Filomat 3:2 (27), 376 377 https://doi.org/.2298/fil7276a Pblished by Faclty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Conditions for Approaching the Origin withot Intersecting the -ais in the Liénard Plane Asadollah Aghajani a, Mohsen Mirafzal a, Donald O Regan b a School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 6844-34, Iran. b School of Mathematics, Statistics and Applied Mathematics National University of Ireland, Galway, University Road, Galway, Ireland. Abstract. We consider the Liénard system in the plane and present general assmptions to obtain some new eplicit conditions nder which this system has or fails to have a positive orbit which starts at a point on the vertical isocline and approaches the origin withot intersecting the -ais. This arises natrally in the eistence of homoclinic orbits and oscillatory soltions. Or investigation is based on the notion of orthogonal trajectories of orbits of the system.. Introdction To stdy the qalitative theory of the Liénard eqation + f () + (), sch as bondedness, oscillation and periodicity of the soltions, reslts are established by eamining the corresponding planar system d dt y F(), dy dt (), where F() : f ()d. Asymptotic and qalitative behavior of this system was stdied by many athors. In the literatre, there are a considerable nmber of reslts on the eistence and niqeness of periodic orbits, homoclinic orbits, oscillation of soltions, center problem and eistence of limit cycles for Liénard and other types of second order differential eqations (see [-22] and the references cited therein). In the stdy of the qalitative behavior of soltions of the Liénard system the notion property (Z + ) is very sefl. We say that system () has property (Z + ) if there eists a point P(, y ) with y F( ) and > sch that the positive semitrajectory of () starting at P approaches the origin throgh only the first qadrant. In this paper, we give some conditions on F() and () nder which the system () has or fails to have property (Z + ). We assme that F and are continos on an open interval I which contains and satisfy smoothness conditions to garantee the eistence and niqeness of soltions of the corresponding initial () 2 Mathematics Sbject Classification. Primary 34C5; Secondary 7K5 Keywords. Liénard system, Homoclinic orbit, Oscillation, Limit cycle Received: 3 December 25; Accepted: 25 May 26 Commnicated by Jelena Manojlović Email addresses: aghajani@ist.ac.ir (Asadollah Aghajani), mirafzal@ist.ac.ir (Mohsen Mirafzal), donal.oregan@nigalway.ie (Donald O Regan)
A. Aghajani et al. / Filomat 3:2 (27), 376 377 3762 vale problems. Also, we assme that F() and () > for which implies the origin is the niqe eqilibrim of (), and hence closed orbits (if any) rotate clockwise arond it. Also, throghot this paper in the reslts related to property (Z + ), we assme that F() >, for > sfficiently small, becase if F() has an infinite nmber of positive zeros clstering at, then the system () obviosly fails to have property (Z + ). The following reslts for property (Z+ ) are the following theorems of Hara [] and Hara-Sgie [6]. Theorem.. (Hara[]) Under the condition that + F() < 2, where G() : ()d, 2G() the Liénard system () fails to have property (Z + ), while it has property (Z+ ) if F() 2 2G(). Theorem.2. (Hara S ie[6]) Sppose that F() 2 2G() h( 2G()), for > sfficiently small, where h(ξ) is a non-negative continos fnction with and h(ξ) ξ ξ 2 is a non-decreasing fnction for ξ > sfficiently small, h(ξ) ξ 2 dξ for some ξ >. Then the system () fails to have property (Z + ). The following reslt gives a sfficient condition for system () to have property (Z + ). Theorem.3. (Hara S ie[6]) Sppose that F() 2 2G() h( 2G()), for > sfficiently small, where h(ξ) is a non-negative continos fnction sch that for ξ > sfficiently small h(ξ) ξ is non-decreasing, and a[h(ξ)] 2 h(ξ) ξ for some a > 4, where H(ξ) ξ 2 d. Then the Liénard system () has property (Z+ ). These reslts are etended by some athors to more general planar dynamical systems of Liénard type, see for eample [, 5, 4]. In this paper, we consider system () and present some new eplicit necessary and sfficient conditions nder which this system has or fails to have property (Z + ). Or reslts improve the above and the eisting reslts in the literatre.
A. Aghajani et al. / Filomat 3:2 (27), 376 377 3763 2. A comparison theorem In this section we give a comparison reslt that will be sed repeatedly in the seqel. First, we state the following reslt of T. Hara [] on property (Z + ). Lemma 2.. (Hara[]) Consider system (). For each point P (, y ) on the vertical isocline in the right halfplane, the positive semitrajectory γ + (P) approaches the origin, withot intersecting the -ais, if and only if there eist a constant and a continos fnction ψ(), sch that ψ() < F() and (ξ) dξ ψ()for < <. F(ξ) ψ(ξ) Using the above lemma, we present a simple comparison theorem. Theorem 2.2. Sppose that the system () corresponding to F and has property (Z + ). If we have F 2 () F () and 2 () (), for all >, then system () corresponding to F 2 and 2 has this property, too. Proof. By assmption the system () corresponding to F and has property (Z + ). Ths from the above lemma, there is a continos fnction ψ() < F () (hence ψ() < F 2 ()) and some > sch that for < < we have 2 (ξ) F 2 (ξ) ψ(ξ) dξ (ξ) dξ ψ(). F (ξ) ψ(ξ) Now sing again the above lemma we have that () has property (Z + ). Consider () with () and F() λ. Then, we have a linear system and, either by elementary reslts or by the above lemma, we easily find that system () has property (Z + ), for λ 2, and that it fails to have property (Z + ), for λ < 2. Now consider the system () as d dt y F() dy dt. (2) The above comparison theorem shows that if + F() < 2, then system (2) fails to have property (Z + ), while it has property (Z+ ) if F() 2. From Theorem.2 of Sgie-Hara [6], we can solve the problem in the case F() 2 as +. Bt this reslt is restricted and cannot solve the problem, when in general we have F() 2 as +. As we shall see, the reslts of this paper solve this problem even in the case lim inf + F() 2 + F().
A. Aghajani et al. / Filomat 3:2 (27), 376 377 3764 3. Main reslts The main idea in proving the main reslts in this paper is to find the family of orthogonal trajectories of system (). Indeed, if this family described by (, y) C and taking w(t) : ( (t), y(t) ), where ( (t), y(t) ) is a soltion of this system, initiate at a point on the vertical isocline and remains at the first qadrant approaching the origin then we mst have w (t) for all t sfficiently large (see Theorem 3.2). Lemma 3.. The orthogonal trajectories of () in the right half-plane are level crves of the fnction (, y) where > is arbitrary. F(ξ) ξ (ξ) ep( )dξ + y ep( Proof. The orbits of system () satisfy the following differential eqation dy d () y F(). Therefore, their orthogonal trajectories have the eqation d dy or eqivalently () y F(), ), >, (3) ( y F() ) d ()dy. (4) In order to make the eqation (4) integrable, we mltiply it by the integration factor µ () ep (, then we get the eact eqation D(, y) where (, y) is given in (3) and the proof is complete. Theorem 3.2. Sppose that for some > we have + ( F(ξ) (ξ) 2) ep ( ξ dξ +. (5) Then the Liénard system () fails to have property (Z + ). Proof. If we assme that the system () has property (Z + ) then there eists a soltion ((t), y(t)) t t <, (t ), hat initiates at the point (, F( )) on the vertical isocline y F() ( > ), remains at the first qadrant and approaches the origin. By calclating d dt along with ((t), y(t)), t t < we get Since d dt d dt d dt + dy y dt () ep ( dy < and dt () <, then (6) gives d dt 2 () (d dt )(dy dt ) ep ( [ d ( dt )2 + ( dy dt )2]. (6) d 2( dt ) ep (,
and by integration from t to t we arrive at ( (t), y(t) ) K t where K F( ) ep ( (t) or eqivalently (t) t F(ξ) (ξ) ep ( ξ ( F(ξ) (ξ) 2) ep ( A. Aghajani et al. / Filomat 3:2 (27), 376 377 3765 2 d dτ ep( (τ) ) dτ 2 (t) ). Now, since y(t) >, (3) together (7) give ξ ) (t) dξ K 2 ep ( dξ K. This contradicts (5) and the proof is complete. Theorem 3.3. Sppose that + ξ ep ( ξ dξ, (7) dξ, ( F(ξ) 2 ) (ξ) dξ +, (8) 2G(ξ) G(ξ) for some >. Then Liénard system () fails to have property (Z + ). Proof. Define for > the fnction () : 2G(), and the mapping Λ : R + R (, 2G( ) ) R by Λ(, y) ((), y) (, v), Then the mapping Λ is a diffeomorphism of the right half-plane onto (, 2G( ) ) that transforms the system () to the following simpler form of the Liénard system d dτ v F () and dv dτ, in which dτ dt () and 2G() F () F ( G ( 2 2 )). Conseqently, we have only to determine whether the system ). Applying Theorem 3.2 to this system, it sffices to have above, instead of (), fails to have property (Z + η + η F () 2 d +. 2 Now, the change of variable 2G() in the integral above and the fact that 2G() as +, give the eqivalent condition (8). The following corollary is an improvement of Theorem.2. Corollary 3.4. Sppose that F() 2 2G() h( 2G()), where is a continos fnction sch that for some > + d +. 2 Then the Liénard system () fails to have property (Z + ).
Proof. We have ( F(ξ) 2G() 2 ) (ξ) G(ξ) dξ 2 A. Aghajani et al. / Filomat 3:2 (27), 376 377 3766 2G() ( ) (ξ) F(ξ) 2 2G(ξ) dξ [2G(ξ)] 3 2 h ( 2G(ξ) ) d ( ) 2G(ξ) 2G() 2G() 2G() d ( ) d. 2 This, together with the fact that 2G() as + gives s + ( F(ξ) 2 ) (ξ) dξ 2G(ξ) G(ξ) + Now, Theorem 3.3 completes the proof. d +. 2 Eample 3.5. Consider the Liénard system () with () and F() 3 2.2 sin 2( ). Here, we have G() 2 2. Writing F() 2 h(), where h() + 2.2 sin2( ), then we have + 2.2 sin 2( ) d 2.. cos ( 2 ) d. d +. + cos d +. Therefore, from Corollary 3.4 system () fails to have property (Z + h() ). Notice that, in this eample the fnction is not non-decreasing in any interval (, ) ( > ), hence we cannot se Theorem.2. Remark 3.6. The method in which F() is compared with 2G() to obtain sfficient conditions nder which the system () has or fails to have property (Z + ) (and other similar properties) was initiated by A.F. Filippov [7] and improved by J. Sgie and T. Hara [, 6]; see also [ 5,, 2 5, 7 2]. In the net reslts we se a new approach and in parallel with the above reslts, we compare F() with 2 (), instead of 2G() (of corse nder some additional assmptions on ()). We start with a simple observation based on Hara s lemma. Proposition 3.7. Sppose that for some >, If we have (), < <. F() > 2 (), < <, then the Liénard system () has property (Z + ). (9) ()
A. Aghajani et al. / Filomat 3:2 (27), 376 377 3767 Proof. Taking ψ() () then by assmptions (9) and () we have () < F() ψ() for < <, and hence (ξ) F(ξ) ψ(ξ) dξ dξ () ψ(), < <. Now, Hara s lemma gives the desired reslt. Proposition 3.8. Sppose that for some >, ep ( d +. () If we have + F() () < 2, then the Liénard system () fails to have property (Z + ). Proof. From (2) we have F(ξ) 2 m <, for some m (, 2) and all ξ > sfficiently small. Ths sing (ξ) (), we get ( F(ξ) + (ξ) 2) ep ( ξ dξ m ep ( d +, and from Theorem 3.2 the proof is complete. Theorem 3.9. Sppose that F and satisfy (2) and +, F() 2 () h ( ep, where h is a nonnegative fnction on (, ) for some >, and satisfies (3) + d +. 2 Then the Liénard system () fails to have property (Z + ). Proof. Using the above assmptions we have ( F(ξ) (ξ) 2) ep ( ξ dξ () () () () ( )( ) ( ξ F(ξ) 2 (ξ) ep dξ (ξ) h ( ξ ep( )) d ( ep ( ξ ) d ( ) d, 2
where () ep ( A. Aghajani et al. / Filomat 3:2 (27), 376 377 3768. From (3) we also have () as +, ths ( F(ξ) ξ 2) ep( + (ξ) and the proof is complete from Theorem 3.2. )dξ + d +, 2 Theorem 3.. Sppose that for > sfficiently small, is differentiable, and () k, +, () ep ( > l >. (4) (5) (6) If we have F() 2 () h ( ep, (7) where is a non-negative continos fnction sch that for > sfficiently small is non-decreasing, (8) and a[h()] 2 where H(ξ) ξ Proof. Let ρ lim + have H( ) ρ for some a > k2 l, (9) 2 d, then the Liénard system () has property (Z+ ). η Define () ep ( < σ < k2 la. h() h(). Then by (8) ρ and ρ for >. If ρ >, then for some > we h(), which contradicts (9). Therefore we have and H() as +. Then there eist > sch that for < <, ). From (5), () as +. Let σ be chosen so that () () > l, h(()) () < σ 2k l and H(()) < σ l. (2) 2k(k + ) In Hara s lemma take ψ() () + k()h(()). Then we have [ () F() ψ() () () h ( ) () kh ( () )] >, ()
for < <. Define the fnction φ() by We have φ() ψ() d h(()) φ() d r() A. Aghajani et al. / Filomat 3:2 (27), 376 377 3769 (ξ) dξ, >. F(ξ) ψ(ξ) [ k(k + ) ( () () ) ( () )h ( () ) H() k () () hence by (2), d d φ() h ( ) () ( σ r() 2 σ 2 k2 ) >, < <, al k 2( () () )H 2( () ) ] () h ( () ), where r() () h(()) k()h(()). Therefore, φ() > for < < and Hara s lemma completes the proof. Eample 3.. To present an eample of a fnction that satisfies all the reqired assmptions appearing in Theorem 3., consider the Liénard system () with () ( + ), on (, ). Here, we have () > and ln 2 () < () + ln 2 () 2 ln 3 () < 6, for (, e ). Also we have Therefore, ln arctan(ln ). lim () ep ( + + and e π 2 >, ep ( d +, so assmptions (4), (5) and (6) are satisfied. Hence, for any fnction F which satisfies (7), (8) and (9) with a sitable h, Liénard system () has property (Z + ). References [] RP. Agarwal, A. Aghajani, V. Roomi, Eistence of Homoclinic orbits for general planer dynamical system of Liénard type, Dyn. Contin. Discret. I. Series A: Mathematical Analysis 8 (22) 267-284. [2] A. Aghajani, A. Moradifam, The Generalized Liénard Eqations, Glasgow Mathematical Jornal 5, 3 (29) 65-67. [3] A. Aghajani, A. Moradifam, Intersection with the vertical isocline in the Liénard plane, Nonlinear Analysis, 68 (28) 3475-3484. [4] A. Aghajani, D. O Regan, V. Roomi, Oscillation of soltions to second-order nonlinear differential eqations of generalized Eler type, Electr. J. Differ. Eq. 23, 85 (23) -3. [5] A. Aghajani, A. Moradifam, On the homoclinic orbits of the generalized Liénard eqations, Applied Mathematics Letters 2, 3 (27) 345-35. [6] L. A. Cherkas, Conditions for a Liénard eqation to have a center, Differential Eqations, 2, 2 (977) 2-26 [7] A.F. Filippov, Sfficient conditions for the eistence of stable limit cycles of second order eqations, Mat. Sbornik 3 (952) 7-8. [8] J.R. Graef, Bondedness and oscillation of soltions of the Liénard eqation, Bll. Amer. Math. Soc. 77 (97) 48-42. [9] S.R. Grace, Oscillation theorems for nonlinear differential eqations of second order, J. Math. Anal. Appl. 7 (992), 22-24. [] M. Gyllenberg, P. Yan, New conditions for the intersection of orbits with the vertical isocline of the Liénard system, Mathematical and Compter Modelling 49, 56 (29) 96-9. [] T. Hara, Notice on the Vinograd type theorems for Liénard system, Nonlinear Anal. 22 (994) 437-444.
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