**10. Discussion and Conclusions**

If we fix the values of *b* and *d*, function *f* , and the set of initial conditions (*S*+ or *S*−), then for all initial conditions from the chosen set, we obtain an identical behaviour at *t* → +∞ (because all solutions with initial conditions from the set *S*<sup>+</sup> (or *S*−) coincide with each other in the segment *t* ∈ [0, *T*], and, therefore, for all *t* ≥ 0).

In Sections 2–9 we derived that the behaviour at *t* → +∞ of the solutions to Equation (2) with the initial conditions from sets *S*<sup>+</sup> and *S*<sup>−</sup> may be only of two types: (1) solutions tend to a constant at *t* → +∞ or (2) we obtain a cycle.

The following generalization of this result takes place.

**Theorem 8.** *If we replace the equality u*(0) = *pR (u*(0) = *pL) with the inequality u*(0) ≥ *pR (u*(0) ≤ *pL, respectively) in the definition of the set S*<sup>+</sup> *(or S*−*, respectively), then the behaviour of the solutions at t* → +∞ *does not change.*

Theorem 8 means that if a solution with initial conditions from *S*+ tends to a constant at *t* → +∞, then a solution with initial conditions satisfying inequality *u*(*s*) ≥ *pR* for all *s* ∈ [−*T*, 0] tends to the same constant at *t* → +∞; if we take initial conditions from *S*<sup>+</sup> and obtain a cycle, then taking initial conditions satisfying inequality *u*(*s*) ≥ *pR*, we get the same cycle (but it may be shifted).

The same result is valid for the set *S*−.

**Proof.** Let us prove that if in the definition of *S*+, we replace equality *u*(0) = *pR* with inequality *u*(0) > *pR*, then the behaviour of the solution does not change.

Denote *u*(0) as *u*0. Since for all *s* ∈ [−*T*, 0], Inequality *u*(*s*) ≥ *pR* holds, then Equation (2) has the form of (4) on the segment *t* ∈ [0, *T*], and the solution has form

$$
\mu\_+(t) = \mu\_0 e^{-t} + \lambda d(1 - e^{-t}).\tag{47}
$$

Two situations are possible:

(1) There exists a time moment *t*<sup>0</sup> > 0 such that expression (47) is greater than *pR* for all *t* ∈ [0, *t*0) and is equal to *pR* at *t* = *t*0;

(2) For all *t* > 0, Expression (47) is greater than *pR*.

If the first situation occurs, then the function *u*+(*t*<sup>0</sup> + *s*) (*s* ∈ [−*T*, 0]) belongs to the set *S*+. All solutions with initial conditions from *S*+ for fixed values *b* and *d* and function *f* have the same behaviour, which is why, in this case, for the considered initial conditions, we have the same behaviour of solutions as for the initial conditions from *S*+.

The second situation is possible only in the case that *d* > 0 (for all *d* ≤ 0 and *u*<sup>0</sup> > *pR*, there exists *t*<sup>0</sup> > 0 such that *u*+(*t*0) = *pR*). In this situation, for all *t* ≥ 0, Equation (2) has the form of (4), and the solution has the form of (47) for all *t* ≥ 0. Expression (47) tends to *λd* at *t* → +∞. Since in all cases where *d* > 0, the solutions with initial conditions from *S*<sup>+</sup>

tend to *λd* at *t* → +∞ (see Sections 2, 4 and 9), then in this situation, for the considered initial conditions, we have the same behaviour of solutions as for initial conditions from *S*+.

The proof of the Theorem for set *S*<sup>−</sup> is absolutely similar as the proof for the set *S*+.

We have studied the nonlocal dynamics of an equation with delay and nonlinearity having simple behaviour at infinity. This type of nonlinearity is interesting because, on one hand, it is a quite general class of functions, and on the other hand, it is a generalization of two important for application types of nonlinearity: compactly supported and piecewise constant nonlinearities. The key assumption that the nonlinear function *F* is multiplied by a large parameter *λ* allows us to construct the asymptotics of all the solutions from the wide sets of initial conditions.

We have studied behaviour at *t* → +∞ of the solutions to (2) for wide sets of initial conditions and conclude that two types of behaviour are possible: (1) the solution tends to a constant or (2) after the pre-period, the solution becomes a cycle.

It is important to mention that it is impossible to obtain such general results using numerical simulation because it is impossible to iterate through all the considered functions *F* and initial conditions. Additionally, even if we take a certain function *F* and initial conditions, the simulation of this equation is a difficult problem, because the parameter *λ* is large.

We have found conditions on signs *b* and *d* under the condition that *bd* = 0 for having a cycle of Equation (2). This cycle has an amplitude of the order *O*(*λ*) and period of the order *O*(1) at *λ* → +∞. We have found conditions on sign *b* (*d*) under condition *d* = 0 (*b* = 0, respectively) for having relaxation cycles of Equation (2). Depending on the properties of the function *f* , this cycle may be sign-changing or sign-preserving.

It is important to mention that most found cycles (see Theorems 1, 3, 6) do not exist in the case of compactly supported nonlinearity [20].

In the future, it will be interesting to study the dynamics of several coupled Equation (2) and to analyse the dependence of the dynamics of the system on the type of coupling.

**Funding:** This research was funded by The Council on Grants of the President of the Russian Federation grant number MK-2510.2022.1.1.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The author declares no conflict of interest. The funder had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
