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Homoclinic and Heteroclinic Motions in Hybrid Systems with Impacts Mehmet Onur Fena, , Fatma Tokmak Fenb ∗ aDepartment of Mathematics, Middle East Technical University, 06800, Ankara, Turkey bDepartment of Mathematics, Gazi University, 06500, Teknikokullar, Ankara, Turkey Abstract 6 1 0 In this paper, we present a method to generate homoclinic and heteroclinic motions in impulsive 2 n systems. We rigorously prove the presence of such motions in the case that the systems are under the a J influence of a discrete map that possesses homoclinic and heteroclinic orbits. Simulations that support 4 the theoretical results are represented by means of a Duffing equation with impacts. 1 ] Keywords: Impulsive systems; Stable and unstable sets; Homoclinic motion; Heteroclinic motion; D C Duffing equation with impacts . n i l n 1 Introduction [ 1 v Impulsive differential equations describe the dynamics of real world processes in which abrupt changes 2 9 occur. Suchequationsplayanincreasinglyimportantroleinvariousfieldssuchasmechanics,electronics, 5 3 biology, neural networks, communication systems, chaos theory and population dynamics [1, 2, 4, 5, 15, 0 . 17, 19, 24, 25, 28]. In this paper, we investigate the existence of homoclinic and heteroclinic motions in 1 0 systems with impulsive effects. 6 1 The main object of the present study is the following impulsive system, : v i X x =A(t)x+f(t,x)+g(t,ζ), t=θ , ′ k r 6 (1.1) a ∆x =B x+J (x)+ζ , |t=θk k k k where θ ,k Z,isastrictlyincreasingsequenceofrealnumberssuchthat θ as k ,A(t)is k k { } ∈ | |→∞ | |→∞ ann ncontinuousmatrixfunction,B areconstantn nrealvaluedmatrices,∆x =x(θ +) x(θ ), × k × |t=θk k − k x(θ +) = lim x(t), the functions f : R Rn Rn and J : Rn Rn are continuous in all their k k t θ+ × → → → k arguments, the function g(t,ζ) is defined by the equation g(t,ζ) = ζ , t (θ ,θ ], and the sequence k k 1 k ∈ − ζ = ζ , k Z, is a solution of the map k { } ∈ ζ =F(ζ ), (1.2) k+1 k ∗Corresponding AuthorTel.: +903123659276,E-mail: [email protected] 1 where the function F :Λ Λ is continuous and Λ is a bounded subset of Rn. Here, R and Z denote the → sets of real numbers and integers, respectively. The system under investigation is a hybrid one, since it combines the dynamics of an impulsive differential equation with a discrete map. Our main objective is to prove rigorously the existence of homoclinic and heteroclinic solutions in the dynamics of (1.1) provided that (1.2) possesses such solutions. Theideaoftheusageofdiscontinuousperturbationstogeneratehomoclinicandheteroclinicmotions insystemsofdifferentialequationswasfirstrealizedinthepapers[3,7]onthebasisoffunctionalspaces. It wasshownin [3]that the chaotic attractorofthe relaysystem,whichwasintroducedin the paper [6], consists of homoclinic solutions. Similar results for impulsive differential equations were obtained in the study [7] by taking advantage of the moments of impulses. Theexistenceofhomoclinicandheteroclinicmotionsinsystemswithimpulseswerealsoinvestigated in the papers [9, 11, 12, 14, 18, 26, 27]. The existence and multiplicity of fast homoclinic solutions for a classofdampedvibrationproblemswithimpulsiveeffectswereinvestigatedin[26]byusingthemountain passtheoremandthe symmetricmountainpasstheoreminthe criticalpointtheory. The mountainpass theoremwasalsoutilizedin[11,18]toshowthepresenceofhomoclinicmotionsinsecondorderimpulsive systems. On the other hand, Wei and Chen [22, 23] considered the existence of heteroclinic cycles in predator-prey systems with Allee effect and state-dependent impulsive harvesting within the scope of their studies. ZhangandLi [27] provedthe existence ofatleastone non-zerohomoclinic solution, which is generated by impulses, under appropriate conditions for a class of impulsive second order differential equations. HanandZhang[14]obtainedtheexistenceofhomoclinicsolutionsforaclassofasymptotically linear or sublinear Hamiltonian systems with impulses by using variational methods. It was mentioned in [14] that no homoclinic solutions existfor the system under investigationwithout impulses. However, in the present study, the emergence of homoclinic and heteroclinic motions are completely provided by theinfluenceofadiscretemapinsteadofimpulsiveeffects. Additionally,ourresultsarevalidforsystems with arbitrary high dimensions. The rest of the paper is organized as follows. In Section 2, we discuss bounded solutions of (1.1), and present sufficient conditions for the existence of homoclinic and heteroclinic motions in the system. Section 3 is devotedfor the main results of the paper. In this part, we show the connectionbetween the stableandunstablesetsoftheimpulsivesystem(1.1)andthediscretemap(1.2),andprovetheexistence of homoclinic and heteroclinic solutions in (1.1). Examples concerning homoclinic and heteroclinic motions in an impulsive Duffing equation are provided in Section 4. Finally, some concluding remarks are given in Section 5. 2 2 Preliminaries In the sequel, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for matrices [16]. Let us denote by U(t,s) the transition matrix of the linear homogeneous system u =A(t)u, t=θ , ′ k 6 (2.3) ∆u =B u(θ ). |t=θk k k The following conditions are required. (C1) det(I+B )=0 for all k Z, where I is the n n identity matrix; k 6 ∈ × (C2) There exists a positive number θ such that θ θ θ for all k Z; k+1 k − ≥ ∈ (C3) There exist positive numbers N and ω such that U(t,s) Ne ω(t s) for t s; − − k k≤ ≥ (C4) There exist positive numbers M , M and M such that f F J sup f(t,x) M , sup F(σ) M , sup J (x) M ; f F k J (t,x) R Rnk k≤ σ Λk k≤ k Z,x Rnk k≤ ∈ × ∈ ∈ ∈ (C5) There exist positive numbers L and L such that f J f(t,x ) f(t,x ) L x x 1 2 f 1 2 k − k≤ k − k for all t R, x ,x Rn, and 1 2 ∈ ∈ J (x ) J (x ) L x x k 1 k 2 J 1 2 k − k≤ k − k for all k Z, x ,x Rn; 1 2 ∈ ∈ L L f J (C6) N + <1; ω 1 e ωθ (cid:18) − − (cid:19) 1 (C7) ω+NL + ln(1+NL )<0. f J − θ LetΘbe the setofallsequencesζ = ζ ,k Z, obtainedbyequation(1.2). Byusingthe resultsof k { } ∈ [1, 21] one can show under the conditions (C1) (C6) that for a fixed sequence ζ Θ the system (1.1) − ∈ possesses a unique bounded on R solution φ (t), which satisfies the following relation, ζ t φ (t)= U(t,s)[f(s,φ (s))+g(s,ζ)]ds+ U(t,θ +)[J (φ (θ ))+ζ ]. (2.4) ζ ζ k k ζ k k Z−∞ −∞X<θk<t 3 One can confirm under the conditions (C1) (C7) that for a fixed sequence ζ Θ, the bounded − ∈ solutionφ (t)attractsallothersolutionsof(1.1),i.e., x(t) φ (t) 0ast foranysolutionx(t) ζ ζ k − k→ →∞ of (1.1). Moreover, M +M M +M f F J F sup φ (t) N + t∈Rk ζ k≤ (cid:18) ω 1−e−ωθ (cid:19) for each ζ Θ. ∈ 3 Homoclinic and heteroclinic motions In this section, first of all, we will describe the stable, unstable and hyperbolic sets as well as the homoclinic and heteroclinic motions for both system (1.1) and the discrete map (1.2). These definitions were introduced in the papers [3, 7]. After that the existence of homoclinic and heteroclinic motions in the dynamics of (1.1) will be proved. Consider the set Θ described in the previous section once again. The stable set of a sequence ζ Θ ∈ is defined as Ws(ζ)= η Θ η ζ 0 as k , k k { ∈ | k − k→ →∞} and the unstable set of ζ is Wu(ζ)= η Θ η ζ 0 as k . k k { ∈ | k − k→ →−∞} The set Θ is called hyperbolic if for each ζ Θ the stable and unstable sets of ζ contain at least one ∈ elementdifferentfromζ.Asequenceη Θishomoclinictoanothersequenceζ Θifη Ws(ζ) Wu(ζ). ∈ ∈ ∈ ∩ Moreover,η Θisheteroclinictothesequencesζ1 Θ,ζ2 Θ,η =ζ1,η =ζ2,ifη Ws(ζ1) Wu(ζ2). ∈ ∈ ∈ 6 6 ∈ ∩ On the other hand, let us denote by A the set consisting of all bounded on R solutions of system (1.1). A bounded solution φ (t) A belongs to the stable set Ws(φ (t)) of φ (t) A if η ζ ζ ∈ ∈ φ (t) φ (t) 0 as t . Besides, φ (t) is an element of the unstable set Wu(φ (t)) of φ (t) η ζ η ζ ζ k − k → → ∞ provided that φ (t) φ (t) 0 as t . η ζ k − k→ →−∞ We say that A is hyperbolic if for each φ (t) A the sets Ws(φ (t)) and Wu(φ (t)) contain at ζ ζ ζ ∈ leastoneelementdifferentfrom φ (t). Asolutionφ (t) A ishomoclinicto anothersolutionφ (t) A ζ η ζ ∈ ∈ if φ (t) Ws(φ (t)) Wu(φ (t)), and φ (t) A is heteroclinic to the bounded solutions φ (t), η ζ ζ η ζ1 ∈ ∩ ∈ φ (t) A, φ (t)=φ (t), φ (t)=φ (t), if φ (t) Ws(φ (t)) Wu(φ (t)). ζ2 η ζ1 η ζ2 η ζ1 ζ2 ∈ 6 6 ∈ ∩ In what follows, we will denote by i((a,b)) the number of the terms of the sequence θ , k Z, k { } ∈ which belong to the interval (a,b), where a and b are real numbers such that a < b. It is worth noting b a that i((a,b)) 1+ − . ≤ θ 4 The connection between the stable sets of the solutions of (1.1) and (1.2) is provided in the next assertion. Lemma 3.1 Suppose that the conditions (C1) (C7) are fulfilled, and let ζ and η be elements of Θ. If − η Ws(ζ), then φ (t) Ws(φ (t)). η ζ ∈ ∈ 1 Proof. Fix an arbitrarypositive number ǫ, anddenote α=ω NL ln(1+NL ). Assume without f J − − θ loss of generality that ǫ 2M . Let γ be a real number such that F ≤ 1 1 NL (1+NL ) NL (1+NL ) f J J J γ 1+N + 1+ + . ≥ ω 1 e ωθ α 1 e αθ (cid:18) − − (cid:19)(cid:18) − − (cid:19) Because the sequence η = η , k Z, belongs to the stable set Ws(ζ) of ζ = ζ , there exists k k { } ∈ { } ǫ ǫ an integer k such that η ζ < for all k k . One can confirm that g(t,η) g(t,ζ) < for 0 k k 0 k − k γ ≥ k − k γ t>θ . k0−1 Making use of the relation t φ (t) φ (t)= U(t,s)[f(s,φ (s)) f(s,φ (s))+g(s,η) g(s,ζ)]ds η ζ η ζ − − − Z−∞ + U(t,θ +)[J (φ (θ )) J (φ (θ ))+η ζ ], k k η k k ζ k k k − − −∞X<θk<t we obtain for t>θ that k0−1 θk0−1 φη(t) φζ(t) 2N(Mf +MF)e−ω(t−s)ds k − k≤ Z−∞ t Nǫ + 2N(MJ +MF)e−ω(t−θk)+ e−ω(t−s)ds γ −∞<θXk≤θk0−1 Zθk0−1 Nǫ t + e−ω(t−θk)+ NLfe−ω(t−s) φη(s) φζ(s) ds γ k − k θk0−X1<θk<t Zθk0−1 + NLJe−ω(t−θk) φη(θk) φζ(θk) k − k θk0−X1<θk<t (3.5) M +M M +M 2N f F + J F e−ω(t−θk0−1) ≤ ω 1 e ωθ Nǫ (cid:18) − − (cid:19)Nǫ + 1 e−ω(t−θk0−1) + 1 e−ω(t−θk0−1+θ) γω − γ(1 e ωθ) − t(cid:16) (cid:17) − − (cid:16) (cid:17) + NLfe−ω(t−s) φη(s) φζ(s) ds k − k Zθk0−1 + NLJe−ω(t−θk) φη(θk) φζ(θk) . k − k θk0−X1<θk<t Define the functions u(t)=eωt φ (t) φ (t) and h(t)=c +c eωt, where η ζ 1 2 k − k c1 =2N Mf +ωMF + M1J +eMωθF eωθk0−1 − Nγǫ eωθωk0−1 + eω1(θk0e−1ω−θθ) (cid:18) − − (cid:19) (cid:18) − − (cid:19) 5 and Nǫ 1 1 c = + . 2 γ ω 1 e ωθ (cid:18) − − (cid:19) The inequality (3.5) implies that t u(t) h(t)+ NL u(s)ds+ NL u(θ ). f J k ≤ Zθk0−1 θk0−X1<θk<t The application of the analogue of the Gronwall’s inequality for piecewise continuous functions yields t u(t) h(t)+ NLf(1+NLJ)i((s,t))eNLf(t−s)h(s)ds ≤ Zθk0−1 + NLJ(1+NLJ)i((θk,t))eNLf(t−θk)h(θk). θk0−X1<θk<t Since the equation t 1+ NLf(1+NLJ)i((s,t))eNLf(t−s)ds Zθk0−1 + NLJ(1+NLJ)i((θk,t))eNLf(t−θk) θk0−X1<θk<t =(1+NLJ)i((θk0−1,t))eNLf(t−θk0−1) is validand(1+NLJ)i((a,b))eNLf(b−a) (1+NLJ)e(ω−α)(b−a) for any realnumbersa andb witha<b, ≤ one can confirm that u(t) c1(1+NLJ)e(ω−α)(t−θk0−1)+c2eωt ≤ t + c NL (1+NL )e(ω α)(t s)eωsds 2 f J − − Zθk0−1 + c2NLJ(1+NLJ)e(ω−α)(t−θk)eωθk θk0−X1<θk<t c1(1+NLJ)e(ω−α)(t−θk0−1)+c2eωt ≤ c NL (1+NL ) + 2 f J eωt 1 e−α(t−θk0−1) α − c NL (1+NL ) (cid:16) (cid:17) + 2 J J eωt 1 e−α(t−θk0−1+θ) . 1 e αθ − − − (cid:16) (cid:17) 6 If we multiply both sides of the last inequality by e ωt, then we obtain that − φη(t) φζ(t) c1(1+NLJ)e−ωθk0−1e−α(t−θk0−1)+c2 k − k≤ c NL (1+NL ) + 2 f J 1 e−α(t−θk0−1) α − c NL (1+NL )(cid:16) (cid:17) + 2 J J 1 e−α(t−θk0−1+θ) 1 e αθ − − − (cid:16) (cid:17) M +M M +M <2N(1+NLJ) f ω F + 1J e ωθF e−α(t−θk0−1) (cid:18) − − (cid:19) Nǫ 1 1 NL (1+NL ) NL (1+NL ) f J J J + + 1+ + . γ ω 1 e ωθ α 1 e αθ (cid:18) − − (cid:19)(cid:18) − − (cid:19) Now, let R>θ be a sufficiently large real number such that k0−1 M +M M +M ǫ 2N(1+NLJ) f ω F + 1J e ωθF e−α(R−θk0−1) ≤ γ. (cid:18) − − (cid:19) For t R, we have ≥ ǫ 1 1 NL (1+NL ) NL (1+NL ) f J J J φ (t) φ (t) < 1+N + 1+ + ǫ. η − ζ γ ω 1 e ωθ α 1 e αθ ≤ (cid:13) (cid:13) h (cid:16) − − (cid:17)(cid:16) − − (cid:17)i (cid:13) (cid:13) (cid:13) (cid:13) Therefore, lim φ (t) φ (t) =0. Consequently, φ (t) Ws(φ (t)). (cid:3) η ζ η ζ t k − k ∈ →∞ In the next lemma, we reveal the connection between the unstable sets of the solutions of (1.1) and (1.2). Lemma 3.2 Suppose that the conditions (C1) (C6) are fulfilled, and let ζ and η be elements of Θ. If − η Wu(ζ), then φ (t) Wu(φ (t)). η ζ ∈ ∈ Proof. Fix an arbitrary positive number ǫ, and let λ be a real number such that N(ω+1 e ωθ) − λ> − . ω(1 e ωθ) N(L (1 e ωθ)+L ω) − − − f − − J Since η = η , k Z, is an element of the unstable set Wu(ζ) of ζ = ζ , there exists an integer k k { } ∈ { } ǫ ǫ k such that η ζ < for all k k . In this case, we have that g(t,η) g(t,ζ) < for t θ . 0 k k− kk λ ≤ 0 k − k λ ≤ k0 By using the relation t φ (t) φ (t)= U(t,s)[f(s,φ (s)) f(s,φ (s))+g(s,η) g(s,ζ)]ds η ζ η ζ − − − Z−∞ + U(t,θ +)[J (φ (θ )) J (φ (θ ))+η ζ ], k k η k k ζ k k k − − −∞X<θk<t 7 one can verify for t θ that ≤ k0 t ǫ φ (t) φ (t) < Ne ω(t s) L φ (s) φ (s) + ds η ζ − − f η ζ k − k k − k λ Z−∞ (cid:16) ǫ (cid:17) + Ne−ω(t−θk) LJ φη(θk) φζ(θk) + k − k λ −∞X<θk<t (cid:16) (cid:17) N ǫ N ǫ L sup φ (t) φ (t) + + L sup φ (t) φ (t) + . ≤ ω ft≤θk0k η − ζ k λ! 1−e−ωθ Jt≤θk0k η − ζ k λ! Therefore, NL NL Nǫ 1 1 f J 1 sup φ (t) φ (t) + . − ω − 1 e ωθ k η − ζ k≤ λ ω 1 e ωθ (cid:18) − − (cid:19)t≤θk0 (cid:18) − − (cid:19) The last inequality implies that sup φ (t) φ (t) <ǫ. Consequently, η ζ k − k t≤θk0 lim φ (t) φ (t) =0, η ζ t k − k →−∞ and φ (t) belongs to Wu(φ (t)). (cid:3) η ζ The main result of the presentpaper is mentioned in the following theorem, which can be provedby using the results of Lemma 3.1 and Lemma 3.2. Theorem 3.1 Under the conditions (C1) (C7), the following assertions are valid. − (i) If η Θ is homoclinic to ζ Θ, then φ (t) A is homoclinic to φ (t) A; η ζ ∈ ∈ ∈ ∈ (ii) If η Θ is heteroclinic to ζ1, ζ2 Θ, then φ (t) A is heteroclinic to φ (t), φ (t) A; η ζ1 ζ2 ∈ ∈ ∈ ∈ (iii) If Θ is hyperbolic, then the same is true for A. The next section is devoted to examples concerning homoclinic and heteroclinic motions in an impulsive Duffing equation. 4 Examples Let us take into account the impulsive Duffing equation 2π x +0.2x +0.81x+0.001x3 =0.7cos t +g(t,ζ), t=θ , ′′ ′ k 3 6 (cid:18) (cid:19) (4.6) ∆x = 0.12x+0.09+ζ , |t=θk − k ∆x = 0.12x +0.015sin(x), ′|t=θk − ′ 8 where θ = 3k, k Z, the function g(t,ζ) is defined through the equation g(t,ζ) = ζ , t (θ ,θ ], k k k 1 k ∈ ∈ − and the sequence ζ = ζ is a solution of the logistic map k { } ζ =F (ζ ), (4.7) k+1 µ k where F (s)=µs(1 s) and µ is a parameter. µ − For 0 < µ 4, the interval [0,1] is invariant under the iterations of (4.7) [10, 13, 20], and the ≤ 1 4s inverses of the function F on the intervals [0,1/2] and [1/2,1] are h (s) = 1 1 and µ 1 2 − − µ (cid:18) r (cid:19) 1 4s h (s)= 1+ 1 , respectively. 2 2 − µ (cid:18) r (cid:19) By using the new variables x =x and x =x one can reduce (4.6) to the system 1 2 ′ x =x , ′1 2 2π x = 0.81x 0.2x 0.001x3+0.7cos t +g(t,ζ), t=θ , ′2 − 1− 2− 1 (cid:18) 3 (cid:19) 6 k (4.8) ∆x = 0.12x +0.09+ζ , 1|t=θk − 1 k ∆x = 0.12x +0.015sin(x ). 2|t=θk − 2 1 Denote by U(t,s) the transition matrix of the linear homogeneous system u =u , ′1 2 u = 0.81u 0.2u , t=θ , ′2 − 1− 2 6 k (4.9) ∆u = 0.12u , 1|t=θk − 1 ∆u = 0.12u . 2|t=θk − 2 One can verify for t>s that i([s,t)) cos 2 (t s) sin 2 (t s) 22 √5 − − √5 − U(t,s)=e−(t−s)/10 25 P  (cid:16) (cid:17) (cid:16) (cid:17) P−1, (cid:18) (cid:19) sin 2 (t s) cos 2 (t s)  √5 − √5 −   (cid:16) (cid:17) (cid:16) (cid:17)  where i([s,t)) is the number of the terms of the sequence θ that belong to the interval [s,t) and k { } 0 1 P = . It can be calculated that U(t,s) Ne ω(t s), t s, where ω = 1/10 and   − − k k ≤ ≥ 2/√5 1/10  −  N =1.17.  For 0<µ 4 the bounded solutions of (4.8) lie inside the compact region ≤ D = (x ,x ) R2 : x 2.8, x 1.4 , 1 2 1 2 ∈ | |≤ | |≤ (cid:8) (cid:9) and the conditions (C1) (C7) arevalid for system(4.8). It is worthnoting that for a periodic solution − 9 ζ = ζ of (4.7) the corresponding bounded solution φ (t) of (4.8) is also periodic. k ζ { } Consider the map (4.7) with µ=3.9. It was demonstrated in [8] that the orbit η = ...,h3(η ),h2(η ),h (η ),η ,F (η ),F2(η ),F3(η ),... , 2 0 2 0 2 0 0 µ 0 µ 0 µ 0 (cid:8) (cid:9) where η0 = 1/3.9, is homoclinic to the fixed point η∗ = 2.9/3.9 of (4.7). Denote by φη(t) and φη∗(t) the bounded solutions of (4.8) corresponding to η and η , respectively. One can conclude by using ∗ Theorem 3.1 that φη(t) is homoclinic to the periodic solution φη∗(t). Figure 1 shows the graphs of the x1 coordinates of φη(t) and φη∗(t). In the figure, the solution φη(t) is represented in blue color, − while φη∗(t) is represented in red color. Figure 1 reveals that φη(t) is homoclinic to φη∗(t), i.e., φη(t) φη∗(t) 0 as t . k − k→ →±∞ 2 1 1 x 0 −1 −20 −15 −10 −5 0 5 10 15 20 25 30 t Figure 1: Homoclinic solution of (4.8). The x1−coordinates of φη(t) and φη∗(t) are shown in blue and red colors, respectively. Thefiguremanifeststhatφη(t)ishomoclinictoφη∗(t). Now, we set µ=4 in equation (4.7). According to [8], the orbit η = ...,h3(η ),h2(η ),h (η ),η ,F (η ),F2(η ),F3(η ),... , 1 0 1 0 1 0 0 µ 0 µ 0 µ 0 (cid:8) (cid:9) e e e e e e e e where η0 = 1/4, is heteroclinic to the fixed points η1 = 3/4 and η2 = 0 of (4.7). Suppose that φηe(t), φ (t)andφ (t)aretheboundedsolutionsof(4.8)correspondingtoη,η1 andη2,respectively. Theorem η1 η2 e 3.1 implies that φηe(t) is heteroclinic to the periodic solutions φη1 and φη2. Figure 2 shows the graphs e of the x1 coordinates of φηe(t), φη1(t) and φη2(t) in blue, red and green colors, respectively. The figure − supports Theorem 3.1 such that φηe(t) converges to φη1(t) as time increases and converges to φη2(t) as time decreases, i.e., φηe(t) is heteroclinic to φη1(t), φη2(t). 1.5 1 1 x 0.5 0 −15 −10 −5 0 5 10 15 20 25 30 t Figure 2: Heteroclinicsolutionof(4.8). Thex1−coordinatesofφηe(t),φη1(t)andφη2(t)arerepresentedinblue,redand greencolors,respectively. Thefigureconfirms thatφηe(t)isheteroclinictotheperiodicsolutionsφη1(t), φη2(t). 10