Stability analysis of Abel’s equation of the first kind

: This paper established su ffi cient conditions for the stability of Abel’s di ff erential equation of the first kind. These conditions explicate the impact of the asymptotic behaviors exhibited by the time-varying coe ffi cients on the overall stability of the system. More precisely, we studied the positivity, the continuation and the boundedness of solutions. Additionally, we investigated the attractivity, the asymptotic stability, the uniform stability and the instability of the system. The results were scrutinized by numerical simulations.


Introduction
In the presence of an additional cubic nonlinearity, the Abel's first-order differential equation bears resemblance to Riccati equations and, consequently, it has the form ẏ (t) = a (t) y 3 (t) + b (t) y 2 (t) + c (t) y (t) + d (t) , where t ≥ t 0 ∈ R, the solution is y(t) ∈ R, and the continuous real functions are a, b, c, d ∈ C 0 (R, R).It is noteworthy that when d is uniformly zero and exactly one of the functions a and b is uniformly zero as well, (1.1) transforms into a Bernoulli equation.The system's widespread importance originates from its significance in numerous applications in physics [13,17], fluids [19], control theory [28], finance [33], cosmology [8], cancer therapy [10], biology [9] and the M-theory [37].Owing to this relevance, numerous mathematical characteristics of the system's states have been studied in the literature.For instance, various analytical solutions have been obtained in [16,18,[29][30][31] under restricted conditions.Due to the nonlinearity of the system, it is arduous to generalize these restricted forms.Therefore, many researchers have chosen to numerically solve the system as in [3][4][5]32].The presence of limit cycles in certain classes of the system has also been conducted in [14,27].
Stability analysis is an essential approach across a wide spectrum of fields within control theory [2,6,24,35] and the qualitative theory of differential equations [11,26,34].This analysis opens the routes for additional knowledge on the behavior patterns of many real-world nonlinear systems [12,15,20,21,36].Despite the aforementioned importance of stability analysis in the various fields, there is no existing study in the literature; according to our knowledge, that considers the stability of the celebrated equation (1.1).Therefore, we have dedicated this study to explore that topic.To delve further into the specifics, we focus on the positivity and boundedness of solutions and study some asymptotic behaviors, including uniform stability, attractivity, asymptotic stability and instability of the system in both its homogeneous and nonhomogeneous forms.For the prior notions, we have obtained precise conditions that are related to the signs and asymptotic behaviors of the time-varying coefficients.To demonstrate the proposed results, numerical simulations have been conducted.
The paper is organized as follows.A compilation of mathematical results obtained from the literature has been presented in section two.Conditions for the positivity of solutions have been introduced in section three.The case when d is uniformly zero has been considered in section four in which the instability and the asymptotic stability are investigated.Section five provides sufficient conditions for the origin attractivity and the state convergence of the system.The conclusion section is included at the end of the paper.

Background results
For a Lebesgue measurable function q : R + → R m , let ∥q∥ ∞ be the essential supremum of |q| on R + where | • | is the Euclidean distance [1].A strictly increasing function γ ∈ C 0 (R + , R + ) is of class K if γ(0) = 0.It belongs to class K ∞ when we have lim s→∞ γ (s) = ∞.Consider the n-dimensional differential equation u (t) = f (t, u (t)), t ≥ t 0 .We assume the continuity of the function f .Furthermore, we assume that f (t, 0) = 0 for every t ≥ t 0 (hence, the origin is an equilibrium point).The origin is uniformly stable if there is some γ of class K and a positive number c; that is independent of t 0 , such that for every initial value with |u 0 | < c, each solution is continuable on [t 0 , ∞) and |u (t)| ≤ γ (|u 0 |) for all t ≥ t 0 .The origin is locally attractive if for every t 0 ∈ R, there is some c > 0; that may depend on t 0 , such that for every |u 0 | < c, each solution is continuable on [t 0 , ∞) with lim t→∞ u (t) = 0.If the prior conditions are satisfied for every u 0 ∈ R m , the origin is globally attractive.On the other hand, the zero solution is asymptotically stable if it is stable and attractive [2].
In the presence of unbounded perturbations, the asymptotic stability of differential equations-based systems is studied in the next theorem.
Then, there exists r > 0 such that for any |y(t 0 )| < r, each solution y(t) is continuable on [t 0 , ∞) and is bounded with |y (t)| ≤ σ c 2 c 1 |y (t 0 )| for every t ∈ [t 0 , ∞) (so that the origin is uniformly stable).Additionally, the origin exhibits asymptotic stability.Lemma 2.1.[22,Theorem 6.1]In addition to the results of the previous theorem, the mapping t → V (t, u (t)) is monotonically decreasing.

Positivity of solutions
Since all of the functions a, b, c and d are continuous, the existence of a continuously differentiable solution of (1.1) is guaranteed.This solution has a maximal interval of existence of the form [t 0 , ω) where ω can be infinite [7].Now, we introduce several conditions for the positiveness of solutions.
Proposition 3.2.Suppose that d(t) ≥ 0 for every t ≥ t 0 .If one of the following sets of conditions is satisfied and y (t 0 ) > 0. Furthermore, suppose that at least one of the functions b and c is nonzero for each t ∈ [t 0 , ∞).
Then, the positivity of solutions is guaranteed.Proof.We show each case individually.
Applying the Lyapunov stability technique on the system under study in its homogeneous form may lead to a differential Lyapunov inequality with an unbounded perturbation.The next theorem proves that; even with the existence of unbounded perturbations, the system is still able to exhibit notions like uniform stability, asymptotic stability and instability depending on the asymptotic properties of the coefficient functions.
Theorem 4.1.We consider the following two distinct results: Then, there is some r > 0 such that when |y (t 0 )| < r, each solution y(t) is continuable on [t 0 , ∞).
for each y(t 0 ) ∈ R, each nontrivial solution y(t) is global with lim t→∞ |y (t)| = ∞ (observe that in this case, (1.1) reduces to a Bernoulli equation).
Proof.Since for every t > t 0 at least one of a(t) and b(t) is nonzero, we have λ (t) > 0 for each t > t 0 .Let V(t) = y2 (t) for each t ∈ [t 0 , ω).We show each case individually.

Subcase (ii-2):
There exists t * > t 0 in a way that V (t) ≤ Ψ (t) for every t ≥ t * .We conclude by (4.4) and the fact that a (•) < 0 that V (t) ≥ 0 for almost all t > t * so that V is nondecreasing.Since y(t) is not the trivial solution, the Lyapunov function is not uniformly zero.Thus, there exists t * > t 0 such that V(t * ) > 0 and, hence, V (t) ≥ V (t * ) for all t ≥ t * .Thus, (4.4) gives V (t) ≥ −2a (t) V 2 (t * ) for all t > t * (noting that a(•) < 0).Therefore, the Fundamental Theorem of Calculus leads to Thus, we get lim t→∞ V (t Subcase (ii-3): Both Subcases (ii-1) and (ii-2) are incorrect.Given ε > 0, the fact that lim t→∞ Ψ (t) = ∞ ensures the existence of T 0 > t 0 such that Ψ (t) > ε for every t ≥ T 0 .In accordance with the present subcase, there must be {t m ≥ T 0 } ∞ m=1 and t ′ m ≥ T 0 ∞ m=1 such that lim m→∞ t m = lim m→∞ t ′ m = ∞ and for each positive integer m, we have This guarantees the presence of two numbers t 1 > T 0 and t 2 > T 0 satisfying t 1 < t 2 , Ψ (t 1 ) < V (t 1 ) and Ψ (t 2 ) > V (t 2 ).By applying the intermediate value theorem on the continuous function χ := V − Ψ and the compact interval [t 1 , t 2 ], we conclude that there exists some T 1 ∈ (t 1 , t 2 ) such that χ (T 1 ) = 0 so that Ψ (T 1 ) = V (T 1 ).We claim that V (t) > ε for every t > T 1 .To prove it, we use the contradiction technique and assume that there is some } is nonempty.Thus, the continuity of V and Ψ imply that ).Therefore, one can verify that V (t) < Ψ (t) for all t ∈ (T 3 , T 2 ).Thus, we get by (4.4) and the fact that a (•) < 0 that V (t) ≥ 0 for all t ∈ (T 3 , T 2 ).This means V is nondecreasing on (T 3 , T 2 ) and, hence, V (T 2 ) ≥ V (T 3 ).Thus, the facts that Ψ (t) > ε, for every t ≥ T 0 , T 3 ∈ S and V (T 2 ) ≤ ε give the contradicted statement ).This finishes the proof of our claim, which states that lim t→∞ V (t) = ∞ so that lim For the case a(t) = t 2 and c(t) = −t for every t ≥ t 0 = 1, we have a(•) 0, c(•) < 0 and so that condition (4.1) is satisfied.Therefore, by Result (i) of Theorem 4.1, we conclude that there is some r > 0 such that when |y (t 0 )| < r, every solution is global and bounded with |y (•)| < |y (t 0 )|, y = 0 exhibits uniform stability and asymptotic stability and the mapping t → |y (t)| is strictly decreasing.This is shown in Figure 2.
For the case a(t) = −t and c(t) = t 2 for all t ≥ t 0 = 1, note that a(•) < 0, c(•) > 0 and b(•) = 0 and, thus, the positivity of solutions is guaranteed for all y(t 0 ) ∈ R by Item (iii) of Proposition 3.2.We have   1, these simulations have been created incorporating the conditions stated in Result (i), which includes the assumption c(•) < 0. Take note that each solution y(t) is continuable on [t 0 , ∞) and converges to zero as t goes to infinity.Moreover, the mapping t → |y (t)| is monotonically decreasing so that |y (t)| < |y (t 0 )| for every t ≥ t 0 .This empathizes the uniform stability and the asymptotic stability, even though this particular case is associated with a Lyapunov inequality involving an unbounded perturbation.  .y(t) versus t for the initial value five.This simulation has been created based on the conditions outlined in Result (ii) of Theorem 4.1, including the assumption a(•) < 0. Observe that the nontrivial solution y(t) is continuable on [t 0 , ∞), nonnegative and diverges to infinity.

Sufficient conditions for the attractivity
The next theorem introduces two sets of conditions for the local and global attractivity of the nonhomogeneous differential equation under study where the system has been transformed into linear and nonlinear nonautonomous differential Lyapunov inequalities with vanishing perturbations.Theorem 5.1.We introduce the following separate results: Furthermore, assume that either lim t→∞ c(t) a(t) = 0 or c (t) < 0 for all t ≥ t 0 , then for every initial value y(t 0 ) ∈ R, each solution y(t) of (1.1) is continuable on [t 0 , ∞) with lim t→∞ y (t) = 0 (so that the origin is globally attractive).(ii) Assume that c (t) < 0 for all t ≥ t 0 , and the initial time t 0 is sufficiently large to satisfy Then there is some r > 0 such that for every |y(t 0 )| < r, each solution y(t) of (1.1) is continuable on [t 0 , ∞) with |y(•)| < r and lim t→∞ y (t) = 0 (so that the origin is locally attractive).Additionally, it is worth mentioning that if a (t) < 0 for every t ≥ t 0 , the conditions (5.2) and ( 5.3) can be relaxed to be lim t→∞ b(t) Proof of Result (i).Consider the case lim t→∞ c(t) a(t) = 0. We get by (1.1) that for all t ∈ (t 0 , ω): where e (t) = 2 max (|b (t)| , |c (t)| , |d (t)|) for each t ≥ t 0 .Given an initial value y(t > 0 so that V(t 0 ) < δ and 3e(t) 2|a(t)| 4 < δ for every t ≥ t 0 and, hence, We conclude by (5.4) that and, thus, using the fact δ > 1 gives so that (see (5.5)) V (t) ≤ 0, for all t ∈ (t 0 , ω) that satisfies 3e (t) Therefore, since V (t 0 ) < δ, we conclude that all assumptions of Proposition 2.1 are fulfilled with 2|a(t)| and g 2 (t) = δ for every t ≥ t 0 .Therefore, V(t) < δ for all t ∈ [t 0 , ω) and thus |y (•)| < r := √ δ.Furthermore, (5.6) implies , for all t ∈ (t , ω) .
Consider the differential equation v (t) = 2a (t) v 2 (t)+3δ 3 2 e (t).We have lim t→∞ e(t) a(t) = 0 by (5.1) and the definition of the function e.Since we have a (t) < 0 for all t ≥ t 0 and As a result, the classical comparison principle along with Theorem 2.2 guarantee that for any y(t 0 ) ∈ R, each solution y(t) is global and the origin is globally attractive.
The system v (t) = 2c (t) v (t) + 3δ 1 2 e (t) has the form of the system mentioned in Theorem 2.2 with 1 2 e (•), and β is the identity mapping.We have Q (t) > 0 because c (t) < 0 for all t ≥ t 0 .Note that by (5.2) and the definition of the function e, one has lim t→∞ E(t) All premises of Theorem 2.2 are fulfilled.Therefore, a comparison principle can show that V (•) ≤ v (•) so that y(t) is global and the zero solution is locally attractive.
Finally, if a (t) < 0 for every t ≥ t 0 , we replace the conditions (5.2) and ( 5. < 1 2 , respectively.For all t ≥ t 0 , equation (1.1) leads for all t ∈ (t 0 , ω) to (5.12) Let δ be a positive number such that Thus, assumption (5.1) is satisfied.Therefore, we have by Result (i) of Theorem 5.1 that for every y(t 0 ) ∈ R, each solution y(t) of (1.1) is global and the origin is globally attractive.This is illustrated in Figure 4.In the subsequent lemma, we utilize Theorem 5.1 to deduce conditions for the convergence of the system's state.
Furthermore, we assume that either lim t→∞ c(t) Then, for every initial value y(t 0 ) ∈ R, each solution y(t) of (1.1) is globally defined and lim t→∞ y (t) = L.
Second; for the case 3L  Thus, assumption (5.13) is satisfied with L = 2 (note that lim t→∞ c(t) a(t) = 3L 2 ) and, hence, Lemma 5.1 guarantees that for any initial value y(t 0 ) ∈ R, the solution y(t) of (1.1) is global and lim t→∞ y (t) = L = 2, as shown in Figure 5. Figure 5.This simulation provides clarity that the solution y(t) is global, nonnegative and converges to L = 2 as t goes to infinity.

Conclusions
Stability analysis of Abel's differential equation of the first kind has been conducted.More precisely, conditions for the positivity of solutions have been derived in Propositions 3.1 and 3.2.Additionally, it has been clarified in section four that applying the Lyapunov technique to the homogenous form of the equation may give a differential inequality with an unbounded perturbation.For the prior case, Theorem 4.1 has derived sufficient conditions for the continuation and boundedness of solutions, the uniform stability, the asymptotic stability and the instability of the equation.In addition, the local/global origin attractivity has been investigated in Theorem 5.1.Based on the results of the aforementioned theorem, conditions have been provided for the state convergence in Lemma 5.1.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Figure 2 .
Figure 2. y(t) versus t for the initial values ±0.2, ±0.4,±0.6, ±0.8.Resulting from Theorem 4.1, these simulations have been created incorporating the conditions stated in Result (i), which includes the assumption c(•) < 0. Take note that each solution y(t) is continuable on [t 0 , ∞) and converges to zero as t goes to infinity.Moreover, the mapping t → |y (t)| is monotonically decreasing so that |y (t)| < |y (t 0 )| for every t ≥ t 0 .This empathizes the uniform stability and the asymptotic stability, even though this particular case is associated with a Lyapunov inequality involving an unbounded perturbation.

Figure 3
Figure 3. y(t) versus t for the initial value five.This simulation has been created based on the conditions outlined in Result (ii) of Theorem 4.1, including the assumption a(•) < 0. Observe that the nontrivial solution y(t) is continuable on [t 0 , ∞), nonnegative and diverges to infinity.

Figure 4 .
Figure 4.The positivity of the global solution and the attractivity of the origin can be readily observed.