A study of travelling, periodic, quasiperiodic and chaotic structures of perturbed Fokas–Lenells model

Abstract

In this paper, a diverse range of travelling wave structures of perturbed Fokas–Lenells model (p-FLM) is obtained by using the extended $({G{'}}/{G^{2}})$-expansion technique. The existence of the obtained solutions is guaranteed by reporting constraint conditions. Then, the governing model is converted into the planer dynamical system with the help of Gallelian transformation. Every possible form of phase portraits is plotted for pertinent parameters, viz. $k, \beta , d_{1}, d_{2}, d_{3}$. We also used the Runge–Kutta fourth-order technique to extract the nonlinear periodic solutions of the considered problem and outcomes are presented graphically. Furthermore, quasiperiodic and chaotic behaviour of p-FLM is analysed for different values of parameters after deploying an external periodic force. Quasiperiodic–chaotic nature is observed for selected values of parameters $k,\beta , d_{1}, d_{2}, d_{3}$ by keeping the force and frequency of the perturbed dynamical system fixed. The sensitive analysis is employed on some initial value problems (IVPs). It is seen that de-sensitisation is present in the perturbed dynamical system while for the same values of parameters, the unperturbed dynamical system has a nonlinear periodic solution.