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New Types of Solutions of Non-linear Fractional Differential Equations

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Advances in Harmonic Analysis and Operator Theory

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 229))

Abstract

Using the Riemann-Liouville and Caputo Fractional Standard Maps (FSM) and the Fractional Dissipative Standard Map (FDSM) as examples, we investigate types of solutions of non-linear fractional differential equations. They include periodic sinks, attracting slow diverging trajectories (ASDT), attracting accelerator mode trajectories (AMT), chaotic attractors, and cascade of bifurcations type trajectories (CBTT). New features discovered include attractors which overlap, trajectories which intersect, and CBTTs.

Mathematics Subject Classification (2010). Primary 47Gxx; Secondary 47Hxx.

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References

  1. V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, HEP, 2011.

    Google Scholar 

  2. V.E. Tarasov, Theoretical Physics Models with Integro-Differentiation of Fractional Order. IKI, RCD, 2011 (in Russian).

    Google Scholar 

  3. R. Herrmann, Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore, 2011.

    Google Scholar 

  4. R. Caponetto, G. Dongola, and L. Fortuna, Fractional Order Systems: Modeling and Control Applications (World Scientific Series on Nonlinear Science Series a). World Scientific, 2010.

    Google Scholar 

  5. I. Petras, Fractional-Order Nonlinear Systems. Springer, 2011.

    Google Scholar 

  6. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London, 2010.

    Google Scholar 

  7. A.C.J. Luo and V. Afraimovich (eds.), Long-range Interaction, Stochasticity and Fractional Dynamics. Springer, 2010.

    Google Scholar 

  8. G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford, 2005.

    Google Scholar 

  9. N. Laskin, G.M. Zaslavsky, Nonlinear fractional dynamics on a lattice with longrange interactions. Physica A 368 (2006), 38–4.

    Article  Google Scholar 

  10. V.E. Tarasov, G.M. Zaslavsky, Fractional dynamics of coupled oscillators with longrange interaction. Chaos 16 (2006), 023110.

    Article  MathSciNet  Google Scholar 

  11. N. Korabel, G.M. Zaslavsky, Transition to chaos in discrete nonlinear Schr¨odinger equation with long-range interaction. Physica A 378 (2007), 223–237.

    Article  Google Scholar 

  12. G.M. Zaslavsky, M. Edelman, V.E. Tarasov, Dynamics of the chain of forced oscillators with long-range interaction: From synchronization to chaos. Chaos 17 (2007), 043124.

    Article  MathSciNet  Google Scholar 

  13. V.E. Tarasov, G.M. Zaslavsky, Fractional dynamics of systems with long-range space interaction and temporal memory. Physica A 383 (2007), 291–308.

    Article  Google Scholar 

  14. I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.

    Google Scholar 

  15. R.R. Nigmatullin, Fractional integral and its physical interpretation. Theoretical and Mathematical Physics 90 (1992), 242–251.

    Article  MathSciNet  Google Scholar 

  16. F.Y. Ren, Z.G. Yu, J. Zhou, A. Le Mehaute, R.R. Nigmatullin, The relationship between the fractional integral and the fractal structure of a memory set. Physica A 246 (1997), 419–429.

    Article  Google Scholar 

  17. W.Y. Qiu, J. Lu, Fractional integrals and fractal structure of memory sets. Phys. Lett. A 272 (2000), 353 358.

    Google Scholar 

  18. R.R. Nigmatullin, Fractional kinetic equations and ‘universal’ decoupling of a memory function in mesoscale region. Physica A 363 (2006), 282–298.

    Article  Google Scholar 

  19. V.E. Tarasov, G.M. Zaslavsky Fractional dynamics of systems with long-range space interaction and temporal memory. Physica A 383 (2007), 291–308.

    Article  Google Scholar 

  20. A. Carpinteri, F. Mainardi, (eds.), Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien, 1997.

    Google Scholar 

  21. B.N. Lundstrom, A.L. Fairhall, M. Maravall, Multiple time scale encoding of slowly varying whisker stimulus envelope in cortical and thalamic neurons in vivo. J. Neuroscience, 30 (2010), 5071–5077.

    Article  Google Scholar 

  22. B.N. Lundstrom, M.H. Higgs, W.J. Spain, A.L. Fairhall, Fractional differentiation by neocortical pyramidal neurons. Nature Neuroscience 11 (2008), 1335–1342.

    Article  Google Scholar 

  23. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Application of Fractional Differential Equations. Elsevier, Amsterdam, 2006.

    Google Scholar 

  24. A.A. Kilbas, B. Bonilla, J.J. Trujillo, Nonlinear differential equations of fractional order is space of integrable functions. Doklady Mathematics 62 (2000), 222–226, Translated from Doklady Akademii Nauk 374 (2000), 445–449. (in Russian).

    Google Scholar 

  25. A.A. Kilbas, B. Bonilla, J.J. Trujillo, Existence and uniqueness theorems for nonlinear fractional differential equations. Demonstratio Mathematica 33 (2000), 583–602.

    MathSciNet  MATH  Google Scholar 

  26. V.E. Tarasov, Differential equations with fractional derivative and universal map with memory. Journal of Physics A 42 (2009), 465102.

    Article  MathSciNet  Google Scholar 

  27. A. Wineman, Nonlinear viscoelastic membranes. Computers and Mathematics with Applications 53 (2007), 168–181.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Wineman, Nonlinear Viscoelastic Solids – A Review. Mathematics and Mechanics of Solids 14 (2009), 300–366

    Article  MathSciNet  MATH  Google Scholar 

  29. F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics, and Epidemics. SIAM, Philadelphia, 1975.

    Google Scholar 

  30. F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology. Springer, New York, 2001.

    Google Scholar 

  31. V. Gafiychuk, B. Datsko, Mathematical modeling of different types of instabilities in time fractional reaction–diffusion systems. Computers and Mathematics with Applications 59 (2010), 1001–1007.

    Article  MathSciNet  Google Scholar 

  32. V. Gafiychuk, B. Datsko, V. Meleshko, D. Blackmore, Analysis of the solutions of coupled nonlinear fractional reaction–diffusion equations. Chaos, Solitons & Fractals 41 (2009), 1095–1104.

    Article  MathSciNet  MATH  Google Scholar 

  33. V. Gafiychuk, B. Datsko, Stability analysis and limit cycle in fractional system with Brusselator nonlinearities. Phys. Let. A 372 (2008), 4902–4904.

    Article  MATH  Google Scholar 

  34. V. Gafiychuk, B. Datsko, V. Meleshko, Analysis of fractional order Bonhoeffer-van der Pol oscillator. Physica A 387 (2008), 418–424.

    Article  Google Scholar 

  35. G.M. Zaslavsky, A.A. Stanislavsky, M. Edelman, Chaotic and pseudochaotic attractors of perturbed fractional oscillator. Chaos 16 (2006), 013102.

    Article  MathSciNet  Google Scholar 

  36. M.S. Tavazoei, M. Haeri, Chaotic attractors in incommensurate fractional order systems. Physica D 237 (2008), 2628–2637.

    Article  MathSciNet  MATH  Google Scholar 

  37. V.E. Tarasov, G.M. Zaslavsky, Fractional equations of kicked systems and discrete maps. J. Phys. A 41 (2008), 435101.

    Article  MathSciNet  Google Scholar 

  38. M. Edelman, V.E. Tarasov, Fractional standard map. Phys. Let. A 374 (2009), 279– 285.

    Article  MathSciNet  MATH  Google Scholar 

  39. V.E. Tarasov, M. Edelman, Fractional dissipative standard map. Chaos 20 (2010), 023127.

    Article  MathSciNet  Google Scholar 

  40. V.E. Tarasov, Discrete map with memory from fractional differential equation of arbitrary positive order. Journal of Mathematical Physics. 50 (2009), 122703.

    Article  MathSciNet  Google Scholar 

  41. B.V. Chirikov, A universal instability of many dimensional oscillator systems. Phys. Rep. 52 (1979), 263–379.

    Article  MathSciNet  Google Scholar 

  42. G.M. Zaslavsky, The simplest case of a strange attractor. Phys. Lett. A 69 (1978), 145–147.

    Article  MathSciNet  Google Scholar 

  43. G.M. Zaslavsky, Kh.-R. Ya. Rachko, Singularities of transition to a turbulent motion. Sov. Phys. JETP 49 (1979), 1039–1044.

    Google Scholar 

  44. A.J. Lichtenberg, M.A. Lieberman, Regular and Chaotic Dynamics. Springer, Berlin, 1992.

    Google Scholar 

  45. G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford, 2005.

    Google Scholar 

  46. E. Ott, Strange Attractors and Chaotic Motions of Dynamical Systems. Rev. Mod. Phys. 53 (1981), 655–671.

    Article  MathSciNet  MATH  Google Scholar 

  47. V. Afraimovich, Sze-Bi Hsu, Lectures on Chaotic Dynamical Systems. Amer. Math. Society. International Press, Providence, 2002.

    Google Scholar 

  48. P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors. Physica D 9 (1983), 189–208.

    Article  MathSciNet  MATH  Google Scholar 

  49. D.A. Russel, J.D. Hanson, and E. Ott, Dimension of strange attractors. PRL 45 (1980), 1175–1178.

    Article  Google Scholar 

  50. F. Haake, Quantum Signatures of Chaos. Springer, Berlin, 2000.

    Google Scholar 

  51. G.M. Zaslavsky, M. Edelman, Superdiffusion in the Dissipative Standard Map. Chaos 18 (2008), 033116.

    Article  MathSciNet  Google Scholar 

  52. Q. Wang and L.-S. Young, From invariant curves to strange attractors. Commun. in Math. Phys. 225 (2002), 275–304.

    Article  MathSciNet  MATH  Google Scholar 

  53. Q. Wang and L.-S. Young, Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Commun. in Math. Phys. 240 (2003), 509–529.

    MathSciNet  MATH  Google Scholar 

  54. M. Edelman, Fractional Standard Map: Riemann-Liouville vs. Caputo. Commun. Nonlin. Sci. Numer. Simul, 16 (2011), 4573–4580.

    Article  MathSciNet  MATH  Google Scholar 

  55. A. Fulinski, A.S. Kleczkowski, Nonlinear maps with memory. Physica Scripta 35 (1987), 119–122.

    Article  MathSciNet  MATH  Google Scholar 

  56. E. Fick, M. Fick, G. Hausmann, Logistic equation with memory. Phys. Rev. A 44 (1991), 2469–2473.

    Article  MathSciNet  Google Scholar 

  57. K. Hartwich, E. Fick, Hopf bifurcations in the logistic map with oscillating memory Phys. Lett. A 177 (1993), 305–310.

    Article  MathSciNet  Google Scholar 

  58. M. Giona, Dynamics and relaxation properties of complex systems with memory. Nonlinearity 4 (1991), 911–925.

    Article  MathSciNet  MATH  Google Scholar 

  59. J.A.C. Gallas, Simulating memory effects with discrete dynamical systems. Physica A 195 (1993), 417–430; Erratum. Physica A 198 (1993), 339–339.

    Google Scholar 

  60. A.A. Stanislavsky, Long-term memory contribution as applied to the motion of discrete dynamical system. Chaos 16 (2006), 043105.

    Article  MathSciNet  Google Scholar 

  61. G. Schmidt, Stochasticity and fixed-point transitions. Phys. Rev. A 22 (1980), 2849– 2854.

    Article  MathSciNet  Google Scholar 

  62. Y. Li, Y.Q. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59 (2010), 1810–1821.

    Article  MathSciNet  MATH  Google Scholar 

  63. M. Edelman, Cascade of bifurcation type trajectories in fractional dynamical systems submitted to Chaos.

    Google Scholar 

  64. G.M. Zaslavsky, M. Edelman, B.A. Niyazov, Self-Similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics. Chaos 7 (1997), 159–181.

    Article  MathSciNet  MATH  Google Scholar 

  65. V.E. Tarasov, Universal electromagnetic waves in dielectrics. J Phys.: Condens. Matter 20 (2008), 175223.

    Google Scholar 

  66. F. Hoppensteadt, A nonlinear renewal equation with periodic and chaotic solutions. SIAM-AMS Proc. 10 (1976), 51–60.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Stern College for Women at Yeshiva University, 245 Lexington Ave., New York, NY, 10016, USA

    Mark Edelman & Laura Anna Taieb

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  1. Mark Edelman
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  2. Laura Anna Taieb
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Correspondence to Mark Edelman .

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Editors and Affiliations

  1. , Departamento de Matemática, Universidade de Aveiro, Campus Universitário de Santiago, Aveiro, 3810-193, Portugal

    Alexandre Almeida

  2. , Departamento de Matemática, Universidade de Aveiro, Campus Universitário de Santiago, Aveiro, 3810-193, Portugal

    Luís Castro

  3. Instituto Superior Tecnico, Depto. Matematica, Universidade Tecnica Lisboa, Av. Rovisco Pais 1, Lisboa, 1049-001, Portugal

    Frank-Olme Speck

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Edelman, M., Taieb, L.A. (2013). New Types of Solutions of Non-linear Fractional Differential Equations. In: Almeida, A., Castro, L., Speck, FO. (eds) Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications, vol 229. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0516-2_8

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