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Dynamics of the fractional oscillator

Author

Listed:
  • Achar, B.N.Narahari
  • Hanneken, J.W.
  • Enck, T.
  • Clarke, T.

Abstract

The integral equation of motion of a simple harmonic oscillator is generalized by taking the integral to be of arbitrary order according to the methods of fractional calculus to yield the equation of motion of a fractional oscillator. The solution is obtained in terms of Mittag–Leffler functions using Laplace transforms. The expressions for the generalized momentum and the total energy of the fractional oscillator are also obtained. Numerical application and the phase plane representation of the dynamics are discussed.

Suggested Citation

  • Achar, B.N.Narahari & Hanneken, J.W. & Enck, T. & Clarke, T., 2001. "Dynamics of the fractional oscillator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 297(3), pages 361-367.
    • Handle: RePEc:eee:phsmap:v:297:y:2001:i:3:p:361-367
    DOI: 10.1016/S0378-4371(01)00200-X
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    Cited by:

    1. Liu, Q.X. & Liu, J.K. & Chen, Y.M., 2017. "An analytical criterion for jump phenomena in fractional Duffing oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 216-219.
    2. Berman, Michael & Cederbaum, Lorenz S., 2018. "Fractional driven-damped oscillator and its general closed form exact solution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 744-762.
    3. Mishra, Shalabh Kumar & Upadhyay, Dharmendra Kumar & Gupta, Maneesha, 2018. "An approach to improve the performance of fractional-order sinusoidal oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 116(C), pages 126-135.
    4. Balachandran, K. & Govindaraj, V. & Rivero, M. & Trujillo, J.J., 2015. "Controllability of fractional damped dynamical systems," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 66-73.
    5. Arthi, G. & Suganya, K., 2021. "Controllability of higher order stochastic fractional control delay systems involving damping behavior," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    6. Arthi, G. & Park, Ju H. & Suganya, K., 2019. "Controllability of fractional order damped dynamical systems with distributed delays," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 74-91.
    7. Rehman, Mujeeb ur & Idrees, Amna & Saeed, Umer, 2017. "A quadrature method for numerical solutions of fractional differential equations," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 38-49.
    8. Drozdov, A.D., 2007. "Fractional oscillator driven by a Gaussian noise," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 237-245.

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