**7. Asymptotics of Solutions in the Case** *b <* **0 and** *d* **= 0**

Firstly, consider initial conditions from *S*+. As in the previous section, in the interval *t* ∈ [0, *T*], the solution has the form of (25), and in the interval *t* ∈ [*T*, 2*T*], it has the form of (26).

If condition (27) holds, then this case is absolutely similar to the case in Section 6, and we obtain the following result.

**Theorem 4.** *Let b* < 0*, d* = 0*, and* (27) *holds. Then for all sufficiently large λ* > 0*, Equation* (2) *has a positive relaxation cycle with the asymptotics* (29) *and period t*<sup>∗</sup> = 2*T* + (1 + *o*(1))ln *λ at λ* → +∞*.*

If condition (30) is true, then there exists an asymptotically small by *λ* value *δ* > 0 such that *u*(*T* + *δ*) = *pL* and *u*(*t*) < *pL* in the interval *t* ∈ (*T* + *δ*, 2*T*]. That is why in the segment *t* ∈ [2*T* + *δ*, 3*T*], the equation has the form of (6), and the solution has the form of (31) in the segment *t* ∈ [2*T*, 3*T*]. One can easily see that under conditions *b* < 0 and (30), Expression (31) is less than *pL* for all *t* > 2*T*. This is why Equation (2) has the form of (6), and the solution has the asymptotics of (31) for all *t* > 3*T*.

Therefore, in the case that *b* < 0 and *d* = 0, if Condition (30) is true, then all solutions with initial conditions from *S*<sup>+</sup> tend to a constant *λb* at *t* → +∞.

Now, consider initial conditions from *S*−. Then, absolutely similarly as in Section 3, the solution has the asymptotics of (7) for all *t* > 0.

Thus, in the case that *b* < 0 and *d* = 0, all solutions with initial conditions from *S*<sup>−</sup> tend to a constant *λb* at *t* → +∞.
