterize complex systems [71–75]. References [1] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36. also see Mohamed El Naschie answers a few questions about this month’s emerging research front in the field of physics, Thomason Essential Science Indicators: http://esi-topics.com/erf/2004/october04-MohamedElNaschie.html.

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