Nonlinear Mathematics and General Nonlinear Sciences

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International Journal of Modern Mathematical Sciences, 2016, 14(1): 63-76 International Journal of Modern Mathematical Sciences ISSN:2166-286X Journal homepage:www.ModernScientificPress.com/Journals/ijmms.aspx Florida, USA Article Nonlinear Mathematics and General Nonlinear Sciences

Yi-Fang Chang

Department of Physics, Yunnan University, Kunming, 650091, China (e-mail: [email protected]; [email protected])

Article history: Received 13 July 2015, Received in revised form 27 January 2016, Accepted 30 January 2016, Published 1 February 2016.

Abstract: First, in nonlinear mathematics, we research the nonlinear geometry, analyze and algebra. Next, the general nonlinear methods and various nonlinear physics are discussed. Further, we search the general nonlinear sciences, in instance, nonlinear astronomy, biology, Earth science, and social sciences, etc. In particular, we research chaos and fractal and their applications.

Keywords: nonlinearity, mathematics, geometry, analyze, algebra, method, physics, science, chaos, fractal

Mathematics Subject Classification (2010): 34A34; 47H30; 70K75; 92B05; 91D10

1. Topological Physics

Linear mathematics corresponds to straight line, linear Descartes coordinates, flat smooth space and time, Euclidean geometry and usual Hilbert space, linear algebra (vector, matrix, linear equations, linear space, etc.)，linear analysis (includes calculus, Laplace transform, Fourier transform, linear operator and operational calculus) and so on. Applications of linear mathematics are very wide, and from this obtain special relativity and quantum mechanics, etc. A type of development of linear mathematics is various types of nonlinear mathematics, which has long process. In 1964 Saaty and Bram published book Nonlinear Mathematics [1], which includes 1) linear and nonlinear transformations, 2) nonlinear algebraic and transcendental equations, 3) nonlinear optimization, nonlinear programming and systems of inequalities, 4) nonlinear ordinary

Int. J. Modern Math. Sci. 2016, 14(1): 63-76 64 differential equations, 5) Introduction to automatic control and the Pontryagin principle, 6) linear and nonlinear prediction theory. Not only applications of nonlinear mathematics are more and more wide, and nonlinearity should be a very important way. In 1988 we proposed: Future sciences must be nonlinear world [2]. In this paper, we discuss nonlinear mathematics, its methods and the general nonlinear sciences.

2. Nonlinear Mathematics

Nonlinear mathematics may be mainly three related aspects: A) Nonlinear geometry. Its basis is curve, non-Descartes coordinates. Nonlinear space corresponds to tensor analysis and curved space, non-Euclidean geometry, differential geometry. The simplest curve is the geodesic line. In this case there have both covariant differential and inverse differential. Combining these results, a possible nonlinear differential operator is:  D  (  C  C Ak  C Ak Ai  C Ak Ai A j  ...)dxl ; (1) x l l lk lki lkij Or it corresponds to an increment operator of Taylor series with m variables:  1       (h  k  ... l )n . (2) n1 n! x y t Spinor and twistor, etc., correspond to be extended. The geometrical theory of nonlinear systems is origin of R.W.Brockett, et al. The method of differential geometry is formed from control theory applied widely in nonlinear systems. We think that it is possibly also a developed direction of linear algebra and quantum mechanics. Usually, the geometrical methods between linear system and nonlinear system exist as follows corresponding relations: space of state corresponds to differential manifold; vector corresponds to vector field; subspace corresponds to sub-manifold and distribution; linear transformation corresponds to differential homeomorphism; method of linear algebra corresponds to Lie algebra. A nonlinear system may be discussed by vector field as well as other tools of differential geometry. It is usually defined in differential manifold, but an important fact is applications of various geometrical tools of vector field and Lie algebra, etc., in sub-manifolds. A special case is the affine nonlinear system, for any local coordinates

  x  f (x)   gi (x)ui , y j  h j (x) , (j=1,2,…). (3)

It is linear for control quantity u, and its dynamic states are completely described by vector fields f(x) and gi (x) .

Linear space corresponds to matrix; nonlinear space corresponds to extensive matrix. Linearity corresponds to general vector, and nonlinearity corresponds to the sum of general tensor with n rank

Tij...kl  X i X j ...X k X l . B). Nonlinear analysis. Its basis is non-standard analysis (NSA) and its development. y  f '(x)x x . (4) Differential dy=f’(x)dx is a linear principal part of increment y , and corresponds to tangent line. Usual  is an infinitesimal, which is a hyperreal number. But, 1) Nonlinear part of differential for very big curve cannot be neglected. Its limit is fold line, which is not differentiable. Probably, it is related with fractal. 2) If function is discontinuous, as with interaction, quantum jump, and quantization, it will combine difference. 3) When x  x  x0 is not infinitesimal, it may develop to Taylor series and power series, etc., they are namely nonlinear transformations, whose linear result is y  f '(x)x when those terms of higher degree are neglected. Further, infinite of Cantor’s different levels should have hyperreal numbers of different levels, which include infinite and infinitesimal of different levels. Differential is tangent line, and corresponds to vector. Extensive differential is curve, and corresponds to tensor and nonlinear equation of the geodesic line: d 2 x   uu  0 . (5) ds2  Vector is developed to tensor. It is related with Lie algebra or Graded Lie algebra (GLA) [3,4]. C). Nonlinear algebra. Present it is mainly nonlinear series, Taylor series and nonlinear function, etc. The solution of set of linear equations may be matrix form. Second order equation corresponds to circular conical curves and surface of second order, and their solutions should be the extensive matrix form. Generally, any nonlinear functions and equations, etc., may be expansion in Taylor series, and may obtain limit cycle, etc. In nonlinear theory there are three well-known results: chaos, fractal and soliton. Tools of nonlinear mathematics include the renormalization group theory, the bifurcation theory of nonlinear equation, the catastrophe theory and so on. Various renormalization equations (Callan, Gell-Mann- Low, Callan-Symanzik, Weinberg, ‘tHooft, Lee, Collin-Macfarlane, Georgi-Politzer, Nishijima- Tomozawa and Lukierski equations) should be unified. Moreover, the linear superposition principle must be developed to the nonlinear Bäcklund transformation, etc [5]. We proved that some nonlinear equations have double solutions of soliton and chaos, in which all nonlinear equations with a soliton solution may derive chaos, but only some equations with a chaos solution have a soliton. The conditions of the two solutions are different. When some parameters are certain constants, the soliton is derived; while these parameters vary in a certain

Copyright © 2016 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2016, 14(1): 63-76 66 region, the bifurcation-chaos appears. It connects a chaotic control probably. We discussed some possible meanings on the double solutions in mathematics, physics, particle theory and neurobiology [6, 7].

3. Nonlinear Method and Nonlinear Operator

General nonlinear method may be additive nonlinear term, or nonlinear factor and coefficient [8], or nonlinear condition, etc.

A linear operator corresponds to a certain matrix, and vector transformation x  T x corresponds to matrix T . A power nonlinear operator corresponds to spatial matrix of n-dimension, and tensor with n rank Tik  ijklTjl . Let ijkl  ijkl , it is spatial matrix, and they may be further developed. Usual operator is defined by Fu=v, and linear operator is:

F(1u  2v)  1Fu  2 Fv . (6) It may be extended to nonlinear operator:

 n F(u)  an (u) . (7) n0 Its inverse element is not alone. Only n=1 term and a=F, it becomes linear operator. Nonlinear operator and nonlinear transformation may be developed. Nonlinear theory may be various different forms. First, they are supposed to Taylor series, and Laurent series form for complex function, i.e., nonlinear mathematics of power series. Linear transformation obeys (1) T(  )  T  T ; (2) T(k)  k(T) , i.e., Tk=kT (commutativity). It may combine Taylor series to develop to 1).

2 3 n T(k)  k(T)  a1T   a2T  ... an1T  , (8) here T is nonlinear operator, or

2 3 n T(k)  k(T)  b1T  b2T ... bn1T , (9)

2 3 n or T(k)  k(T)  c1k   c2k  ... cn1k  . (10) They are nonlinear, respectively, for T,  or k. It may introduce a parameter  as power expanded. For instance, this combines perturbation theory of quantum mechanics, and derive:

2 2 H  (H0  H)  E  (E0  E1   E2 ...)  E( 0  1    2 ...). (11) Here both H and  are nonlinear. 2). Assume that a nonlinear operation. First, define nonlinear algebra as Taylor series type:

2 3 n T(k)  k(T)  A1F  A2 F  A3F ... An F . (12)

It is a non-commutative algebra as commutative relation of Taylor type developed to non-commutative operator Tk  kT . Further, n may not be integer, and be fraction, real number, complex number, etc. Even n may be various other forms, as exponential, logarithmic, triangle, hyperbolic types and so on. As an exemplar, nonlinear algebra may include commutative relation Tk  kT =A and anti- commutative relation Tk  kT  A' and unified Graded Lie algebra form, like A'A  2kT . Nonlinearity corresponds possibly to the extension of number field [9]. Arithmetic is linear, but power, extract, exponential, logarithmic, triangle, hyperbolic functions and their inverse functions, etc., are nonlinear operations. In Heaviside’s operational calculus p, the linear differential operator of m order is:

2 m Q(p)  a0  a1 p  a2 p ... am p . (13) For nonlinear operational calculus, n in pn is not integer. When n=1/2, it may be still linear transformation, and be related with calculus of fraction order:

t 2 d t p 1/ 2  2 ；p 1/ 2 f (t)   t  f ( )d ； (14)   dt 0

1 1 d t 1 p1/ 2  ；p1/ 2 f (t)   f ( )d . (15) t  dt 0 t  Both have differential, and have integral. Nonlinear simplified method may be applied to: 1) Partial differential equation becomes ordinary differential equation, such as only for time-evolution. 2) Continuous systems are discrete, and become difference equation. 3) Dimension of model decreases. Moreover, wavelet analysis may be applied to various nonlinear theories. Based on a brief review on developments of number system, we proposed a new developed pattern. The quaternion is extended to a matrix form aI+bC+cB+dA. They form usually a ring. But some fields may be composed of some special 2-rank, even n-rank matrices, for example, three matrices aI+bC, aI+cB, aI+dA and so on. It is a new type of hypercomplex number fields. Moreover, the physical applications and possible meaning of the new number system are researched [9]. We discussed the topological physics and the general topological sciences [10]. Field, ring, group, set and number theory, probability, topology, etc., should be universal for nonlinear and linear theories.

4. Nonlinear Physics

In 1990 Chaohao Gu, et al., edited Nonlinear Physics [11], which includes: 1) Integrable systems: Hamiltonian structure, symmetries, Bäcklund and Darboux transformations; 2) Finite dimensional dynamical systems; 3) Quantum aspects and statistical mechanics; 4) Physical

Copyright © 2016 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2016, 14(1): 63-76 68 phenomena; and other topics, in example, nonlinear evolution equations, dispersive effects, from soliton theory to string theory, the evolution equation: 1 u  u  u 2u  3uu2  u 4u . (16) t xxx xx x 3 x In physics general relativity is nonlinear theory, in which space-time is curved. We proved that gravitational wave must be nonlinear wave [12]. All classical theories have almost nonlinear extensions: mechanics extends to various nonlinear mechanics, elastic mechanics, hydrodynamics, acoustics, and thermodynamics, etc. Usual wave theory extends to nonlinear oscillation and wave. There are nonlinear electrodynamics and nonlinear optics, and nonlinear quantum mechanics [13-20], etc. Even there is Journal of Nonlinear Mathematical Physics. In a word, various theories all may extend and develop to nonlinear theories. Further, we researched general nonlinear quantum theory [5,21], which is related with possible violations of the Pauli exclusion principle [22-25]. For nonlinear quantum theory of Taylor series type, assume that

2 3 n1 qi p j  p j qi  iij (  a1  a2 ... an ) , (17) corresponding operator is:   2  3 ˆ p  i(  a1 2  a2 3  ...). (18) x x x Traditional method is that nonlinear effects as perturbations deal with approximation step by step. Nonlinear thermodynamics develops to nonlinear statistical physics. It and nonlinear interactions possess both relations: one aspect produces stochastic chaos solutions, and another aspect corresponds to nonequilibrium state. They tend to disorder bifurcation and chaos, or produce new order dissipation structure and self-organization. Nonlinear waves have the recurrence. Nonlinear dissipation structures, as concentration, etc., show often periodicity. Both should be consistent.

1 Nonlinear function f (, xi ) cannot determine single valued inverse mapping f (, xi ) and inverse function x  ( f , ) , both are not invertible. It corresponds to the evolutional equation. Nonlinear stochastic Schrödinger equation is also Ginzburg-Landau (G-L) equation with time: q  q q  q3  F . (19) While another G-L equation is: 1 (i  eA) 2  a  b | |2   0, (20) 2m It is similar to nonlinear Klein-Gordon equation with interaction. Geometry, analysis and algebra of nonlinear theory correspond physically to that curved space- time, gravitational field and other interactions cannot be neglected, NSA corresponds to microscopic

Copyright © 2016 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2016, 14(1): 63-76 69 structures, and Taylor series correspond to perturbation theory. Nonlinear theory is related with superstring and loop gravity quantum theory, etc. Therefore, they are unified theory of gravity and quantum mechanics. Twistor is related with nonlinear graviton and monopole. In Yang-Mills theory and other nonlinear theories, instanton is also nonlinear result. We proposed a new type of soliton equation, whose solutions may describe some statistical distributions, for example, Cauchy distribution, normal distribution and student’s t distribution, etc. Further, from an extension of this type of equation we obtained the exponential distribution, and the Fermi-Dirac distribution in quantum statistics. Moreover, by using the method of the soliton-solution, the nonlinear Klein-Gordon equation and nonlinear Dirac equations may derive Bose-Einstein and Fermi-Dirac distributions, respectively, and both distributions may be unified by the nonlinear equation [6]. Based on the universal wave-particle duality, along an opposite direction of the developed quantum mechanics, we used a method where the wave quantities frequency v and wave length  are replaced on various mechanical equations, and derived some new results. It is called the mechanical wave theory. From this we derived new operators which represent more physical quantities, and proposed some nonlinear equations and their solutions, which may be probably applied to quantum theory [26]. Physical nonlinearity is related with interaction, interference each other, nonlinear superposition, etc. We should investigate general nonlinear theory, nonlinear interaction and unification of nonlinear fields.

5. General Nonlinear Sciences

Strogatz discussed systematically that nonlinear dynamics and chaos and their applications in physics, biology, chemistry and engineering [27]. A notable result of nonlinear theory is chaos, which not only is observed in some phenomena of nature, and is applied to explain many problems in natural sciences and social sciences, and may obtain quantitative or semiquantitative predictions. For example, chaos explains three-body problem, turbulence, and the Great Red Spot of Jupiter, etc. Dilapidation in material science is namely a chaos produced by nonlinear interactions, and total material will destroy when dilapidations are enlarged to many places. Nonlinear systems may derive soliton and chaos, and form structures, self-organizations and fractals with self-similarity, and possess limit cycle, periodicity and attractor. Linear systems only produce a relation with direct ratio.

In astronomy and cosmology, general relativity is nonlinear theory. From this Universe, and its change and evolution are very rich and colorful. Based on the basic equations of a rotating disk on the nebula, we applied the qualitative analysis theory of nonlinear equation, and obtained a nonlinear dynamical model of formation of binary stars [28]. Under certain conditions a pair of singular points results in the course of evolution, which corresponds to the binary stars. Under other conditions these equations give a single central point, which corresponds to a single star. This method and model may be extended and developed. Steinitz and Farbiash established the correlation between the spins (rotational velocities) in binaries, and show that the degree of spin correlation is independent of the components separation. Such a result might be related for example to Zhang’s nonlinear model for the formation of binary stars from a nebula [29]. Further, based on the hydrodynamics and hydromagnetics of nebula, from Alfver equation of the cosmical electrodynamics [30] we discussed the formation of binary stars by the qualitative analysis theory [31]. The base of the most exact evolutionary theory of large scale structures is general relativity, we calculated 2+1 dimensional plane equations of gravitational field, and discussed the evolutions of disk nebula by the qualitative analysis theory, in which the binary stars or single star are formed for different conditions. This is the most exact model of formation of binary stars [32]. Moreover, based on the Lorenz model derived from the equations of hydrodynamics of nebula, we discussed the formation of binary stars by the qualitative analysis theory of nonlinear equation. Here the two wings in the Lorenz model form just the binary stars, whose Roche surface is result of evolution under certain condition. The nonlinear interaction plays a crucial role, and is necessary condition of the formation of binary stars and of multiple stars [28,31,32]. Based on the nonlinear physical theory, using the method of the soliton-solution, we derived the fermion probability density equation [33]: d / dE  a(1 2) , (21) which corresponds to the Dirac equation. Then we extend the chaos theory, in which the period bifurcation is equivalent to the particle production. So this extended chaos theory can be used for description of the multiparticle production and the extensive air showers at high energy. Let the parameter takes a suitable value, the quantitative results will be obtained. Many properties of the multiparticle production and of the chaos theory are universal. Based on the nonlinear equations of the density wave theory, the evolutionary direction and the observable conditions on spiral galaxies may be derived by the qualitative analysis theory [8,33]. Based on the geodynamics, earthquakes take place when the momentum-energy excess a faulting threshold due to the movement of the fluid layer under the rock layer and the transport and accumulation of the momentum. Based on the nonlinear equations of fluid mechanics, we discussed the nonlinear dynamics of earthquake, and derived a simplified nonlinear solution of momentum,

Copyright © 2016 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2016, 14(1): 63-76 71 which correspond to the accumulation of the energy. Otherwise, chaos equation is obtained, in which chaos corresponds to earthquake, which shows complexity on seismology, and impossibility of exact prediction of earthquakes. But, combining Carlson-Langer model and Gutenberg-Richter relation, we derived approximately the magnitude-period formula of the earthquake:

(a0 a)(b0M 0 bM ) T 10 T0 . (22) From this some results can be calculated quantitatively [34,35]. For example, we forecasted a series of earthquakes, especially in 2019 in California. Combining the Lorenz model, we discussed the earthquake migration to and fro, and applied the qualitative analysis theory. Moreover, many external causes for earthquake are merely the initial conditions of this nonlinear system. We researched some new ways, and proposed the topology of earthquake [35]. Based on the inseparability and correlativity of the biological systems, we proposed the nonlinear whole biology and four basic hypotheses [36,37]. It may unify reductionism and holism, structuralism and functionalism. Further, the loop quantum theory, which constitutes a very small discontinuous space, as new method is applied to biology. From this we proposed the model of protein folding and lungs. In the model we applied some known results, and obtain four approximate conclusions. Moreover, the medical meaning of the theory is discussed briefly [37]. Based on the neural synergetics and its basic equations [38,39], we derived quantitatively the Lorenz equations and Lorenz mode1 of brain, whose two wings correspond to two hemispheres of brain, and two hemispheres jump about, which describes thinking. It shows that life lies in cooperation in chaos. It is also nonlinear neurobiology. Combining some known theories in neurobiology and the elastic hypothesis of memory, etc., the physical neurobiology is proposed [40]. In biology the quasispecies equation is:

n  xi   x j f jQji x j , (i=0,1,…,n) (23) j0 It is a nonlinear biological equation. The replication equation is:   xi  xi[ f j (x) (x)], (i=0,1,…,n). (24) It and Lotka-Volterra equation are equivalence, such the theoretical ecology and the evolutionary game theory are completely contacted. Different sensation systems are usually independent each other. We open out the potential of blind children, and found through a period training of time, some children by touch or nose or ear can distinguish different colors, even some figures and numbers. Therefore, we proposed a hypothesis: The neural excitable cell is continuously induced and excited, then grow out new synapse and dendrite, so the feeling system, hearing system, smell system, etc., may joint to visual system, and form new neural

Copyright © 2016 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2016, 14(1): 63-76 72 circuit, and achieve finally a transformation among vision and other sensations. Further, we proposed some possible tests, for example, for trained mammal, etc., and researched possible theories. It is know synaptic plasticity, and is also a testable application of the nonlinear whole neurobiology. This may build a bridge between modern science and traditional culture, religion [41]. We discussed fractal, chaos and soliton in nonlinear biology and neurobiology, in which soliton may keep the integrality and veracity of information in neural transfer. The nonlinear mechanism of memory is researched. Based on the extensive quantum theory in which the formulations are the same with the quantum mechanics and only quantum constant h is different, we proposed the extensive quantum biology [42]. We discussed biofield and some nonlinear theories in biology, in which chaos and its application to cancer are researched. Fractal and complex dimension in biology are searched. Nonlinear biothermodynamics and in which possible entropy decrease are investigated. We proposed the matrix representations of hypercycle theory. Its fuzzy element corresponds to that each element is fractal. NeuroQuantology should be nonlinearity and quantization, and may relate to quantum chaos, quantized matrix, etc [43]. We proposed that quantum elements of DNA are A-T and G-C, and researched the extensive quantum theory of DNA, so corresponding quantum theory and its many mathematical methods are applied to DNA and molecular biology. From this we discussed symmetry and supersymmetry of DNA, and the quantum theory and equations of DNA, in particular, the SU(2) gauge theory and some solutions of equation. Further, we proposed the string theory of DNA and general biological string. Some solutions and functions of these theories may describe probably DNA, biological things and their motions [44]. We derived the Lorenz model from the synergetic equations, which may describe the change between two species. By a way of the adiabatic approximation, different models of population dynamics are obtained. Further, we researched various simplified results in the model and their ecological meaning by the qualitative analysis theory of the nonlinear equations, etc. From this it points out two outlets on the protection of rare species. The ecological synergetics promulgates deeply a complex nonlinear relation between competition and cooperation on different species. Then the general nonlinear evolutional equations of ecosystem are searched, and the human crises and our outlets are discussed [45]. In society there are much nonlinear phenomena. We proposed the nonlinear whole sociology and its four basic laws, and the nonlinear theory of economic growth. We discussed generally the four variables and the eight aspects in social physics, and searched social thermodynamics and the five fundamental laws of social complex systems, and researched different relations among social elements, the moderate degree on the entropy production in systems and on the input negative entropy flow for open systems [46]. Based on the synergetics [38,39], we proposed the social synergetics, and the four

Copyright © 2016 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2016, 14(1): 63-76 73 basic theorems, in which theorem of perfect correlation on humanity is researched mathematically [47,48]. From the synergetic equations, we obtained the equations on the rule of law, and proved mathematically that a society of the rule of law cannot lack any aspect for three types of the legislation, the administration and the judicature. Otherwise, we proposed an equation of corruption, and discuss quantitatively some threshold values for a social system into corruption. Further, from synergetics we obtain the Lorenz model, which may be a visualized two-party mechanism as a type of stable structure in democracy [48]. In social sciences the scientific base of democracy is a way of unblocked information and an open system, and is a process from random and chaos, through fluctuation and bifurcation, to order and consistency. It is also a base of the modern society, and can form the self-organized structures [47,48]. Any society needs to develop science and has innovation, it must be learning democracy. It is a process through discussion (chaos) to order. Therefore, Prigogine and Stengers proposed order out of chaos [49]. Briggs and Peat pointed out that chaos not only reflects in science, and incarnates much regions of society and nature, in which wholeness is core of chaos, i.e., chaotic holism [50]. Based on the sameness for men or any elements in the social systems, we searched the social thermodynamics, and possible entropy decrease in social sciences. Using the similar formulas of the preference relation and the utility function, we proposed the confidence relations and the corresponding influence functions that represent various interacting strengths of different families, cliques and systems of organization. This produces a multiply connected topological economics. Further, we discussed the binary periods of the political economy by the complex function and the elliptic functions. Using the nonlinear equations of hydrodynamics we researched the formulations of the binary and multiple centers in various social systems [51]. We proposed the developed directions of society and a unification of simplicity and complexity from a new tree-field representation [52]. The causality is a common basis of various natural sciences, Buddhism and some social sciences. Everyone possesses fate and luck. 2015 is anniversary centenary on general relativity, which shows that matter and movement determine the space-time of everyone. It as a universal physical representation of causality is a great contribution to modern social sciences. Fate is various innate fields and surroundings. Luck is acquired activity and fortune, and they are changeable. Both aspects may be influence each other. We proposed the social extensive general relativity and electrodynamics, etc. The solutions of these theories may describe some moving orbits of life and society [53]. It is ubiquitous that evolutions of nonlinear systems are sensitively determined to initial conditions, which even can affect historical course. “The times produce their heroes, and heroes produce their times.” This exhibits unification between inevitability and chanciness in history. The times are big

Copyright © 2016 by Modern Scientific Press Company, Florida, USA Int. J. Modern Math. Sci. 2016, 14(1): 63-76 74 surroundings and conditions of historical evolution, while chance and hero, etc., are various occasional factors of happened historical events. Another well-known result of nonlinear theory is fractal with self-similarity. It shows in management science, democracy, art and literature, etc. Further, we extended the fractal dimension D into the complex dimension in both aspects of mathematics and physics [54]. The representation of complex dimension may be:

Dz  D  iT . (25) When the complex dimension is combined with relativity, whose dimensions are three real spaces and one imaginary time, it expresses a change of the fractal dimension with time or energy, etc., and exists in the fractal’s description of meteorology, seismology, medicine and the structure of particle, etc [8,52]. We discussed the fractal relativity, which connects with self-similarity of the Universe and an extensive quantum theory [55,56]. Combining the quaternion, etc., we introduced the high dimensional time:

ict ic1t1  jc2t2  kc3t3 . (26) It may derive the arrow of time and irreversibility. Then the fractal dimensional time is obtained, and space and time possess completely symmetry. The higher dimensional, fractal, complex and super- complex space-time theory covering all might be constructed preliminarily [55,56].

6. Conclusion

Nonlinear interactions may derive two results: A) Various structures, as dissipation structures, self-organization, cooperation, soliton, fractal, hypercycle and so on. B) Various changes, as phase transformation, bifurcation, chaos, catastrophe and so on. Both must be combined, world may form various complex nonlinear structures. Here chaos is very important in dynamical systems [57,58]. In a word, all complex systems must be nonlinear. Nonlinearity is the only way of developed sciences [2]. Heisenberg claimed that physical phenomena are all nonlinear. Social phenomena, etc., are even more nonlinear. But, these may often become linear approximation. Nonlinear mathematics and nonlinear physics should develop to general nonlinear sciences, which include nonlinear chemistry, nonlinear astronomy, nonlinear biology, nonlinear Earth science, and nonlinear social sciences. Their special examples are fractal sciences and chaos sciences, etc., as well as various branches of different regions. Nonlinear sciences lay stress on wholeness, and they are often also nonlinear whole sciences, and correspond to system theory and other theories across all domains. Various nonlinear sciences may use for reference each other, and are developed together.

References

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