Stability and Trajectories Analysis of a Fractional Generalization of Simple Pendulum Dynamic Equation

In this paper, we present the dynamics of the simple pendulum by using the fractional-order derivatives. Equations of motion are proposed for cases without input and external forcing. We use the fractional-order Euler-Lagrange equations to obtain the fractional-order dynamic equation of the simple pendulum. We perform equilibria analysis, indicate the conditions where stability dynamics can be observed for both integer and fractional-order models. Finally, phase diagrams have been plotted to visualize the effect of the fractional-order derivatives.

[1]  Igor Podlubny,et al.  Mittag-Leffler stability of fractional order nonlinear dynamic systems , 2009, Autom..

[2]  I. Ozkol,et al.  Classical and Fractional-Order Analysis of the Free and Forced Double Pendulum , 2010 .

[3]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[4]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[5]  D.Baleanu,et al.  Lagrangians with linear velocities within Riemann-Liouville fractional derivatives , 2004 .

[6]  Ivo Petras,et al.  A note on the fractional-order Volta’s system , 2010 .

[7]  Xavier Moreau,et al.  An overview of the CRONE approach in system analysis, modeling and identification, observation and control , 2008 .

[8]  Dumitru Baleanu,et al.  FRACTIONAL-ORDER TWO-ELECTRIC PENDULUM , 2012 .

[9]  Weihua Deng,et al.  Short memory principle and a predictor-corrector approach for fractional differential equations , 2007 .

[10]  Mohammad Saleh Tavazoei,et al.  Limitations of frequency domain approximation for detecting chaos in fractional order systems , 2008 .

[11]  Dumitru Baleanu,et al.  On exact solutions of a class of fractional Euler–Lagrange equations , 2007, 0708.1433.

[12]  I. Podlubny.  Fractional differential equations , 1998 .

[13]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[14]  Om P. Agrawal,et al.  Formulation of Euler–Lagrange equations for fractional variational problems , 2002 .

[15]  Denis Matignon,et al.  Generalized fractional di erential and di erence equations : stability properties and modelling issuesDenis , 1998 .

[16]  Dumitru Baleanu,et al.  Fractional Bateman—Feshbach Tikochinsky Oscillator , 2014 .

[17]  Yangquan Chen,et al.  Computers and Mathematics with Applications Stability of Fractional-order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–leffler Stability , 2022 .

[18]  A. Hanyga,et al.  Nonlinear differential equations with fractional damping with applications to the 1dof and 2dof pendulum , 2005 .

[19]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[20]  L. Dorcak.  Numerical Models for the Simulation of the Fractional-Order Control Systems , 2002 .

[21]  D.Baleanu,et al.  Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives , 2005, hep-th/0510071.

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

[23]  Mathieu Moze,et al.  On stability of fractional order systems , 2008 .

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

[25]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[26]  Saptarshi Das,et al.  Intelligent Fractional Order Systems and Control - An Introduction , 2012, Studies in Computational Intelligence.