New Scenario for Transition to Slow 3-D Turbulence Part I.Slow 1-D Turbulence in Nikolaevskii System

1. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 DOI : 10.14810/ijrap.2015.4102 21 NEW SCENARIO FOR TRANSITION TO SLOW 3-D TURBULENCE PART I.SLOW 1-D TURBULENCE IN NIKOLAEVSKII SYSTEM. J. Foukzon Israel Institute of Technology,Haifa, Israel Abstract: Analyticalnon-perturbative study of thethree-dimensional nonlinear stochastic partialdifferential equation with additive thermal noise, analogous to thatproposed by V.N.Nikolaevskii [1]-[5] to describelongitudinal seismic waves, ispresented. Theequation has a threshold of short-waveinstability and symmetry, providing longwavedynamics.New mechanism of quantum chaos generating in nonlineardynamical systemswith infinite number of degrees of freedom is proposed. The hypothesis is said,that physical turbulence could be identifiedwith quantum chaos of considered type. It is shown that the additive thermal noise destabilizes dramatically the ground state of theNikolaevskii system thus causing it to make a direct transition from a spatially uniform to a turbulent state. Keywords: 3D turbulence, chaos, quantum chaos,additive thermal noise,Nikolaevskii system. 1.Introduction In the present work a non-perturbative analyticalapproach to the studying of problemof quantum chaos in dynamical systems withinfinite number of degrees of freedom isproposed.Statistical descriptions of dynamical chaos and investigations of noise effects on chaoticregimes arestudied.Proposed approach also allows estimate the influence of additive (thermal)fluctuations on the processes of formation ofdeveloped turbulence modes in essentially nonlinearprocesses like electro-convection andother. A principal rolethe influence ofthermalfluctuations on thedynamics of some types of dissipative systems inthe approximate environs of derivation rapid of ashort-wave instability was ascertained. Impotentphysicalresults follows from Theorem 2, is illustrated by example of 3D stochastic model system: , ,

2. + ∆ − 1 + ∆

3. , ,

4. + , ,

5. + , ,

6. + , ,

7. , ,

8. + +,

9. − ,

10. = 0, ∈ ℝ (1.1) , 0,

11. = 0, ,

12. = !#$,%

13. !$!$'!$(!% , ≪ 1,0 +, , = 1,2,3, (1.2)

14. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 22 which was obtained from thenon-stochastic3.Nikolaevskiimodel: !/$,%,0

15. !% + ∆ − 1 + ∆

16. , ,

17. + 1 !/$,%,0

18. !$ + !/$,%,0

19. !$' + !/$,%,0

20. !$( 2 , ,

21. + ,

22. (1.3) which is perturbed by additive “small” white noise ,

23. . And analytical result also illustrated by exampleof1.stochasticmodel system !/4$,%,0

24. !% + ∆ − 1 + ∆

25. , ,

26. + !/4$,%,0

27. !$ , ,

28. + ,

29. − ,

30. = 0, ∈ ℝ (1.4) , 0,

31. = 0, ,

32. = !'#$,%

33. !$!% , ≪ 1,0 , (1.5) which was obtained from thenon-stochastic 1DNikolaevskiismodel: !/$,%,0

34. !% + ∆ − 1 + ∆

35. , ,

36. + !/$,%,0

37. !$ , ,

38. + ,

39. = 0, , 0,

40. = 0, ∈ ℝ, (1.6) , 0,

41. = 0.(1.7) Systematic study of a different type of chaos at onset ‘‘soft-mode turbulence’’based onnumerical integration of the simplest 1DNikolaevskii model (1.7)has been executed by many authors [2]-[7].There is an erroneous belief that such numerical integration gives a powerful analysisismeans of the processes of turbulence conception, based on the classical theory ofchaos of the finite-dimensional classical systems [8]-[11]. Remark1.1.However, as it well known, such approximations correct only in a phase ofturbulence conception, when relatively small number of the degrees of freedom excites. In general case, when a very large number of the degrees of freedom excites, well known phenomena of thenumerically induced chaos, can to spoils in the uncontrollable wayany numerical integration[12]-[15] Remark1.2.Other non trivial problem stays from noise roundoff error in computer computation using floatingpoint arithmetic [16]-[20].In any computer simulation the numerical solution is fraught with truncation by roundoff errors introduced by finite-precision calculation of trajectories of dynamical systems, where roundoff errors or other noise can introduce new behavior and this problem is a very more pronounced in the case of chaotic dynamical systems, because the trajectories of such systems exhibitextensivedependence on initial conditions. As a result, a small random truncation or roundoff error, made computational error at any step of computation will tend to be large magnified by future computationalof the system[17]. Remark1.3.As it well known, if the digitized or rounded quantity is allowed to occupy the nearest of a large number of levels whose smallest separation is 56, then, provided that the original quantity is large compared to 56 and is reasonably well behaved, theeffect of the quantization or rounding may betreated as additive random noise [18].Bennett has shown that such additive noise is nearly white, withmean squared value of 56 /12[19].However the complete uniform white-noise model to be valid in the sense of weak convergence of probabilistic measures as the lattice step tends to zero if the matrices of realization of the system in the state space satisfy certain nonresonance conditions and the finite-dimensional distributions of the input signal are absolutely continuous[19]. The method deprived of these essential lacks in general case has been offered by the author in papers

42. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 23 [23]-[27]. Remark1.4.Thus from consideration above it is clear thatnumerical integration procedure ofthe1D Nikolaevskii model (1.6)-(1.7) executed in papers [2]-[7]in fact dealing withstochastic model (1.4)-(1.5).There is an erroneous the point of view,that a white noise with enough small intensity does not bringanysignificant contributions in turbulent modes, see for example [3]. By this wrong assumptions the results of the numerical integration procedure ofthe1D Nikolaevskii model (1.6)-(1.7) were mistakenly considered and interpretedas a veryexact modeling the slow turbulence within purely non stochastic Nikolaevskii model (1.6)-(1.7).Accordingly wrongconclusionsabout that temperature noisesdoes not influence slowturbulence have been proposed in [3].However in [27] has shown non-perturbativelythat that a white noise with enough small intensity can to bring significant contributions in turbulent modes and even to change this modes dramatically. At the present time it is generally recognized thatturbulence in its developed phase has essentiallysingular spatially-temporal structure. Such asingular conduct is impossible to describeadequately by the means of some model system of equations of a finite dimensionality. In thispoint a classical theory of chaos is able todescribe only small part of turbulencephenomenon in liquid and another analogous sof dynamical systems. The results of non-perturbative modeling ofsuper-chaotic modes, obtained in the present paper allow us to put out a quite probablehypothesis: developed turbulence in the realphysical systems with infinite number of degreesof freedom is a quantum super-chaos, at that the quantitative characteristics of this super-chaos, iscompletely determined by non-perturbativecontribution of additive (thermal) fluctuations inthe corresponding classical system dynamics [18]-[20]. 2.Main Theoretical Results We study the stochastic 7-dimensionaldifferential equation analogous proposed byNikolaevskii [1] to describe longitudinal seismicwaves: !/4$,%,0,8

43. !% + ∆ − 1 + ∆

44. , , , 9

45. + , , , 9

46. ∑ ; !/4$,%,0,8

47. !$ = + + ,

48. − , , 9

49. = 0, (2.1) ∈ ℝ= , , 0, , 9

50. = 0, ,

51. = !?@#$,%

52. !$!$'…!$?!% , 0 +, , = 1, … , 7.(2.2) The main difficulty with the stochasticNikolaevskii equationis that the solutions do not take values in an function space but in generalized functionspace. Thus it is necessary to give meaning to the non-linear terms $B , , , 9

53. , , = 1, … , 7 because the usual product makes no sense for arbitrary distributions. We deal with product of distributions via regularizations, i.e., we approximate the distributions by appropriate way and pass to the limit.In this paper we use the approximation of the distributions by approach of Colombeaugeneralized functions [28]. Notation 2.1.We denote byCℝ= × ℝE

54. the space of the infinitely differentiable functionswith compact supportin ℝ= × ℝEandbyC′ℝ= × ℝE

55. its dual space.Let ℭ = Ω, Σ, µ

56. be a probability space. We denote byGthe space of allfunctionsH: Ω → C′ℝ= × ℝE

57. such that 〈H, L〉is a random variable for allL ∈ Cℝ= × ℝE

58. .Theelements ofGare called random generalized functions. Definition 2.1.[29].We say that a random field Nℜ,

59. | ∈ ℝE, ∈ ℝ=Q isa spatiallydependent semimartingale if for each ∈ ℝ= , Nℜ,

60. | ∈ ℝEQ is asemimartingale in relation to the same filtration Nℱ%| ∈ ℝEQ. If ℜ,

61. is a S∞ -function of and continuous inalmost everywhere,it is called aS∞ -semimartingale.

62. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 24 Definition 2.2.We say that that , , , 9

63. ∈ G is a strong generalized solution(SGS) of the Eq.(2.1)-(2.2) if there exists asequence ofS∞ -semimartingales, , , T, 9

64. , T ∈ 0,1 such that there exists i

65. , , , 9

66. =VWX lim[→6, , , T, 9

67. inC′ℝ= × ℝE

68. almost surely for 9 ∈ Ω, ii

69. $B , , , T, 9

70. =VWX lim[→6$B , , , T, 9

71. , , = 1, … , 7almost surely for 9 ∈ Ω, iii

72. for all L ∈ Cℝ= × ℝE

73. iv

74. 〈%, , , 9

75. , L〉 − 〈∆ − 1 + ∆

76. , , , 9

77. , L〉 − ] ; 2 〈$B , , , 9

78. , L〉 = + − 〈,

79. , L〉 + + ^ _ ℝ? ^ L ,

80. ∞ 6 _`%,

81. = 0, ∈ ℝEalmost surely for 9 ∈ Ω, and where `%,

82. = !?#$,%

83. !$!$'…!$? , v

84. , , , 9

85. = 0almost surely for 9 ∈ Ω. However in this paper we use the solutionsofstochastic Nikolaevskii equation only in the sense of Colombeaugeneralized functions [30]. Remark2.1.Note that from Definition 2.2it is clear that any strong generalized solution, , , 9

86. of the Eq.(2.1)-(2.2) one can to recognized as Colombeaugeneralized function such that , , , 9

87. =VWX a, , , T, 9

88. b [ #

89. By formula #

90. one can todefine appropriate generalized solutionof the Eq.(2.1)-(2.2) even if a strong generalized solutionof the Eq.(2.1)-(2.2) does not exist. Definition 2.3.Assumethata strong generalized solution of the Eq.(2.1)-(2.2) does not exist.We shall say that: (I)Colombeaugeneralized stochastic process a, , , T, 9

91. b [ is a weak generalized solution (WGS) of the Eq.(2.1)-(2.2) orColombeausolutionof the Eq.(2.1)-(2.2) iffor all L ∈ Cℝ= × ℝE

92. andfor allT ∈ 0,1 i

93. 〈, , , T, 9

94. , %L〉 − 〈∆ − 1 + ∆

95. , , , T, 9

96. , L〉 − ] ; 2 〈$B , , , T, 9

97. , L〉 = + + + 〈,

98. , L〉 + ^ _ ℝ? ^ L ,

99. ∞ 6 _`%,

100. = 0, ∈ ℝEalmost surely for 9 ∈ Ω, ii

101. , , , T, 9

102. = 0almost surely for 9 ∈ Ω.

103. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 25 (II)Colombeaugeneralized stochastic process a, , , T, 9

104. b [ is aColombeau-Ito’ssolutionof the Eq.(2.1)-(2.2) if for all L ∈ Cℝ=

105. and for allT ∈ 0,1 i

106. 〈%, , , T, 9

107. , L〉 + 〈∆ − 1 + ∆

108. , , , T, 9

109. , L〉 ] ; 2 〈$B , , , T, 9

110. , L〉 = + − −〈,

111. , L〉 − ^ L

112. ℝ? ,

113. _ = 0, ∈ ℝEalmost surely for 9 ∈ Ω, ii

114. , , , T, 9

115. = 0almost surely for 9 ∈ Ω. Notation 2.2.[30]. Thealgebra of moderate element we denote byℰeℝ=.The Colombeau algebra of the Colombeau generalized functionwe denote by fℝ=

116. . Notation 2.3.[30].We shall use the following designations. If g ∈ fℝ=

117. it representatives will be denoted byhi, their values on L = jL[

118. k,T ∈ 0,1will be denoted by hiL

119. and it pointvalues at ∈ ℝ= will be denoted hiL,

120. . Definition 2.4.[30].Let l6 = l6ℝ=

121. be the set of all L ∈ .ℝ=

122. such that^ L

123. _ = 1. Let ℭ = Ω, Σ, µ

124. be a probability space.Colombeau random generalized function this is a map g: Ω → fℝ=

125. such that there is representing functionhi: l6 × ℝ= × Ω with the properties: (i) for fixedL ∈ l6ℝ=

126. the function , 9

127. → hiL, , 9

128. is a jointly measurable on ℝ= × Ω; (ii) almost surely in 9 ∈ Ω,the function L → hiL, . , 9

129. belongs to ℰeℝ= and is a representative of g; Notation 2.3.[30]. TheColombeaualgebra ofColombeau random generalized functionis denoted byfΩℝ=

130. . Definition 2.5.Let ℭ = Ω, Σ, µ

131. be a probability space. Classically, a generalizedstochastic process on ℝ= is a weakly measurable mapn: Ω → .′ℝ=

132. denoted by n ∈ .Ω ′ ℝ=

133. . If L ∈ l6ℝ=

134. , then (iii) n9

135. ∗ L

136. = 〈n9

137. , L−.

138. 〉 is a measurable with respect to9 ∈ Ω and (iv) smooth with respect to ∈ ℝ= and hence jointly measurable. (v) Also jn9

139. ∗ L

140. k ∈ ℰeℝ=. (vi) Therefore hpL, , 9

141. = n9

142. ∗ L

143. qualifies as an representing function for an element offΩℝ=

144. . (vii) In this way we have an imbedding C′ℝ=

145. → fΩℝ=

146. . Definition 2.6.Denote by qH

147. = qℝ=E

148. ↾ H the space of rapidly decreasing smooth functions on H = ℝ= × 0, ∞

149. . Letℭ = Ω, Σ, µ

150. with (i) Ω = q′H

151. , ii

152. Σ- the Borels-algebra generated by the weak topology. Therefore there is unique probability measure t on Ω, Σ

153. such that u _t9

154. exp y〈9, L〉 = exp z− 1 2 ‖φ‖|'}

155. ~ for all L ∈ qH

156. . White noise9

157. with the support in H is the generalized process 9

158. : Ω → C′ℝ=E

159. such that: (i) L

160. = 〈9

161. , L〉 = 〈9, L ↾ H〉 (ii) L

162. = 0, (iii) L

163. = ‖φ‖|'}

164. .Viewed as a Colombeau random generalized function, it has a representative (denoting on variables in ℝ=E by ,

165. ): h€L, , , 9

166. = 〈9, L−, −

167. ↾ H〉, which vanishes if is less than minus the diameter of the support ofL.Therefore is a zero on ℝ= × −∞, 0

168. in fΩℝ=E

169. . Note that its variance is the Colombeau constant:h€ L, , , 9

170. = ^ _ ^ |L − , − ‚

171. | _‚ ∞ 6 ℝ? .

172. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 26 Definition 2.7.Smoothedwith respect to ℝ= white noise j[,

173. k[ the representativeh€L, , , 9

174. withL ∈ l6ℝ=

175. × C′ℝ=

176. ,such thatL = jL[

177. k,T ∈ 0,1 andL[ = Tƒ= L a $ [ b

178. . Theorem 2.1.[25]. (Strong Large Deviation Principle fo SPDE)(I) Letj[, , , , 9

179. k[ ,T ∈ 0,1be solution of theColombeau-Ito’sSPDE [26]: !j/„$,%,0,,8

180. k „ !% + ∆ − 1 + ∆

181. j[, , , , 9

182. k[ + aa…[j[, , , , 9

183. kb [ b ∑ ; †…[ a !/„$,%,0,,8

184. !$ b‡ [ = ; + j[,

185. k[ − j[, , 9

186. k[ = 0, (2.3) ∈ ℝ= , [, , , , 9

187. ≡ 0, + 0, , = 1, … , 7.(2.4) Here: (1)j…[Š

188. k [ ∈ ‹ℝ

189. , ‹ℝ

190. the Colombeau algebra of Colombeau generalized functions and …6Š

191. = Š. (2) j[,

192. k[ ∈ fℝ=E

193. . (3)[,

194. is a smoothedwith respect to ℝ= white noise. (II)Leta[,ŒŒ, , , , 9

195. b [ ,T ∈ 0,1 be solution of the Colombeau-Ito’s SDE[26]: a/„,Œ$Œ,%,0,,8

196. b „ % + ∆Œ − 1 + ∆Œ

197. a[,ŒŒ, , , , 9

198. b [ + zz…[ a[,ŒŒ, , , , 9

199. b~ [ ~ ∑ ; †…[ a /„,Œ@,j$Œ@Ž,,%,0,,8kƒ/„,Œj$Œ,,%,0,,8k b‡ [ = ; + j[Œ,

200. k[ − j[Œ, , 9

201. k[ = 0, (2.5) Œ ∈ ‘ ⊂ ℎ” ∙ ℤ= , Œ = —, … , —=

202. ∈ ℤ= , |Œ| = ∑ —+ = ; , ŒE,; = j˜ , … , ˜ + ℎ”, … , ˜? k, −™ ≤ —; ≤ ™,(2.6) [,ŒŒ, 0, , , 9

203. ≡ 0, Œ ∈ ‘.(2.7) Here Eq.(2.5)-Eq.(2.7) is obtained from Eq.(2.3)-Eq.(2.4) byspatialdiscretizationon finite lattice‘ ℎ” → 0if™ → ∞ and ∆Œ-is a latticed Laplacian[31]-[33]. (III)Assume that Colombeau-Ito’s SDE (2.5)-(2.7)is a strongly dissipative.[26]. (IV) Letℜ, , , ›

204. be the solutions of thelinear PDE: !ℜ$,%,0,œ

205. !% + ∆ − 1 + ∆

206. ℜ, , , ›

207. + › ∑ ; !ℜ$,%,0,œ

208. !$ = ; − ,

209. = 0, › ∈ ℝ,(2.8) ℜ, 0, , ›

210. = 0. (2.9) Then liminf[→6|[, , , , 9

211. − ›| ≤ ℜ, , , ›

212. . (2.10)

213. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 27 Proof.The proof based on Strong large deviations principles(SLDP-Theorem) for Colombeau-Ito’ssolution of the Colombeau-Ito’s SDE, see [26],theorem 6.BySLDP-Theorem one obtain directlythe differential master equation (see [26],Eq.(90)) for Colombeau-Ito’s SDE(2.5)-(2.7): _ ag[,ŒŒ, ,

214. b [ _ + ∆Œ − 1 + ∆Œ

215. ag[,ŒŒ, ,

216. b [ + +›Œ ∑ ; †…[ a i„,Œ@,j$Œ@Ž,,%,0kƒi„,Œj$Œ,,%,0kE b‡ [ = ; + j[Œ,

217. k[ + ŸT

218. = 0,(2.11) g[,ŒŒ, 0,

219. = −›Œ. (2.12) We set now ›Œ ≡ › ∈ ℝ. Then from Eq.(2.13)-Eq.(2.14) we obtain _ ag[,ŒŒ, , , ›

220. b [ _ + ∆Œ − 1 + ∆Œ

221. ag[,ŒŒ, , , ›

222. b [ + +› ∑ ; †…[ a i„,Œ@,j$Œ@Ž,,%,0,œkƒi„,Œj$Œ,,%,0,œk b‡ [ = ; + j[Œ,

223. k[ + T

224. = 0,(2.13) g[,ŒŒ, 0, , ›

225. = −›. (2.14) From Eq.(2.5)-Eq.(2.7) and Eq.(2.13)-Eq.(2.14) by SLDP-Theorem (see see[26], inequality(89)) we obtain the inequality liminf[→6 1¡[,ŒŒ, , , , 9

226. − ›¡ 2 ≤ g[,ŒŒ, , , ›

227. , T ∈ 0,1.(2.15) Let us consider now the identity |[, , , , 9

228. − ›| = ¡¢[, , , , 9

229. − [,ŒŒ, , , , 9

230. £ + ¢[,ŒŒ, , , , 9

231. − ›£¡ . (2.16) Fromtheidentity (2.16) bythetriangle inequality we obtaintheinequality |[, , , , 9

232. − ›| ≤ ¡[, , , , 9

233. − [,ŒŒ, , , , 9

234. ¡ + ¡[,ŒŒ, , , , 9

235. − ›¡ . (2.17) From theidentity (2.17) by integration we obtain theinequality |[, , , , 9

236. − ›| ≤ ≤ ¡[, , , , 9

237. − [,ŒŒ, , , , 9

238. ¡ + ¡[,ŒŒ, , , , 9

239. − ›¡ . (2.18) From theidentity (2.18) by theidentity (2.15) for all T ∈ 0,1we obtain theinequality |[, , , , 9

240. − ›| ≤ ≤ ¡[, , , , 9

241. − [,ŒŒ, , , , 9

242. ¡ + g[,ŒŒ, , , ›

243. .(2.19)

244. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 28 In the limit ™ → ∞ fromthe inequality we obtain the inequality |[, , , , 9

245. − ›| ≤ ≤ limsup”→∞¡[, , , , 9

246. − [,ŒŒ, , , , 9

247. ¡ + limsup”→∞g[,ŒŒ, , , ›

248. . (2.20) Wenote that limsup”→∞¡[, , , , 9

249. − [,ŒŒ, , , , 9

250. ¡ = 0. (2.21) Therefore from (2.20) and (2.21) we obtain the inequality |[, , , , 9

251. − ›| ≤ limsup”→∞g[,ŒŒ, , , ›

252. (2.22) In the limit ™ → ∞ from Eq.(2.13)-Eq.(2.14) for any fixed T ≠ 0, T ≪ 1, we obtain the differential master equationforColombeau-Ito’s SPDE (2.3)-(2.4) _jg[, , , ›

253. k[ _ + ∆ − 1 + ∆

254. jg[, , , ›

255. k[ + +› ∑ ; †…[ a !i„$,%,0,œ

256. !$ b‡ [ = ; + j[,

257. k[ + T

258. = 0,(2.23) g[, , , ›

259. = −›. (2.24) Therefore from the inequality (2.22) followsthe inequality |[, , , , 9

260. − ›| ≤ g[, , , ›

261. . (2.25) In the limit T → 0from differential equation (2.23)-(2.24)we obtain the differential equation (2.8)-(2.9)and it is easy to see that lim[→6g[, , , ›

262. = ℜ, , , ›

263. . (2.26) From the inequality (2.25) one obtainthe inequality liminf[→6|[, , , , 9

264. − ›| ≤ lim[→6g[, , , ›

265. = ℜ, , , ›

266. .(2.27) From the inequality (2.27) and Eq.(2.26) finally we obtainthe inequality liminf[→6|[, , , , 9

267. − ›| ≤ ℜ, , , ›

268. . (2.28) The inequality (2.28) finalized the proof. Definition 2.7.(TheDifferential Master Equation)The linear PDE: !ℜ$,%,0,œ

269. !% + ∆ − 1 + ∆

270. ℜ, , , ›

271. + › ∑ ; !ℜ$,%,0,œ

272. !$ = ; − ,

273. = 0, › ∈ ℝ,(2.29) ℜ, 0, , ›

274. = 0 (2.30), We will call asthe differential master equation.

275. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 29 Definition 2.8.(TheTranscendental Master Equation)Thetranscendental equation ℜj, , , ›, ,

276. k = 0, (2.31) wewill call asthe transcendentalmaster equation. Remark2.2.We note that concrete structure of the Nikolaevskii chaos is determined by the solutions›, ,

277. variety bytranscendentalmaster equation(2.31).Master equation (2.31)is determines by the only way some many-valued function›, ,

278. which is the main constructive object, determining thecharacteristics of quantum chaos in the corresponding model of Euclidian quantum fieldtheory. 3.Criterion of the existence quantum chaos in Euclidian quantum N-model. Definition3.1.Let , , , 9

279. be the solution of the Eq.(2.1). Assume that for almost all points,

280. ∈ ℝ= × ℝE(in the sense of Lebesgue–measureonℝ= × ℝE), there exist a function ,

281. such that lim→6 §a, , , 9

282. − ,

283. b ¨ = 0. (3.1) Then we will say that afunction ,

284. is a quasi-determined solution (QD-solution of the Eq.(2.). Definition3.2. Assume that there exist a setℌ ⊂ ℝ= × ℝEthat is positive Lebesgue–measure, i.e.,tℌ

285. 0 and ∀,

286. ,

287. ∈ ℌ → ¬∃lim→6¢ , , , 9

288. £°,(3.2) i.e., ,

289. ∈ ℌ imply that the limit: lim→6¢ , , , 9

290. £does not exist. Then we will say thatEuclidian quantum N-model has thequasi-determined Euclidian quantum chaos (QD-quantum chaos). Definition3.3.For each point,

291. ∈ ℝ= × ℝEwe define a set ℜ ±, ,

292. ° ⊂ ℝ by the condition: ∀›¢› ∈ ℜ ±, ,

293. ° ⟺ ℜ, , , ›

294. = 0£.(3.3) Definition3.4.Assume that Euclidian quantum N-model(2.1) has the Euclidian QD-quantum chaos. For each point ,

295. ∈ ℝ= × ℝE we define a set-valued functionℜ ±,

296. : ℝ= × ℝE → 2ℝ by the condition: ℜ ±, ,

297. = ℜ ±, ,

298. °(3.4) We will say thattheset -valued functionℜ ±, ,

299. is a quasi-determinedchaotic solution(QD-chaotic solution)of the quantum N-model.

300. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 Pic.3.1.Evolution of at point Pic.3.2.The spatial structure of instant Theorem3.1.Assume that,

301. such that 7 ∈ ³, + ∈ ℝE, , = 1, … QD-chaotic solutions. Definition3.5.For each point , (i) E, ,

302. = limsup→6 (ii) ƒ, ,

303. = liminf→6 (iii) €, ,

304. = E, , , Definition3.7. (i) Function E, ,

305. is called International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 Evolution of QD-chaotic solutionℜ ±, ,

306. in time ∈ 0,10 = 3. ∈ 0,10, = −10ƒ , s = 10 , ´ = 1.1. spatial structure ofQD-chaoticsolutionℜ ±, ,

307. at instant = 3, = −10ƒ , s = 10 , ´ = 1.1.

308. = s sin´ ∙

309. Then for all values of parameters7, , s … , 7, ∈ −1,1, ´ ∈ ℝ= , s ≠ 0, quantum N-model (2.1) has the

310. ∈ ℝ= × ℝE we define the functions such that: 6, , , 9

311. , , , , 9

312. , 9

313. − ƒ, , , 9

314. . is calledupper boundof the QD-quantum chaosat point ,

315. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 30 s, +, , = 1, … , 7 model (2.1) has the

316. .

317. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 (ii) Function ƒ, ,

318. is called lower bound of the (iii) Function €, ,

319. is called Definition3.8. Assume now that limsup%→∞€, ,

320. = €,

321. * Then we will say thatEuclidian quantum N finitewidthat point ∈ ℝ= . Definition3.9.Assume now that limsup%→∞€, ,

322. = €,

323. = Then we will say that Euclidian quantum N width at point ∈ ℝ= . Pic.3.3.TheQD-quantum chaos of the asymptotically infinite width at point International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 is called lower bound of the QD-quantum chaosat point ,

324. is called width oftheQD-quantum chaosat point,

325. . that

326. * ∞.(3.5) Then we will say thatEuclidian quantum N-model has QD-quantum chaos of the asymptotically

327. = ∞. (3.6) Then we will say that Euclidian quantum N-model has QD-quantum chaos of the asymptotically quantum chaos of the asymptotically infinite width at point = 3. = 0.1 10 , ´ = 1. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 31

328. . quantum chaos of the asymptotically quantum chaos of the asymptotically infinite 1, = 10, s =

329. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 32 Pic.3.4.The fine structure of the QD-quantum chaos of the asymptotically infinite width at point = 3, = 0, = 10, s = 10µ , ´ = 1, ∈ 10µ , 10µ + 10ƒ, › ∈ −0.676,0.676. Definition3.10. For each point ,

330. ∈ ℝ= × ℝE we define the functions such that: (i) ℜ ±E, ,

331. = supℜ ±, ,

332. °, (ii) ℜ ±ƒ, ,

333. = infℜ ±, ,

334. °, (iii) ℜ ±€, ,

335. = ℜ ±E, ,

336. − ℜ ±ƒ, ,

337. . Theorem3.2. For each point ,

338. ∈ ℝ= × ℝEis satisfiedtheinequality ℜ ±€, ,

339. ≤ €, ,

340. .(3.7) Proof. Immediately follows by Theorem2.1 and Definitions 3.5, 3.10. Theorem3.3. (Criterion of QD-quantum chaos in Euclidian quantum N-model) Assume that mes,

341. |ℜ ±€, ,

342. 0° 0.(3.8) Then Euclidian quantum N-model has QD-quantum chaos. Proof. Immediately follows bytheinequality(3.7)and Definition3.2. 4. Quasi-determined quantum chaos and physical turbulencenature. In generally accepted at the present time hypothesiswhatphysical turbulencein the dynamical systems with an infinite number of degrees of freedom really is, thephysical turbulence is associated with a strangeattractors, on which the phase trajectories of dynamical system reveal the knownproperties of stochasticity: a very high dependence on the initial conditions, whichis associated with exponential dispersion of the initially close trajectories and bringsto their non-reproduction; everywhere the density on the attractor almost of all thetrajectories a very fast decreaseoflocal auto-correlation function[2]-[9] Φx, τ

343. = 〈 ¸,

344. ¸, ¹ +

345. 〉,(4.1) Here ¸,

346. = ,

347. − 〈,

348. 〉, 〈

349. 〉 = lim}→∞〈

350. 〉}, 〈

351. 〉} = 1 H u } 6

352. . In contrast with canonical numerical simulation, by using Theorem2.1 it is possible to study non-perturbativelythe influence of thermal additive fluctuationson classical dynamics, which in the consideredcase is described by equation (4.1). The physicalnature of quasi-determined chaosis simple andmathematically is associated withdiscontinuously of the trajectories of the stochastic process, , , 9

353. on parameter . In order to obtain thecharacteristics of this turbulence, which is a very similarlytolocal auto-correlation function (3.1) we define bellowsomeappropriatefunctions.

354. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 33 Definition 4.1.The numbering function™ ,

355. of quantum chaos in Euclidian quantum N-modelis defined by ™,

356. = cardℜ ±,

357. °.(4.2) Here by cardN¼Q we denote the cardinality of a finite set¼,i.e., the number of its elements. Definition 4.2.Assume now that a set ℜ ±,

358. °is ordered be increase of its elements. We introduce the functionℜ ±;,

359. , y = 1, … , ™,

360. which value at point,

361. , equals the y-th element ofa set ℜ ±,

362. °. Definition 3.3.The mean value function,

363. ofthe chaotic solution ℜ ±,

364. at point ,

365. is defined by ,

366. = j™,

367. k ƒ ∑ ℜ ±;,

368. ”$,%

369. ; .(4.3) Definition 3.4.The turbulent pulsations function ∗,

370. of the chaotic solution ℜ ±,

371. at point ,

372. is defined by ∗,

373. = ½j™,

374. k ƒ ∑ ¡ℜ ±;,

375. − ,

376. ¡ ”$,%

377. ; .(4.4) Definition3.5.Thelocal auto-correlation function is definedby Φ, ¹

378. = lim}→∞〈 ¸, ¹

379. ¸, ¹ +

380. 〉} = lim}→∞ } ^ ¸ } 6 ,

381. ¸, ¹ +

382. _ ,(4.5) ¸,

383. = ,

384. − ¾

385. , ¾

386. = lim}→∞ } ^ } 6 ,

387. _ .(4.6) Definition 3.5.Thenormalized local auto-correlation function is defined by Φ¿, ¹

388. = Φ$,À

389. ΦÁ,6

390. .(4.7) Let us consider now 1DEuclidian quantum N-model corresponding to classical dynamics !' !$' § − a1 + !' !$'b ¨ ,

391. + !/$,0

392. !$ ,

393. − s sin´ ∙

394. = 0,(4.8) Corresponding Langevin equation are [34]-[35]: , ,

395. + ∆ − 1 + ∆

396. , ,

397. + , ,

398. , ,

399. − −s sin´

400. = ,

401. , 0∆= !' !$',(4.9) , 0,

402. = 0, ,

403. = !'#$,%

404. !$!% .(4.10) Corresponding differential master equation are

405. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 !ℜ$,%,0,œ

406. !% + ∆ − 1 + ∆

407. ℜ, , ℜ, 0, , ›

408. = −›.(4.12) Corresponding transcendental master equation NÂÃÄÅ∙$

409. ƒWÁÆ%∙ÇÅ

410. ÂÃÄÅ$ƒœ∙È∙%

411. Q∙œ∙È∙ Ç'Å

412. Eœ'∙È'∙Å' É´

413. = ´ − ´ − 1

414. .(4.14) We assume now that É´

415. = 0.Then from Eq.(4.13) for a NÂÃÄÅ∙$

416. ƒÂÃÄÅ$ƒœ∙È∙%

417. Q∙œ∙È∙Å œ'∙È'∙Å' + œ Ê = 0 Ncos´ ∙

418. − cos´ − › ∙ ∙

419. Q The result of calculation using transcendental is presented by Pic.4.1 and Pic.4.2. Pic.4.1.Evolution of QD-chaotic solution International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 , , ›

420. + › !ℜ$,%,0,œ

421. !$ − s sin´

422. = 0,(4.11) Corresponding transcendental master equation (2.29)-(2.30) are ∙Å + NÄÌ¿Å∙$

424. ÄÌ¿Å$ƒœ∙È∙%

425. Q∙ÇÅ

426. Ç'Å

427. Eœ'∙È'∙Å' + œ Ê = 0,(4.13) Then from Eq.(4.13) for all ∈ 0, ∞

428. we obtain 0,or(4.14)

429. Q ∙ s ∙ ƒ ∙ ´ƒ › 0.(4.15) transcendental master equation (4.15) the corresponding function is presented by Pic.4.1 and Pic.4.2. chaotic solutionO ±10 , ,

430. in time ∈ 0, 10, ∆ 0.1, s 10 , 1, , ∆› 0.01 International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 34 master equation (4.15) the corresponding function O ±, ,

431. 0, ´ 1,

432. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 Pic.4.2.The spatial structure ofQD The result of calculation using master equation(4. presented by Pic.4.3 and Pic.4.4 Pic.4.3.The EuclidianquantumN-model Pic.4.4.The development Euclidian quantum N-model at point Let us calculate now corresponding of calculation using Eq.(4.7)-Eq.(4.7) International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 QD-chaoticsolutionO ±, ,

433. at instant 10 , 0, ´ 1, ∆ 0.1, ∆› 0.01. calculation using master equation(4.13) the correspondingfunction 4. Thedevelopment of temporal chaotic regime of1D model at point 1, ∈ 0, 10. 10ƒÍ , s 10 , 1, The developmentof temporal chaotic regimeof 1D model at point 1, ∈ 0, 10, 10ƒÍ , s 5 ∙ 10Ï , 1 correspondingnormalized local auto-correlation functionΦ¿ Eq.(4.7) is presented by Pic.4.5 and Pic.4.6. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 35 = 1, s 10 , unction O Ð, ,

434. is , ´ 1. 1, ´ 1. , ¹

435. .The result

436. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 Pic.4.5. Normalized local auto ∈ Pic.4.6.Normalized local auto ∈ 0 Inpaper [7]the mechanism of the onset of chaos and its relationship to the characteristics of the spiral attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg equation(4.14). Numerical data are compared with experimental results. !Ñ$,%

437. !% y9

438. Ò,

439. 1 |Ò Ò International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 Normalized local auto-correlation function Φ¿1, ¹

440. ∈ 0,50, 10ƒÍ , s 10 , 1, ´ 1. Normalized local auto-correlation functionΦ¿1, ¹

441. . 0,100, 10ƒÍ , s 5 ∙ 10Ï , 1, ´ 1. the mechanism of the onset of chaos and its relationship to the characteristics of the spiral attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg . Numerical data are compared with experimental results. ,

442. |

443. Ò,

444. Ó !'Ñ$,%

445. !$ , (4.14) 0,

446. 0, ÒÔ,

447. 0, ∈ 0, Ô, Ô 50. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 36 the mechanism of the onset of chaos and its relationship to the characteristics of the spiral attractors are demonstrated for inhomogeneous media that can be modeled by the Ginzburg– Landau

448. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 Pic.4.7. Normalized local auto However as pointed out above (see for stochastic model !Ñ$,%

449. !% y9

450. Ò,

451. 1 |Ò Ò 5.The order of the phase transition turbulent state at instant In order to obtain the character of the phase transition a spatially uniform to a turbulent state Oj, , , ›, ,

452. k 0. (5.1) Bydifferentiation the Eq.(5.1) one obtain Oj$,%,0,œ$,%,0

453. k 0 !Oj$,%,0,œ$,%,0

454. k !œ œ$ 0 FromEq.(5.2) one obtain œ$,%,0

455. 0 a !Oj$,%,0,œ$,%,0

456. k !0 b ∙ a !Oj Let us consider now 1DEuclidian quantum N transcendental master equation Eq.(4.13) ›one obtain International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 Normalized local auto-correlation functionΦ¿25, ¹

457. [7]. However as pointed out above (see Remark1.1-1.4 ) such numerical simulation in factgives ,

458. |

459. Ò,

460. Ó !'Ñ$,%

461. !$ √,

462. , ≪ 1, (4.15) 0,

463. 0, ÒÔ,

464. 0, ∈ 0, Ô, Ô 50. phase transitionfrom a spatially uniformstate at instant Ö × Ø. In order to obtain the character of the phase transition (first-order or second-order on parameter a spatially uniform to a turbulent stateat instant × 0one can to use the master equation () of the form one obtain k $,%,0

465. 0 !Oj$,%,0,œ$,%,0

466. k !0 0. (5.2) a j$,%,0,œ$,%,0

467. k !œ b ƒ .(5.3) Euclidian quantum N-model given byEq. (4.9)-Eq. (4.10). From corresponding transcendental master equation Eq.(4.13)by differentiation the equation Eq.(4.13)with respect to International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 37 givesnumerical data state to a on parameters , ´) from one can to use the master equation () of the form Eq. (4.10). From corresponding with respect to variable

468. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 38 Oj, , , ›, ,

469. k › Ncos´ ∙

470. exp ∙ É´

471. cos´ › ∙ ∙

472. Q ∙ ∙ ´ É´

473. › ∙ ∙ ´ Ncos´ ∙

474. exp ∙ É´

475. sin´ › ∙ ∙

476. Q ∙ › ∙ ∙ ∙ ´ É´

477. › ∙ ∙ ´ Ncos´ ∙

478. exp ∙ Ép

479. cos´ › ∙ ∙

480. Q ∙ 2 ∙ › ∙ ∙ ´ É´

481. › ∙ ∙ ´ Nsin´ ∙

482. exp ∙ É´

483. cos´ › ∙ ∙

484. Q ∙ ∙ ∙ É´

485. É´

486. › ∙ ∙ ´ NÄÌ¿Å∙$

488. ÄÌ¿Å$ƒœ∙È∙%

489. Q∙œ∙ÇÅ

490. ∙È'∙Å' Ç'Å

491. Eœ'∙È'∙Å'' Ê . (5.4) From Eq.(5.4)for a sufficiently small × 0 one obtain 1 !Oj$,%,0,œ$,%,0

492. k !œ 2 %×6 Ê .(5.5) From master equation Eq.(4.13) one obtainby differentiation the equation Eq.(4.13)with respect to variable one obtain O, , , ›

493. Ù ∙ ÇÅ

494. 0 exp ∙ É´

495. cos´ › ∙ ∙

496. Ú ∙ › ∙ ∙ ´ É´

497. › ∙ ∙ ´ Ncos´ ∙

498. exp ∙ É´

499. cos´ › ∙ ∙

500. Q ∙ 2 ∙ ÇÅ

501. 0 ∙ › ∙ ∙ ´ É´

502. › ∙ ∙ ´ Ù ∙ ÇÅ

503. 0 exp ∙ É´

504. sin´ › ∙ ∙

505. Ú ∙ É´

506. É´

507. › ∙ ∙ ´ Nsin´ ∙

508. exp ∙ É´

509. sin´ › ∙ ∙

510. Q ∙ ÇÅ

511. 0 É´

512. › ∙ ∙ ´ NÄÌ¿Å∙$

514. ÄÌ¿Å$ƒœ∙È∙%

515. Q∙Ç'Å

516. ∙ ÛÜÝ

517. ÛÞ Ç'Å

518. Eœ'∙È'∙Å'' (5.6) From Eq.(5.6)for a sufficiently small × 0one obtain 1 !Oj$,%,0,œ$,%,0

519. k !0 2 %×6 a ∙ ÇÅ

520. 0 b ÇÅ

521. ∙Å∙ÄÌ¿Å∙$

522. Ç'Å

523. Eœ'∙È'∙Å' ´´ 1

524. ÇÅ

525. ÄÌ¿Å∙$

526. Ç'Å

527. Eœ'∙È'∙Å',(5.7) Therefore from Eq.(5.3),Eq. (5.5) and Eq.(5.7) one obtain

528. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 39 ℑ, ,

529. %×6 1 % œ$,%,0

530. 0 2 %×6 × s´´ 1

531. ÄÌ¿Å∙$

532. ÇÅ

533. sign´ ∙

534. (5.8) In the limit → 0 from Eq. (5.8) one obtain ,

535. = lim%→6 œ$,%,0

536. %0 = s´´ − 1

537. ÄÌ¿Å∙$

538. ÇÅ

539. sign´ ∙

540. ,(5.9) and where É´

541. = ´ − ´ − 1

542. = ´ á

543. , á

544. = − ´ − 1

545. . FromEq. (5.9) follows that limâ0

546. →6@ ,

547. = +∞,(5.10) limâ0

548. →6ã ,

549. = −∞.(5.11) FromEq. (5.10)-(5.11) follows second orderdiscontinuity of the quantity , ,

550. at instant = 0. Therefore the system causing it to make a direct transitionfrom a spatially uniformstate≈6, 0,

551. = 0 to a turbulent statein an analogous fashion to the second-order phase transition inquasi-equilibrium systems. 6.Chaotic regime generatedby periodical multi-modes external perturbation. Assume nowthat external periodical force

552. has the followingmulti-modes form

553. = − ∑ sä å ä sin´ä

554. .(6.1) Corresponding transcendental master equation are , , , ›

555. = ] sä Ncos´ä ∙

556. − exp ∙ É´

557. cos´ä − › ∙ ∙

558. Q ∙ › ∙ ∙ ´ä É´ä

559. + › ∙ ∙ ´ä å ä + + ∑ sä NÄÌ¿Åæ∙$

560. ƒWÁÆ%∙ÇÅæ

561. ÄÌ¿Åæ$ƒœ∙È∙%

562. Q∙ÇÅæ

563. Ç'Åæ

564. Eœ'∙È'∙Åæ ' å ä + › = 0, É´

565. = ´ − ´ − 1

566. .(6.2) Let us consider the examples of QD-chaotic solutions with a periodical force:

567. = −s ∑ sin a ä$ ˜ b å ä .(6.3)

568. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 40 Pic.6.1.Evolution of QD-chaotic solution ±10 , ,

569. in time ∈ 7 ∙ 10 , 10µ, ∆ = 0.1, ç = 1, — = 100, = −1, ´ = 1, s = 10 , = 1, ∆› = 0.01. Pic.6.2.The spatial structure ofQD-chaoticsolution ±, ,

570. at instant = 10 , ∈ 1.4 ∙ 10 , 2.5 ∙ 10, ç = 1, — = 100, = −1, ´ = 1, s = 10 , = 1, ∆ = 0.1, ∆› = 0.01. Pic.6.2.The spatial structure ofQD-chaoticsolution ±, ,

571. at instant = 5 ∙ 10 , ∈ 1.4 ∙ 10 , 2.5 ∙ 10, ç = 1, — = 100, = −1, ´ = 1, s = 10 , = 1, ∆ = 0.1, ∆› = 0.01. 7.Conclusion A non-perturbative analytical approach to the studying of problemof quantum chaos in dynamical systems withinfinite number of degrees of freedom isproposed and developed successfully.It is shown that the additive thermal noise destabilizes dramatically the ground state of the system thus causing it to make a direct transition from a spatially uniform to a turbulent state.

572. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 41 8.Acknowledgments A reviewer provided important clarifications. References [1] Nikolaevskii, V. N.(1989). Recent Advances inEngineering Science, edited by S.L. Kohand C.G. Speciale.Lecture Notes in Engineering, No. 39(Springer - Verlag. Berlin. 1989), pp. 210. [2] Tribelsky, M. I., Tsuboi, K. (1996).Newscenario to transition to slow turbulence. Phys.Rev. Lett. 76 1631 (1996). [3] Tribelsky, M. I. (1997). Short-wavelegthinstability and transition in distributed systems with additional symmetryUspekhifizicheskikhnauk (Progresses of the Physical Studies)V167,N2. [4] Toral, R., Xiong, J. D. Gunton, and H.W.Xi.(2003) Wavelet Description of the Nikolaevskii Model.Journ.Phys.A 36, 1323 ( 2003). [5] Haowen Xi, Toral, R,.Gunton, D, Tribelsky M.I. (2003).ExtensiveChaos in the NikolaevskiiModel.Phys. Rev. E.Volume: 61,Page:R17,(2000) [6] Tanaka, D., Amplitude equations of Nikolaevskii turbulence, RIMS KokyurokuBessatsu B3(2007), 121–129 [7] Fujisaka,H.,Amplitude Equation ofHigher-Dimensional Nikolaevskii Turbulence Progress of Theoretical Physics, Vol. 109, No. 6, June 2003 [8] Tanaka, D.,Bifurcation scenario to Nikolaevskii turbulence in small systems, http://arxiv.org/abs/nlin/0504031v1DOI: 10.1143/JPSJ.74.222 [9] Anishchenko, V. S.,Vadivasova, T. E.,Okrokvertskhov, G. A. and Strelkova,G. I., Statistical properties of dynamical chaos, Physics-Uspekhi(2005),48(2):151 http://dx.doi.org/10.1070/PU2005v048n02ABEH002070 [10] Tsinober, A., The Essence of Turbulence as a Physical Phenomenon,2014, XI, 169 pp., ISBN 978-94-007-7180-2 [11] Ivancevic,V. G.,High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction 2007, XV, 697 p. [12] Herbst,B. M. and Ablowitz, M. J.,Numerically induced chaos in the nonlinear Schrödinger equation,Phys. Rev. Lett. 62, 2065 [13] Mark J. Ablowitz and B. M. Herbst,On Homoclinic Structure and Numerically Induced Chaos for the Nonlinear Schrodinger Equation,SIAM Journal on Applied Mathematics Vol. 50, No. 2 (Apr., 1990), pp. 339-351 [14] Li, Y., Wiggins, S., Homoclinic orbits and chaos in discretized perturbed NLS systems: Part II. Symbolic dynamics,Journal of Nonlinear Science1997, Volume 7, Issue 4, pp 315-370. [15] Blank,M. L., Discreteness and Continuity in Problems of Chaotic Dynamics,Translations of Mathematical Monographs1997; 161 pp; hardcoverVolume: 161ISBN-10: 0-8218-0370-0 [16] FADNAVIS,S., SOME NUMERICAL EXPERIMENTS ON ROUND-OFF ERROR GROWTH IN FINITE PRECISION NUMERICAL COMPUTATION.HTTP://ARXIV.ORG/ABS/PHYSICS/9807003V1 [17] Handbook of Dynamical Systems, Volume 2Handbook of Dynamical Systems Volume 2, Pages 1086 (2002)Edited by Bernold Fiedler ISBN: 978-0-444-50168-4 [18] Gold, Bernard, and Charles M. Rader. Effects of quantization noise in digital filters. Proceedings of the April 26-28, 1966, Spring joint computer conference. ACM, 1966. [19] Bennett, W. R., Spectra of Quantized Signals,Bell System Technical Journal, vol. 27, pp. 446-472 (July 1948) [20] I. G. Vladimirov, I. G.,Diamond,P.,A Uniform White-Noise Model for Fixed-Point Roundoff Errors in Digital Systems,Automation and Remote ControlMay 2002, Volume 63, Issue 5, pp 753-765 [21] Möller,M. Lange,W.,Mitschke, F., Abraham, N.B.,Hübner, U.,Errors from digitizing and noise in estimating attractor dimensionsPhysics Letters AVolume 138, Issues 4–5, 26 June 1989, pp. 176–182 [22] Widrow, B. and Kollar, I., Quantization Noise: Round off Error in Digital Computation, Signal

573. International Journal of Recent advances in Physics (IJRAP) Vol.4, No.1, February 2015 42 Processing,Control, and Communications,ISBN: 9780521886710 [23] Foukzon, J.,Advanced numerical-analytical methods for path integral calculation and its application to somefamous problems of 3-D turbulence theory. New scenario for transition to slow turbulence. Preliminary report.Meeting:1011, Lincoln, Nebraska, AMS CP1, Session for Contributed Papers. http://www.ams.org/meetings/sectional/1011-76-5.pdf [24] Foukzon, J.,New Scenario for Transition to slow Turbulence. Turbulencelike quantum chaosin three dimensional model of Euclidian quantumfield theory.Preliminary report.Meeting:1000, Albuquerque, New Mexico, SS 9A, Special Session on Mathematical Methods in Turbulence. http://www.ams.org/meetings/sectional/1000-76-7.pdf [25] Foukzon, J.,New scenario for transition to slow turbulence. Turbulence like quantum chaos in three dimensional modelof Euclidian quantum field theory.http://arxiv.org/abs/0802.3493 [26] Foukzon, J.,Communications in Applied Sciences, Volume 2, Number 2, 2014, pp.230-363 [27] Foukzon, J., Large deviations principles of Non-Freidlin-Wentzell type,22 Oct. 2014. http://arxiv.org/abs/0803.2072 [28] Colombeau, J., Elementary introduction to new generalized functions Math. Studies 113, North Holland,1985 [29] CATUOGNO,P., AND OLIVERA C.,STRONG SOLUTION OF THE STOCHASTIC BURGERS EQUATION, HTTP://ARXIV.ORG/ABS/1211.6622V2 [30] Oberguggenberger, M.,.F.Russo, F.,Nonlinear SPDEs: Colombeau solutions and pathwise limits, HTTP://CITESEERX.IST.PSU.EDU/VIEWDOC/SUMMARY?DOI=10.1.1.40.8866 [31] WALSH, J. B.,FINITE ELEMENT METHODS FOR PARABOLIC STOCHASTIC PDE’S, POTENTIAL ANALYSIS, VOLUME 23, ISSUE 1, PP 1-43.HTTPS://WWW.MATH.UBC.CA/~WALSH/NUMERIC.PDF [32] Suli,E.,Lecture Notes on Finite ElementMethods for Partial DifferentialEquations, University of Oxford, 2000.http://people.maths.ox.ac.uk/suli/fem.pdf [33] Knabner, P.,Angermann, L.,Numerical Methods forElliptic and Parabolic PartialDifferential Equations, ISBN 0-387-95449-Xhttp://link.springer.com/book/10.1007/b97419 [34] DIJKGRAAF, R., D. ORLANDO, D. ORLANDO, REFFERT, S., RELATING FIELD THEORIES VIA STOCHASTIC Quantization,Nucl.Phys.B824:365-386,2010DOI: 10.1016/j.nuclphysb.2009.07.018 [35] MASUJIMA,M.,PATH INTEGRAL QUANTIZATION AND STOCHASTIC QUANTIZATION,EDITION: 2ND ED. 2009ISBN-13: 978-3540878506

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