NON-AUTONOMOUS BIFURCATION IN IMPULSIVE SYSTEMS

This is the first paper which considers non-autonomous bifurcations in impulsive differential equations. Impulsive generalizations of the non-autonomous pitchfork and transcritical bifurcation are discussed. We consider scalar differential equation with fixed moments of impulses. It is illustrated by means of certain systems how the idea of pullback attracting sets remains a fruitful concept in the impulsive systems. Basics of the theory are provided. Asymptotic behavior of fixed points and analysis of bifurcation is of great importance in the qualitative theory of differential equations. In autonomous ordinary differential equations this theory is well developed. As in the autonomous systems, non-autonomous bifurcation is understood as a qualitative change in the structure and stability of the invariant sets of the system. However, to implement this concept in non-autonomous systems, locally defined notions of attractive and repulsive solutions are needed. There are currently qualitative studies which are devoted to non-autonomous bifurcation theory by treating attractors called pullback attractors [11, 12, 23, 25, 26, 29, 31, 35]. The theory of pullback attraction is not concerned with the asymptotic behavior of the solution as t → ∞ for fixed t0, but as t0 → −∞ for fixed t [8,11,13,15–18,25,28,30,32,33]. This approach requires the discussion of bifurcation in non-autonomous differential equations by defining various types of stability and instability. Investigation of states of dynamical systems which are not constant in time leads to non-autonomous problems in the form of the equation of perturbed motion. If this model depends on parameters, it is the 2010 Mathematics Subject Classification. 34A37, 34C23, 34D05, 37B55, 37G35, 34D45, 37C70, 37C75.

Asymptotic behavior of fixed points and analysis of bifurcation is of great importance in the qualitative theory of differential equations.In autonomous ordinary differential equations this theory is well developed.As in the autonomous systems, non-autonomous bifurcation is understood as a qualitative change in the structure and stability of the invariant sets of the system.However, to implement this concept in non-autonomous systems, locally defined notions of attractive and repulsive solutions are needed.There are currently qualitative studies which are devoted to non-autonomous bifurcation theory by treating attractors called pullback attractors [11,12,23,25,26,29,31,35]. The theory of pullback attraction is not concerned with the asymptotic behavior of the solution as t → ∞ for fixed t 0 , but as t 0 → −∞ for fixed t [8, 11, 13, 15-18, 25, 28, 30, 32, 33].This approach requires the discussion of bifurcation in non-autonomous differential equations by defining various types of stability and instability.
Investigation of states of dynamical systems which are not constant in time leads to non-autonomous problems in the form of the equation of perturbed motion.If this model depends on parameters, it is the non-autonomous differential equations with impulses.We present three systems which illustrate the given definitions.The first system (Section 3) is the impulsive extension of a non-autonomous pitchfork bifurcation, ẋ = a(t)x − b(t)x 3 , ( In Theorem 1 we have obtained impulsive extension for the results of Caraballo and Langa in [11] and Langa et al. in [29].Next, in Section 4, we investigate the non-autonomous transcritical bifurcation in the impulsive system ẋ = a(t)x − b(t)x 2 , In particular, in Theorem 2 and Theorem 3, we give impulsive extension for results of Langa et al. in [31] for equation (2).Finally, in Section 5 we consider bifurcation in the non-order-preserving system ẋ = a(t)x − b(t)x 3 , ( In the conclusion part, we summarize the results and consider how the theory might be further developed in a systematic way.

Preliminaries
In this section we introduce concepts of attractive and repulsive solutions, which are used to analyze asymptotic behavior of impulsive nonautonomous systems.This paper is concerned with systems of the type ẋ = f (t, x), where ∆x| (6).In this paper, we treat only scalar impulsive differential equations such that ϕ(t, t 0 , x 0 ) is continuable on R. Solutions are unique both forwards and backwards in time and EJQTDE, 2014 No. 74, p. 3 J i (x) is order-preserving so that the whole system ( 6) is order-preserving, i.e., x 0 > y 0 ⇒ x(t, t 0 , x 0 ) > y(t, t 0 , y 0 ) for all t, t 0 ∈ R allowing x(t) or y(t) to be ±∞ if necessary.We say that the function ϕ : R → R n is from the set P C(R, θ), where • ϕ is left continuous on R; • it is continuous everywhere except possibly points of θ where it has discontinuities of the first kind.Developing the theory for non-autonomous impulsive differential equations by following the same route as for autonomous systems poses a problem.Indeed, for generic non-autonomous system we would not expect to find any fixed points: if x 0 is the fixed point, then this would require that f (x 0 , t) = 0 and J i (x 0 ) = 0 for all i ∈ Z and t ∈ R. Instead, we replace fixed points to the notion of a complete trajectory.The piecewise continuous map x : R → G is said to be a complete trajectory if X(t, t 0 )x(t 0 ) = x(t) for all t, t 0 ∈ I where X(t, t 0 ) is the solution operator for (6).We investigate appearances and disappearances of complete trajectories that are stable and unstable in the pullback sense.Note that complete trajectories are particular examples of invariant sets.A time varying family of sets Σ(t) is invariant if ϕ(t, t 0 , Σ(t 0 )) = Σ(t) for all t, t 0 ∈ R.That is, if x(t 0 ) ∈ Σ(t 0 ), then ϕ(t, t 0 , x(t 0 )) ∈ Σ(t).In order to study non-autonomous bifurcation with impulses we should define corresponding concepts of stability.In this paper, we use Hausdorff semi-distance between sets A and B as dist(A, B) = sup a∈A inf b∈B d(a, b) 1.1.Attraction.Asymptotic properties of continuous dynamics and dynamics with discontinuity are the same.Therefore, we shall use notion of pullback attracting sets without any change from [8, 11, 13, 15-18, 24, 25, 28, 30, 32, 33, 35].In autonomous system, an invariant set Σ is attracting if there exists a neighborhood N of Σ such that dist(ϕ(t, 0, x 0 ), Σ) → 0 as t → ∞ for all x 0 ∈ N (7) where initial time is not important, we may take it arbitrary.For this case it is true that X(t, t 0 ) = X(t − t 0 , 0).The concept of attraction for EJQTDE, 2014 No. 74, p. 4 autonomous systems is equivalent to the existence of a neighborhood N of Σ for each fixed t ∈ R, dist(ϕ(t, t 0 , x 0 ), Σ) → 0 as t 0 → −∞ for all x 0 ∈ N. (8) This is the idea of pullback attraction [24,33], which does not involve running time backwards.Instead, we consider taking measurements in an experiment now (at time t) which began at some time in the past (at time t 0 < t).That is, we are interested in asymptotic behavior as t 0 → −∞ for fixed t.
Pullback attraction is a natural tool to study non-autonomous systems because it provides us to consider asymptotic behavior without having to consider sets Σ(t) that are moving, since final time is fixed.This approach has many applications in stochastic differential equations [17,18], ODEs [24,25] and PDEs [14,16,32].Definition 1. [24] An invariant set Σ(•) is called (locally) pullback attracting if for every t ∈ R there exists a δ(t) > 0 such that if It is crucial that δ is not allowed to depend on t 0 , otherwise every invariant set would be pullback attracting due to continuous dependence on initial conditions.If lim t 0 →−∞ dist(ϕ(t, t 0 , x 0 ), Σ(t)) = 0 for every t ∈ R and every x 0 ∈ R n then Σ(•) is said to be globally pullback attracting.1.2.Stability.The above discussion helps to define asymptotic stability, which has two parts.One of them is attraction and another one is stability.In this part, we define stability in non-autonomous case in the pullback sense.Definition 2. [29] An invariant set Σ(•) is pullback stable if for every t ∈ R and ϵ > 0 there exists a δ(t) > 0 such that for any t 0 < t, x 0 ∈ N (Σ(t 0 ), δ(t)) implies that ϕ(t, t 0 , x 0 ) ∈ N (Σ(t), ϵ).
An invariant set Σ(•) is said to be locally (globally) pullback asymptotically stable if it is pullback stable and locally (globally) pullback attracting.As in the scalar non-autonomous differential equations, pullback EJQTDE, 2014 No. 74, p. 5 attraction implies pullback stability for complete trajectories of scalar impulsive systems.Lemma 1. [31] Let y(t) be a complete trajectory in a non-autonomous scalar impulsive differential equation that is locally pullback attracting; then, this trajectory is also pullback stable.
The proof of this lemma, given by Langa et al. in [31], is the same for impulsive systems.This lemma allows us to consider only pullback attraction properties of complete trajectories rather than their pullback stability properties.
1.3.Instability.Local pullback instability is defined as the converse of pullback stability.An invariant set Σ(•) is called locally pullback unstable if it is not pullback stable, i.e., if there exists a t ∈ R and ϵ > 0 such that for each δ > 0, there exists a t 0 < t and x 0 ∈ N (Σ(t 0 ), δ) such that dist(ϕ(t, t 0 , x 0 ), Σ(t)) > ϵ.However, we make use of the idea "unstable set" defined by Crauel for the random dynamical systems which is more natural concept from a dynamics point of view.
We say that Σ(•) is asymptotically unstable if for some t we have Since we always have Σ(t) ⊂ U Σ(t) when Σ(•) is invariant, the last definition says that Σ(t) is a strict subset of U Σ(t) .In this case we will say that U Σ(t) is non-trivial.The power of this definition comes from the following result.

Proposition 1. [29] If Σ(•) is asymptotically unstable then it is also locally pullback unstable and cannot be locally pullback attracting.
This result proven by Langa et al. in [29] is valid for impulsive systems.Most ideas of instability are related to the behavior of solutions ϕ(t) as t → −∞.Note that the idea of the asymptotic instability defined above is a time-reversed definition of 'forward attraction'.Alternatively, EJQTDE, 2014 No. 74, p. 6 it is possible to define instability as a time-reversed version of pullback attraction.Definition 4. [31] An invariant set Σ(•) is (locally) pullback repelling if it is (locally) pullback attracting for time-reversed system, i.e., if for every t ∈ R and every

The pitchfork bifurcation
In this section, we study generalization of the system (1) with fixed moments of impulses.Consider the system ẋ = a(t)x − b(t)x 3 , (10a) where a, b ∈ P C(R, θ).Assume that there exist constants A, B, C and D such that for i ∈ Z and t ∈ R. We suppose that there exist positive numbers θ and θ such that Moreover, there exists the limit By means of substitution y = 1 x 2 , the system ( 10) is converted to the linear impulsive system In what follows, we discuss the system ( 15) to analyze the system (10).Since c i ̸ = 0, the transition matrix of the associated homogeneous part of ( 15), according to [1], is the following: Proof.By relation (14), there exists T such that if t − s ≥ T, then Consequently, by means of ( 11) and ( 13), it is true that The lemma is proved.
Theorem 1. Assume that (11), ( 12) and ( 14) hold for the system (10).Then, for γ < 0 the origin is globally asymptotically pullback stable, and for γ > 0 the origin is asymptotically unstable and there appear positive and negative, β(t, γ) and −β(t, γ) respectively, locally asymptotically pullback complete trajectories such that Proof.Equation (10b) can be rewritten as . To show that equation ( 10) is order-preserving, it is sufficient that the jump EJQTDE, 2014 No. 74, p. 8 equation satisfies x(θ i +) > y(θ i +) for x(θ i ) > y(θ i ).In other words, we must show that , one can check that f ′ (x) > 0. Since uniqueness is assumed and the equation is order-preserving, for x 0 ̸ = 0 we have x(t) ̸ = 0. Therefore, by substitution y = 1 x 2 , we see that the solution of the system (10), according to [1,34], satisfies the integral equation ) By means of ( 14), one can see that the asymptotic behavior of y(t, t 0 , y 0 ) depends on the sign of γ.
If γ > 0, then from (18) it follows that y(t, t 0 , y 0 ) → 0 as t → ∞ implying that all solutions are unbounded as t → ∞.However, as t 0 → −∞ we have lim The last equation implies that . By means of (13) and Lemma 2, one can show that Thus, β 2 (t, γ) is bounded both from above and from below.To check that β(t, γ) is a complete trajectory, it would be enough to check that EJQTDE, 2014 No. 74, p. 9 To show that η(t) satisfies the equation jumps, we note for fixed j it is true that Construction of β(t, γ) ensures that it is pullback attracting.Thus, Lemma 1 implies that β(t, γ) is pullback stable.Moreover, since the system ( 10) is order-preserving, for γ > 0 all trajectories with x 0 > 0 are pullback attracted to β(t, γ) and all trajectories with x 0 < 0 are pullback attracted to −β(t, γ) as it is illustrated in Figure 1.By means of (18), it EJQTDE, 2014 No. 74, p. 10 follows that converges to 0 as t → −∞ implying that origin is asymptotically unstable.
Remark 1.We do not consider formal impulsive analogue of equation ( 1), since it is not possible to find explicit solution of the system (20).
Example 1.Let a(t) ≡ a, c i ≡ c, and θ i = ih for the system (10) with h > 0. That is, Then γ = 2a − 1 h ln c.By means of y = 1 x 2 , the system ( 21) is converted to the linear impulsive system ẏ = −2ay + 2b(t), ∆y| t=ih = (c − 1)y + d i . ( Asymptotic behavior of ( 22) depends on the sign of 2a − 1 h ln c = γ, and results of Theorem 1 are true for the system (21).If, in particular, c = 1 and d i = 0, then there is no equation of jumps in the system (21).Moreover, γ = 2a so that the asymptotic behavior of ( 22) depends on the sign of a. Thus, results of Theorem 1 are generalizations of the results obtained in the studies of Langa et al. in [29] and Caraballo and Langa in [11].EJQTDE, 2014 No. 74, p. 11

The transcritical bifurcation
Consider the impulsive system ẋ = a(t)x − b(t)x 2 , (23a) where R, θ).Differently from the previous section, the function a can be unbounded.However, as in the previous section, we suppose that there exist positive numbers θ and θ such that EJQTDE, 2014 No. 74, p. 12 θ ≤ θ i+1 − θ i ≤ θ, and there exists the limit The functions b and d i are asymptotically positive as t → −∞, i.e., there exist constants b and d such that b(t) ≥ b > 0 for all t ≤ T − , and By means of substitution x = 1 y , the system ( 23) is converted to the linear impulsive differential equation The transition matrix of the associated homogeneous part of the system (26), according to [1], is (27) Assume that there exists a γ 0 > 0 such that for all t ∈ R, i ∈ Z, 0 < γ < γ 0 , and lim inf for all −γ 0 < γ < 0.
Theorem 2. Assume that the above conditions hold for equation (23).Then, for −γ 0 < γ < 0 the origin is locally pullback attracting in R; and for 0 < γ < γ 0 the origin is asymptotically unstable and the trajectory x γ (t) is locally pullback attracting.
Proof.Equation (23b) can be rewritten as To show that ( 23) is order-preserving, it is enough to show that the jump equation satisfies y for equation (23), we see that the solution of the impulsive system (26), according to [1,34], satisfies the integral equation Transforming backwards we have By means of ( 24), one can see that the asymptotic behavior of ( 31) depends on the sign of γ.
Consider the case when γ > 0.
If x 0 > 0, then as t 0 → −∞, (31) implies that lim as long as the solution exists on the interval [t 0 , t] .To ensure the existence, it is sufficient to have for τ ∈ [t 0 , t].Let us show that (33) holds if we require x 0 < (1+α t )x γ (t 0 ) for some α t > 0.
No. 74, p. 14 for all t 0 ≤ τ ≤ t.Taking into account the assumption (25), it suffices to show that the last expression holds for any τ from the interval [T − , t].This can be done by choosing α t > 0 appropriately.Hence, choosing δ(t) = α t m γ and implementing Definition 1, it follows that x γ (t) is locally pullback attracting.
Since x(t) ≡ 0 and x γ (t) are solutions and the system is order-preserving, any solution with 0 < x 0 < x γ (t 0 ) exists for all t ≤ t 0 .Moreover, assumption (28) Thus, from equation (31) and relation (24), it follows that x(t, t 0 , x 0 ) → 0 as t → −∞, which implies that the origin is asymptotically unstable.
If x 0 < 0, then for t 0 sufficiently large and negative x(τ, t 0 , x 0 ) blow up for some τ ≥ t 0 .To see this, note that Y (t, t 0 )x −1 0 is negative and tends to zero as t 0 → −∞, while and bounded below.As a result, x(τ, t 0 , x 0 ) → −∞ in a finite time as the denominator of (31) tends to zero for some τ ≥ t 0 .
For x 0 > 0, it is sufficient to show that for τ ∈ [t 0 , t].By means of (25), inequality (34) is satisfied if Because of assumption (24), for t 0 small enough Y (τ, t 0 ) is bounded below on (−∞, T − ].Thus, ( 34) is satisfied provided that EJQTDE, 2014 No. 74, p. 15 For x 0 < 0 the argument requires condition (29), which implies that there exists a µ t such that Y (τ, t 0 ) for all t 0 ≤ µ t .Now, it is sufficient to show that which has the right-hand side of this expression is bounded below by mγ 2 using (37).Therefore, for each t there exists a µ t such that if t 0 ≤ µ t and |x 0 | is sufficiently small, the solution exists on [t 0 , t] and, hence, the origin is locally pullback attracting.The theorem is proved.
Next, we want to formulate an impulsive extension of the system (23), which is related to the forward attraction.We assume that the functions b and d i are asymptotically positive as t → ∞, and the 'balance condition' (28) is valid.That is, b(t) ≥ b > 0 for all t ≥ T + , and for all t ∈ R, 0 < γ < γ 0 .
EJQTDE, 2014 No. 74, p. 16 Proof.If γ < 0, the origin is locally forward attracting when x 0 is sufficiently small, since condition (39) implies that inf If γ > 0, the trajectory x γ (t) is locally forward attracting.To see this, we notice that ) . Therefore, Using the balance condition (40) with x 0 > 0 implies that Condition (39) guarantees that the integral and the sum in the numerator and denominator are positive for t sufficiently large.So, from the last expression it follows that lim sup Therefore, any solution with x 0 > 0 is bounded as t → ∞.Hence, from (42) it follows that x γ (t) is forward attracting as long as solutions exist.
EJQTDE, 2014 No. 74, p. 17 Next, we show that solutions dot not blow up for The last expression is positive for sufficiently small α t 0 because of the assumption (39).Therefore, x γ (t) is locally forward attracting.
Remark 2. In this paper, we do not consider the formal impulsive analogue of ( 2 since it is not possible to find explicit solution of the system (43).
Example 2. Let a(t) ≡ a, c i ≡ c, and θ i = ih for the system (23) with 4. Bifurcation in the non-order-preserving system In the continuous differential equations requiring uniqueness implies that a system is order-preserving.However, in impulsive systems order- preservation is violated even for the scalar case if we do not impose any EJQTDE, 2014 No. 74, p. 19 condition on the jump equation.In this section, we want to consider a non-order-preserving system and analyze bifurcation phenomena.Let us consider the system (5) which differs only by the jump equation from the system (10).We assume the same conditions for (5) as for (10).The impulsive equation of ( 5) can be rewritten as x(θ i +) = − x(θ i ) , one can check that f ′ (x) < 0. Although uniqueness of solutions is assumed, the system ( 5) is non-order-preserving due to the jump equations.However, by means of transformation y = 1 x 2 , the system ( 5) is also transformed into the system (15).Therefore, the results of Theorem 1 are also true for the system (5).Exceptionally, since the system ( 5) is non-order-preserving, for γ > 0 all trajectories of the system (5) are in the neighborhood of |β(t, γ)| and alternatively change their position from neighborhood the of β(t, γ) to the neighborhood of −β(t, γ) as it is shown in Figure 2.

Conclusion
The pitchfork and the transcritical bifurcations are considered for nonautonomous impulsive differential equations.Explicitly solvable models with the specific equations of jump have been considered.This allowed us to categorize one-dimensional bifurcations in impulsive systems which are order-preserving.Moreover, the non-order-preserving system is studied.
This theory could be developed in many ways.One can consider formal impulsive analogues for the pitchfork bifurcation for the system (20), and corresponding formal impulsive analogue for the system (43), for the transcritical bifurcation without finding explicit solution similarly to that done in [35].Non-autonomous saddle-node bifurcation remains unconsidered even for one-dimensional impulsive systems.Finally, general theory of higher-dimensional bifurcation results with impulses has to be developed.
by introducing the transformation EJQTDE, 2014 No. 74, p. 13 x = 1 ) EJQTDE, 2014 No. 74, p. 18 Then γ = a − 1 h ln c.By means of y = 1 x , the system (21) is converted to the linear impulsive system ẏ = −ay + b(t), ∆y| t=ih = (c − 1)y + d i .(45) Asymptotic behavior of (45) depends on the sign of γ, and results of Theorem 2 and Theorem 3 are true for the system (44).If c = 1 and d i = 0, then γ = a and there is no equation of jumps in the system (44).