Oscillation of Non-Linear Systems Close to Equilibrium Position in the Presence of Coarse-Graining in Time and Space

One considers the motion of nonlinear systems close to their equilibrium positions in the presence of coarse-graining in time on the one hand, and coarse-graining in time on the other hand. By considering a coarse-grained time as a time in which the increment is not dt but rather (dt)c > dt, one is led to introduce a modeling in terms of fractional derivative with respect to time; and l ikewise for coarse-graining with respect to the space variable x. After a few prerequisites on fractional calculus via modified Riemann-Liouville derivative, one examines in a de tailed way the solutions of fractional linear differential equations in this framewor k, and then one uses this result in the linearization of nonlinear systems close to their equil ibrium positions.

[1]  L. Nottale.  Fractal space-time and microphysics , 1993 .

[2]  Kiran M. Kolwankar,et al.  Hölder exponents of irregular signals and local fractional derivatives , 1997, chao-dyn/9711010.

[3]  G. Jumarie.  Maximum Entropy, Information Without Probability and Complex Fractals: Classical and Quantum Approach , 2000 .

[4]  Guy Jumarie,et al.  On the representation of fractional Brownian motion as an integral with respect to (dt)alpha , 2005, Appl. Math. Lett..

[5]  Guy Jumarie,et al.  From Self-Similarity to Fractional Derivative of Non-Differentiable Functions via Mittag-Leffler Function , 2008 .

[6]  Kiran M. Kolwankar,et al.  Local Fractional Fokker-Planck Equation , 1998 .

[7]  M. E. Naschie,et al.  A review of E infinity theory and the mass spectrum of high energy particle physics , 2004 .

[8]  Vo Anh,et al.  Scaling laws for fractional diffusion-wave equations with singular data , 2000 .

[9]  Dumitru Baleanu,et al.  Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives , 2005 .

[10]  Malgorzata Klimek,et al.  Lagrangean and Hamiltonian fractional sequential mechanics , 2002 .

[11]  Dumitru Baleanu,et al.  Lagrangian Formulation of Classical Fields within Riemann-Liouville Fractional Derivatives , 2005 .

[12]  B. Øksendal,et al.  FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE , 2003 .

[13]  Guy Jumarie,et al.  Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions , 2009, Appl. Math. Lett..

[14]  B. Mandelbrot,et al.  A CLASS OF MICROPULSES AND ANTIPERSISTENT FRACTIONAL BROWNIAN MOTION , 1995 .

[15]  B. Øksendal,et al.  Stochastic Calculus for Fractional Brownian Motion and Applications , 2008 .

[16]  Fractional Brownian motion with complex variance via random walk in the complex plane and applications , 2000 .

[17]  Guy Jumarie,et al.  Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker–Planck equations , 2004 .

[18]  Guy Jumarie,et al.  On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion , 2005, Appl. Math. Lett..

[19]  Guy Jumarie,et al.  A Fokker-Planck equation of fractional order with respect to time , 1992 .

[20]  Guy Jumarie.  Further Results on the Modelling of Complex Fractals in Finance, Scaling Observation and Optimal Portfolio Selection , 2002 .

[21]  A. El-Sayed,et al.  Fractional-order diffusion-wave equation , 1996 .

[22]  M. Naschie.  Gravitational instanton in Hilbert space and the mass of high energy elementary particles , 2004 .

[23]  L. Decreusefond,et al.  Stochastic Analysis of the Fractional Brownian Motion , 1999 .

[24]  Robert Carroll,et al.  On the Quantum Potential , 2004 .

[25]  D. Baleanu,et al.  Fractional Euler—Lagrange Equations of Motion in Fractional Space , 2007 .

[26]  Laurent Nottale,et al.  The scale-relativity program , 1999 .

[27]  Laurent Nottale,et al.  Scale-relativity and quantization of the universe I. Theoretical framework , 1997 .

[28]  Laurent Nottale,et al.  Scale relativity and fractal space-time: applications to quantum physics, cosmology and chaotic systems. , 1996 .

[29]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[30]  M. E. Naschie,et al.  Elementary prerequisites for E-infinity . (Recommended background readings in nonlinear dynamics, geometry and topology) , 2006 .

[31]  Philippe Sainty.  Construction of a complex‐valued fractional Brownian motion of order N , 1992 .

[32]  G. Jumarie.  A norandom variational approach to stochastic linear quadratic Gaussian optimization involving fractional noises (FLQG) , 2005 .

[33]  Real quantum cybernetics , 1987 .

[34]  Dumitru Baleanu,et al.  The Hamilton formalism with fractional derivatives , 2007 .

[35]  Jacky Cresson,et al.  Quantum derivatives and the Schrödinger equation , 2004 .

[36]  Andrzej Hanygad,et al.  Multidimensional solutions of time-fractional diffusion-wave equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[37]  T. Osler.  Taylor’s Series Generalized for Fractional Derivatives and Applications , 1971 .

[38]  G. Jumarie.  Stochastic differential equations with fractional Brownian motion input , 1993 .

[39]  E. Barkai,et al.  Fractional Fokker-Planck equation, solution, and application. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[41]  Guy Jumarie,et al.  Probability calculus of fractional order and fractional Taylor's series application to Fokker-Planck equation and information of non-random functions , 2009 .

[42]  Nabil T. Shawagfeh,et al.  Analytical approximate solutions for nonlinear fractional differential equations , 2002, Appl. Math. Comput..

[43]  O. Agrawal,et al.  Fractional hamilton formalism within caputo’s derivative , 2006, math-ph/0612025.

[44]  M. Naschie.  On Penrose view of transfinite sets and computability and the fractal character of E-infinity spacetime , 2005 .

[45]  G. Jumarie.  SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION , 2001 .

[46]  D. Baleanu,et al.  Fractional Hamilton’s equations of motion in fractional time , 2007 .

[47]  W. Wyss,et al.  THE FRACTIONAL BLACK-SCHOLES EQUATION , 2000 .

[48]  Dumitru Baleanu,et al.  On fractional Schrdinger equation in a -dimensional fractional space , 2009 .

[49]  G. Ord,et al.  Entwined paths, difference equations, and the Dirac equation , 2002, quant-ph/0208004.

[50]  M. Naschie.  Non-linear dynamics and infinite dimensional topology in high energy particle physics , 2003 .

[51]  Benoit B. Mandelbrot,et al.  Alternative micropulses and fractional Brownian motion , 1996 .

[52]  Guy Jumarie,et al.  Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDE and Taylor’s series of nondifferentiable functions , 2007 .

[53]  Quantum derivatives and the Schr€ o equation , 2004 .