Finding the closed-form solutions of dissipative oscillatory systems

Saba Irum; Imran Naeem

doi:10.1038/s41598-022-08418-3

Finding the closed-form solutions of dissipative oscillatory systems

Sci Rep. 2022 Mar 22;12(1):4825. doi: 10.1038/s41598-022-08418-3.

Authors

Saba Irum^#¹, Imran Naeem^#²

Affiliations

¹ Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan. saba.irum@lums.edu.pk.
² Department of Mathematics, Lahore University of Management Sciences, Lahore, Pakistan.

^# Contributed equally.

Abstract

This paper shows how to use the approximate Hamiltonian approach for the non-conservative system not capable of possessing Hamiltonian. Using the approximate Hamiltonian method for a non-conservative system is not possible in general. We propose a way to obtain the closed-form solutions for such systems. We use the approximate dual Hamiltonian method to construct the first integrals and closed-form solutions of the Van der Pol equation. First the solutions of the initial value VdP equation is obtained using approximate dual Hamiltonian method. Then a good agreement is observed in the comparison between the numerical results and the results through approximate dual Hamiltonian method. Finally, we use the approximate dual Hamiltonian method to find the dual Hamiltonian and first integrals of the forced Van der Pol oscillator and Liénard system. These significant results can be applied to any Van der Pol equation.