Anharmonic effects on the dynamic behavior’s of Klein Gordon model’s

•Theoretical nonlinear Klein Gordon model with anharmonic, cubic and quartic interactions between nearest neighbor particles immersed in a parametrized on-site substrate potential is presented.•Illustration of theoretical model by introducing a discrete nonlinear electrical transmission line.•The mathematical model which derive is a new class of differential equations possessing several key parameters and ranging from many singular straight lines.•Dynamical system methods are used to discuss bifurcations of phase portraits and vector fields for each orbit of phase portraits with corresponding conditions.•All possible exact parametric representations of solutions are compute and phenomena such as the formation of cracks originating from dislocations that are observed in semiconductor heterostructures can now have a beginning of analytical explanations. This work completes and extends the Ref. Tchakoutio Nguetcho et al. (2017), in which we have focused our attention only on the dynamic behavior of gap soliton solutions of the anharmonic Klein-Gordon model immersed in a parameterized on-site substrate potential. We expand our work now inside the permissible frequency band. These considerations have crucial effects on the response of nonlinear excitations that can propagate along this model. Moreover, working in the allowed frequency band is not only interesting from a physical point of view, it also provides an extraordinary mathematical model, a new class of differential equations possessing vital parameters and vertical singular straight lines. The dynamics around these singularities gives new informations of great interest, such as to better understand the break up (rupture) of a stretched polymer chain by pulling and its relation to soliton destruction and which until now had no mathematical explanations yet. The mathematical model of the differential equation we obtain corresponds, as an equivalent experimental model, to the nonlinear transmission electrical line introduced in 2009 in Ref. Yemélé and Kenmogné(2009), whose equations were corrected but not entirely resolved in 2016 in Ref. Yamgoué and Pelap(2016). Our dynamic study thus presents a theoretical prediction for their experimental set up. An extensive account of the bifurcation theory for modulated-wave propagation is given. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we not only make a total inventory of the solutions that can include