Analytical solutions of nonlinear system of fractional-order Van der Pol equations

  • Shankar Rao MunjamEmail author
  • Rajeswari Seshadri


The double-well, in-phase and out-of-phase periodic solutions of the system of fractional-order Van der Pol equations and the exact solution of the nonlinear fractional-order Van der Pol equations with independent initial profiles are investigated in this paper. The influence of two main physical parameters such as angular frequency and the amplitude are included for the study. In addition, the difference between autonomous (i.e., \(f=1, g=M=0\)) and the non-autonomous (i.e., \(f=1, g\ne 0, M=0\)) nonlinear oscillators as well as the double-well VDPDO (i.e., \(f<0, g>0\)) cases is analysed. It is found that the variations in in-phase and out-of-phase periodic solutions and convergence rate strongly depend on the initial conditions with fractional orders. The effect of the physical parameters on phase portrait and the time history curves for various values of fractional orders are plotted and discussed.


Van der Pol equations Periodic solutions Fractional derivatives Phase portrait 



The first author Shankar Rao Munjam gratefully acknowledges the postdoctoral fellowship (Reg. No. 179838), Shanghai Jiao Tong University, Shanghai, China, for providing the financial assistance. The referee’s comments which led to an improvement of the paper are also acknowledged.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Naval Architecture, Ocean and Civil EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Department of MathematicsPondicherry UniversityPondicherryIndia

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