Local dynamics of a second order equation with large exponentially distributed delay and considerable friction

We study local dynamics of a nonlinear second order differential equation with a large exponentially distributed delay in the vicinity of the zero solution under the condition $\gamma>\sqrt{2}$. The parameter $\gamma$ can be interpreted as a friction coefficient. We find such parameter values that critical cases in the stability problem are realized. We show that the characteristic equation for zero solution stability can have arbitrary many roots in the vicinity of imaginary axis. So, the critical case of an infinite dimension is realized. We construct normal forms analogues to describe dynamics of the origin equation. We formulate results about the correspondence of solutions of received PDE and second order DDE with a large exponentially distributed delay. The received asymptotic formulas allow us to evidently find characteristics of origin problem local regimes that are close to the zero solution and also to obtain domains of parameters and initial conditions, where the appearance of any given-type solution is possible.