**Corollary 3.** *On the time segment t* ∈ [*t*<sup>1</sup> + 2*T*, *t*2]*, a solution of system* (3) *has form* (12)*.*

It follows from Formulas (24) and (25) that we obtain an operator of translation along the trajectories that map our set of initial conditions *IC*(*i*, *k*, *x*) to a set *IC*(¯*i*, ¯ *k*, *x*¯). Thus, at the point *t*2, we return to the initial situation with replacement *k*, *x*, *i*, and *t*<sup>1</sup> by ¯ *k*, *x*¯, ¯*i*, and *t*2. If we do the same steps as in this section and in all the next iterations, Assumptions 1 and 2 and non-degeneracy condition (17) in the case *<sup>γ</sup>* <sup>&</sup>gt; 0 (non-degeneracy condition (20) in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0, respectively) hold (with new values *k* = *kn*, *x* = *xn*, *i* = *in* and replacing *t*<sup>1</sup> with *tn* (*n* = 2, 3, ...)), then, from an operator of translation along the trajectories, we obtain a map on *in*, *kn*, and *xn*. This map determines

dynamics of the system (3) because on the segments *t* ∈ [*tn*, *tn*+1] solution satisfies Formulas (9), (10) amd (12) with *i* = *in*, *k* = *kn*, *x* = *xn*, *t*<sup>1</sup> = *tn*.

In the next two sections, we construct an exact form of maps on *i* = *in*, *k* = *kn*, and *x* = *xn* in the case *<sup>γ</sup>* <sup>&</sup>gt; 0 (see Section 3) and in the case <sup>−</sup><sup>1</sup> <sup>2</sup> < *γ* < 0 (see Section 4) and using dynamical properties of these maps clarify asymptotics of solution on the intervals *t* ∈ [*tn*, *tn*+1] (*n* = 2, 3, . . .).
