Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations

Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations Oleg Makarenkov ∗ , Paolo Nistri † Dept. of Mathematics, Voronezh State University, Voronezh, Russia e-mail: omakarenkov@kma.vsu.ru arXiv:0709.4418v1 [math.CA] 27 Sep 2007 Dip. di Ingegneria dell’ Informazione, Università di Siena, 53100 Siena, Italy e-mail: pnistri@dii.unisi.it Abstract In this paper we consider a class of planar autonomous systems having an isolated limit cycle x0 of smallest period T > 0 such that the associated linearized system around it has only one characteristic multiplier with absolute value 1. We consider two functions, defined by means of the eigenfunctions of the adjoint of the linearized system, and we formulate conditions in terms of them in order to have the existence of two geometrically distinct families of T −periodic solutions of the autonomous system when it is perturbed by nonsmooth T −periodic nonlinear terms of small amplitude. We also show the convergence of these periodic solutions to x0 as the perturbation disappears and we provide an estimation of the rate of convergence. The employed methods are mainly based on the theory of topological degree and its properties that allow less regularity on the data than that required by the approach, commonly employed in the existing literature on this subject, based on various versions of the implicit function theorem. Keywords: planar autonomous systems, limit cycles, characteristic multipliers, nonsmooth periodic perturbations, periodic solutions, topological degree. 1. Introduction Loud in [23] provided conditions under which the perturbed system of ordinary differential equations ẋ = ψ(x) + εφ(t, x, ε), (1) ψ ∈ C 2 (Rn , Rn ), φ ∈ C 1 (R × Rn × [0, 1], Rn ) (2) where and φ is T -periodic with respect to time, has, for sufficiently small ε > 0, a T -periodic solution which tends to a T -periodic limit cycle x0 of the unperturbed system ẋ = ψ(x) (3) as ε → 0. The limit cycle x0 satisfies the property that the linearized system ẏ = ψ ′ (x0 (t))y ∗ Supported (4) by a President of Russian Federation Fellowship for Scientific Training Abroad, by the Grant VZ-010 of RF Ministry of Education and U.S.CRDF, and by RFBR Grants 07-01-00035, 06-01-72552, 05-01-00100. † Supported by the national research project PRIN “Control, Optimization and Stability of Nonlinear Systems: Geometric and Topological Methods”. Corresponding author. 1 has only one characteristic multiplier with absolute value 1. Here and in the following by C i (Rm , Rn ) we denote the vector space of all continuous functions acting from Rm to Rn having i-th continuous derivatives. The main tool employed by Loud is the following, so-called bifurcation, function Z T hz0 (τ ), φ(τ − θ, x0 (τ ), 0)i dτ, f0 (θ) = (5) 0 where z0 is a T -periodic solution of the adjoint system of (4) ∗ ż = − (ψ ′ (x0 (t))) z, (6) here A∗ denotes the transpose of the matrix A. Specifically, Lemma 2 in [23] states that in order that system (1) has a T -periodic solution xε such that xε (t − θ0 ) → x0 (t) as ε → 0 (7) it is necessary that θ0 ∈ R be a zero of the equation f0 (θ) = 0. (8) If (8) is satisfied for some θ = θ0 and f0′ (θ0 ) 6= 0, i.e. θ0 is simple, then by ([23], Theorem 1) for all sufficiently small ε > 0 system (1) possesses a T -periodic solution xε satisfying kxε (t − θ0 ) − x0 (t)k ≤ εM, (9) where M > 0 is a constant. These results are also consequences of general results stated by Malkin in [26]. The function f0 has been widely employed to treat different problems concerning periodic solutions of system (1) with ε > 0 small. We quote in the sequel some papers from the relevant bibliography devoted to this subject. In [23] Loud also considered the case when (8) is identically satisfied, i.e. f0 (θ) = 0 for any θ ∈ [0, T ], to treat this case he introduced a new function which plays the role of f0 and he showed that if this function has a simple zero θ0 then there exists a family of T -periodic solutions to (1) satisfying (7) (see also [22]). Moreover in [23] it is also considered the case when θ0 is not a simple zero of f0 , and the problem of the existence of T -periodic solutions to (1) is associated with the problem of the existence of roots of a certain quadratic equation. The case when the limit cycle x0 of system (3) is not isolated, in particular, when the unperturbed system is Hamiltonian, and the case when the characteristic multiplier of system (4) is not simple have been considered by many authors. If system (3) is not necessary autonomous and it has a multi-parameterized family of T -periodic solutions, then existence of T -periodic solutions of the perturbed system satisfying (15) was proved by Malkin [26]. Melnikov [29] treated the case when the limit cycle is not isolated and the limit cycles near x0 are of different periods (see also Loud [24] and Kac [14]) and he showed that the simple zeros of suitably defined bifurcation functions fm,n , m, n ∈ N, called Melnikov subharmonic functions, generate periodic solutions in a neighborhood of x0 whose periods are in m : n ratio with respect to the periods of the perturbation term. Finally, Rhouma and Chicone [34] have considered the case when 1 is not a simple multiplier of the linearized system, to deal with the problem of existence of T -periodic solutions they introduced a new two variables bifurcation function f0 whose simple zeros determine families of T -periodic solutions satisfying (7). These theoretical results have been then developed in different directions: Hausrath and Manásevich [10], (see also [11]), found a class of T -periodic perturbations φ for which the subharmonic Melnikov function f1,1 2 has at least two simple zeros, obtaining the existence of at least two families of T -periodic solutions to (1) satisfying (7). Makarenkov in [27] provided useful formulas to calculate simple zeros of Malkin’s bifurcation function in case when the function φ is sinusoidal in time. Tkhai [37] and Lazer [20] developed Malkin’s and Melnikov’s approaches respectively to study the existence of periodic solutions to (1) satisfying (7) and possessing some additional symmetry properties that represent relevant features in the applications. Farkas in [13] investigated the existence of the so-called D-periodic solutions to (1) which are not necessarily periodic but having periodic derivative. Greenspan and Holmes in [8] and Guckenheimer and Holmes in [7] applied the method of subharmonic Melnikov’s functions to a variety of practical problems, a number of applications of Malkin’s bifurcation function can be found in the book of Blekhman [1]. In all the previous papers, to show the existence of T -periodic solutions for ε > 0 small, several formulations of the implicit function theorem have been employed. Therefore, condition (2) is the common assumption of these papers (sometimes it is even required more regularity on ψ and φ). The persistence of the limit cycle x0 under less restrictive regularity assumptions than (2) is studied only for the cases when system (3) is linear, in this case the modified averaging methods developed by Mitropol’sksii [30] and Samoylenko [35] can be applied as well as the coincidence degree theory introduced by Mawhin, see, for instance, ([28], Theorem IV.13); Hamiltonian, see M. Henrard and F. Zanolin [12]; or piecewise differentiable, see Kolovskiı̆ [17] and Šteı̆nberg [36]. In the present paper we assume that the linearized system (4) has only one characteristic multiplier equal to 1 and ψ ∈ C 1 (R2 , R2 ), φ ∈ C(R × R2 × [0, 1], R2 ). (10) By combining the function f0 with the analogously defined function Z s hz1 (τ ), φ(τ − θ, x0 (τ ), 0)i dτ, f1 (θ, s) = s−T where z1 is an eigenfunction of system (6) corresponding to the characteristic multiplier ρ∗ 6= 1, we give conditions in Theorem 3 for the existence of T -periodic solutions to (1) satisfying (7). Although, as we have mentioned before, in many papers it was proved the existence of two or more families of T -periodic solutions to (1) converging to x0 in the sense of (7), it was not guaranteed that these families do not coincide geometrically, namely if one is just a shift in time of the other. In this paper our results ensure the existence of at least two geometrically distinct families of T -periodic solutions to (1) satisfying (7). Moreover, since property (9) is a consequence of the application of the implicit function theorem it is not anymore guaranteed under our conditions (10). However, we will show in Theorem 1 that under conditions (10) the following property holds εM1 |f1 (θ0 , t)| ≤ kxε (t − θε (t)) − x0 (t)k + o(ε) ≤ εM2 |f1 (θ0 , t)| for any ε ∈ (0, ε0 ) and any t ∈ [0, T ], (11) where 0 < M1 < M2 and θε (t) → θ0 as ε → 0 uniformly with respect to t ∈ [0, T ]. The introduction of the function f1 , as shown by (11) and Corollaries 1 and 2 of this paper, gives a new qualitative information about the convergence (7) with respect to (9) and it is a contribution to the problem posed by Hale and Tb́oas in [9] concerning the behavior of the periodic solutions of a second order periodically perturbed autonomous system when the perturbation disappears. We would like also to remark, that Loud in [23] provided a precise information about the way of convergence of xε to x0 by means of the representation xε (t) = x0 (t + θ0 ) + εy(t + θ0 ) + o(ε), 3 where the function y is a suitably chosen solution of system (60) of this paper with ξ = x0 (0), see ([23], formulas 1.3 and 2.11). In order to prove the existence of T -periodic solutions to (1) satisfying (7) under assumptions (10) we make use of the topological degree theory. Specifically, for ε > 0, we consider the integral operator Gε : C([0, T ], R2 ) → Rt Rt C([0, T ], R2 ) given by (Gε x)(t) = x(T ) + 0 ψ(x(τ ))dτ + ε 0 φ(τ, x(τ ), ε)dτ, t ∈ [0, T ], here C([0, T ], R2 ) is the Banach space of all the continuous functions defined on [0, T ] with values in R2 equipped with the sup-norm. We also consider the Leray-Schauder degree d(I − Gε , WU ), see Brown ([4], §9), of the compact vector field I − Gε  with respect to the open set WU = x ∈ C([0, T ], R2 ) : x(t) ∈ U, for any t ∈ [0, T ]} , where U is an open set of R2 . We will provide conditions in terms of the functions f0 and f1 ensuring that d(I −Gε , WU0 ) 6= d(I −Gε , WUε ) for all ε > 0 sufficiently small, where U0 is the interior of the limit cycle x0 and Uε ⊂ U0 (or U0 ⊂ Uε ) is a suitably defined family of sets such that Uε → U0 as ε → 0. To do this we use a result by Capietto, Mawhin and Zanolin ([5], Corollary 1) which, under our assumptions, states that d(I − G0 , WUε ) = 1 for ε > 0 sufficiently small. Then in Theorem 2, by means of a result due to Kamenskii, Makarenkov and Nistri ([15], Theorem 2), we conclude that there exists a continuous vector field F : R2 → R2 with F (x0 (θ)) = f0 (θ)ẋ0 (θ) + f1 (θ, θ)y1 (θ), for any θ ∈ [0, T ], (12) here y1 denotes the eigenfunction of (4), corresponding to the characteristic multiplier ρ 6= 1, such that for all ε > 0 sufficiently small we have that d(I − Gε , WU0 ) = dB (F, U0 ), where dB (F, U0 ) is the Brouwer degree of F on U0 , see e.g. Brown ([4], §8). In our case the integer dB (F, U0 ) can be easily calculated since it is equal to the Poincaré index of x0 with respect to the vector field F multiplied by +1 or −1 according with the orientation of x0 , see Lefschetz ([21], Ch. IX, §4). Furthermore, as it was observed by Bobylev and Krasnoselskii in [2], for any small neighborhood Bδ (∂WU0 ) of the boundary ∂WU0 of WU0 , we have that d(I − G0 , Bδ (∂WU0 )) = 0 and so one cannot directly apply Leray-Schauder fixed point theorem for studying the existence of T -periodic solutions to (1) satisfying (7). The paper is organized as follows. Theorem 1 of Section 2 states property (11) for the T -periodic solutions of system (1), in Theorem 2 we prove the coincidence degree formula d(I − Gε , WU0 ) = dB (F, U0 ) for ε > 0 small. Finally, we give the main result of the paper: Theorem 3 which states the existence of at least two geometrically distinct families of T -periodic solutions to (1) satisfying (7). In Section 3 we provide an example which shows how formula (12) can be used for the practical calculation of dB (F, U0 ). In fact, under quite general conditions on f0 and f1 , we show that if φ(t, ξ) = −φ(t + T /2, ξ) for any t ∈ [0, T ] and ξ ∈ R2 , then dB (F, U0 ) ∈ {0, 2}. Finally, we outline some methods for calculating the eigenfunctions y1 , z0 and z1 . 2. Main results. Through the paper we assume the following condition: (A0 )− system (3) has a limit cycle x0 with smallest period T > 0 and the linearized system (4) has only one characteristic multiplier equal to 1. In what follows we provide the notations that we will use in the proofs of the results of this Section. By y1 we denote the eigenfunction of (4) corresponding to the characteristic multiplier ρ 6= 1 (clearly ẋ0 is the eigenfunction of (4) corresponding to the characteristic multiplier 1). Moreover, z0 and z1 will denote the 4 eigenfunctions of (6) corresponding to the characteristic multipliers 1 and ρ∗ 6= 1 respectively. (a1 , a2 ) is the matrix whose columns are the vectors a1 , a2 ∈ R2 , (a1 , a2 )∗ denotes the transpose of (a1 , a2 ), Ω(·, t0 , ξ) is the solution of system (3) satisfying x(t0 ) = ξ, Ω′ξ (·, t0 , ξ) is the derivative of Ω(·, t0 , ξ) with respect to the third variable, U0 is the interior of the limit cycle x0 of system (3), ∂U0 is the boundary of U0 , [v]i , i = 1, 2, is the i-th component of vector v ∈ R2 . a k b will indicate that the vectors a, b ∈ R2 are parallel, a⊥ denotes the vector |ha, bi| a ∈ R2 rotated of π/2 clockwise and ∠(a, b) = arccos is the angle between the two vectors a, b ∈ R2 . kak · kbk By o(ε), ε > 0, we will denote a function, which may depend also on other variables having the property that o(ε) → 0 as ε → 0 uniformly with respect to the other variables when they belong to a bounded set. ε Finally, let t, r ∈ R and let h(t, r) be the vector of R2 given by h(t, r) = x0 (t) + r z0 (t)⊥ . kz0 (t)⊥ k (13) Define the function (t, r) → I(t, r) as follows I(t, r) = Ω(T, 0, h(t, r)). (14) It is easily seen that, for any t ∈ [0, T ], the curve r → I(t, r) intersects the limit cycle x0 at the point I(t, 0) = x0 (t). The following theorem states a property similar to (9) in the case when the autonomous system (3) is perturbed by nonsmooth functions φ. Theorem 1. Assume conditions (10). Assume that, for all sufficiently small ε > 0, system (1) has a T -periodic solution xε satisfying xε (t − θ0 ) → x0 (t) as ε → 0, (15) for any t ∈ [0, T ], where θ0 ∈ [0, T ]. Then there exist constants 0 < M1 < M2 , ε0 > 0 and r0 ∈ (0, 1] such that εM1 |f1 (θ0 , t)| ≤ kxε (t − θε (t)) − x0 (t)k + o(ε) ≤ εM2 |f1 (θ0 , t)| f or any ε ∈ (0, ε0 ) and any t ∈ [0, T ], (16) where θε (t) → θ0 as ε → 0 uniformly with respect to t ∈ [0, T ], and xε (t − θε (t)) ∈ I(t, [−r0 , r0 ]), t ∈ [0, T ]. To prove Theorem 1 we need some preliminary lemmas. Lemma 1. For any t ∈ R we have  (ẋ0 (t) y1 (t))∗ (z0 (t) z1 (t)) =  hẋ0 (0), z0 (0)i 0 0 hy1 (0), z1 (0)i  . (17) Proof. By Perron’s lemma [32] (see also Demidovich ([6], Sec. III, §12) for any t ∈ R we have     h ẋ (t), z (t)i h ẋ (t), z (t)i h ẋ (0), z (0)i h ẋ (0), z (0)i 0 0 0 1 0 0 0 1 = . (ẋ0 (t) y1 (t))∗ (z0 (t) z1 (t)) :=  hy1 (t), z0 (t)i hy1 (t), z1 (t)i hy1 (0), z0 (0)i hy1 (0), z1 (0)i Thus, in particular, hẋ0 (0), z1 (0)i = hẋ0 (T ), z1 (T )i . On the other hand ẋ0 (0) = ẋ0 (T ) and z1 (T ) = ρ∗ z1 (0), ρ∗ 6= 1, thus hẋ0 (0), z1 (0)i = 0. Analogously, since y1 (T ) = ρy1 (0), ρ 6= 1, and z0 (0) = z0 (T ), we have that hy1 (0), z0 (0)i = 0. 5 Lemma 2. Under the assumptions of Theorem 1 there exist r0 ∈ (0, 1] and α0 ∈ [0, π/2) such that ∠(I(t, r) − x0 (t), ẋ0 (t)⊥ ) < α0 for any t ∈ [0, T ] and any r ∈ [−r0 , r0 ]. (18) Proof. Assume the contrary, hence there exist sequences {tn }n∈N ⊂ [0, T ], tn → t0 as n → ∞, {rn }n∈N ⊂ (0, 1], rn → 0 as n → ∞ such that ∠(I(tn , rn ) − x0 (tn ), ẋ0 (tn )⊥ ) → π/2 as n → ∞. (19) We have   z0 (tn )⊥ z0 (tn )⊥ I(tn , rn ) − x0 (tn ) = Ω T, 0, x0 (tn ) + rn + o(rn ). − x0 (tn ) = rn Ω′ξ (T, 0, x0 (tn )) ⊥ kz0 (tn ) k kz0 (tn )⊥ k (20) By Theorem 2.1 of [18] it follows that Ω′ξ (T, 0, h(t, 0)) = Y (T, t) where Y (·, t) is the fundamental matrix for the system ẏ(τ ) = ψ ′ (x0 (τ + t))y(τ ), (21) satisfying Y (0, t) = I, thus Ω′ξ (T, 0, h(t, 0))y1 (t) = ρy1 (t). On the other hand from Lemma 1 we have y1 (t) k z0 (t)⊥ , (22) therefore Ω′ξ (T, 0, h(t, 0))z0 (t)⊥ = ρz0 (t)⊥ and (20) can be rewritten as follows I(tn , rn ) − x0 (tn ) = ρrn Hence z0 (tn )⊥ + o(rn ). kz0 (tn )⊥ k (23)  o(rn ) z0 (tn )⊥ ⊥ , ẋ (t ) + 0 n kz0 (tn )⊥ k rn . ∠(I(tn , rn ) − x0 (tn ), ẋ0 (tn )⊥ ) = arccos o(rn ) z0 (tn )⊥ ⊥k · k ẋ (t ) + ρ 0 n kz0 (tn )⊥ k rn  ρ Without loss of generality we may assume hẋ0 (0), z0 (0)i = 1 (24) and so ∠(I(tn , rn ) − x0 (tn ), ẋ0 (tn )⊥ ) → arccos kz0 (t0 )⊥ k 1 · kẋ0 (t0 )⊥ k as n → ∞ contradicting (19). Therefore there exist r0 ∈ (0, 1] and α0 ∈ [0, π/2) satisfying (18). We can now prove the following. Lemma 3. Under the assumptions of Theorem 1 there exists ε0 > 0 such that for any ε ∈ (0, ε0 ) and any t ∈ [0, T ] we have xε (t − θε (t)) ∈ I(t, [−r0 , r0 ]) where θε (t) = θ0 − ∆ε (t), ∆ε (t) ∈ [t − T 2,t + T 2] and ∆ε (t) → 0 as ε → 0 uniformly in t ∈ [0, T ]. Moreover, there exists M > 0 such that kxε (t − θε (t)) − x0 (t)k ≤ εM for any ε ∈ (0, ε0 ) 6 and any t ∈ [0, T ]. (25) Proof. First of all observe that r0 > 0 given by Lemma 2 can be chosen to satisfy I(t, [−r0 , r0 ]) ∩ x0 ([0, T ]) = {x0 (t)}, for any t ∈ [0, T ]. (26) From (18) of Lemma 2 and (15) we have that there exists ε0 > 0 such that I(t, [−r0 , r0 ]) ∩ xε ([0, T ]) 6= ∅ for   any ε ∈ (0, ε0 ) and any t ∈ [0, T ]. Hence, for any ε ∈ (0, ε0 ) and t ∈ [0, T ] there exists ∆ε (t) ∈ t − T2 , t + T2 such that xε (t − θ0 + ∆ε (t)) ∈ I(t, [−r0 , r0 ]). (27) ∆ε (t) → 0 as ε → 0 (28) We claim that uniformly with respect to t ∈ [0, T ]. In fact, assume the contrary, thus there exist sequences {εn }n∈N ⊂ (0, ε0 ), εn → 0 as n → ∞, and {tn }n∈N , tn → t0 ∈ [0, T ] as n → ∞, such that ∆εn (tn ) → ∆0 6= 0 and xεn (tn − θ0 + ∆εn (tn )) ∈ I(tn , [−r0 , r0 ]). Since from (26) we have I(tn , [−r0 , r0 ]) ∩ x0 ([0, T ]) = {x0 (tn )} then xεn (tn − θ0 + ∆εn (tn )) → x0 (t0 ) as n → ∞. (29) xεn (tn − θ0 + ∆εn (tn )) → x0 (∆0 + t0 ). (30) Applying (15) we have From (29) and (30) we conclude that x0 (t0 ) = x0 (∆0 + t0 ),  where ∆0 ∈ t0 − contradiction. T 2 , t0 +  T 2 (31) , since T is the smallest period of x0 it follows from (31) that ∆0 = 0, which is a Pick any τ ∈ [0, T ], in what follows we show that the shifts t → ∆ε (t) have the property that the convergence of xε (τ + t − θε (t)) to x0 (τ + t) is of order ε > 0, where θε (t) = θ0 − ∆ε (t), and thus the claim of Lemma 3 is proved. For this consider the change of variables νε (τ, t) = Ω(0, τ, xε (τ + t − θε (t))) in system (1). It is clear that xε (τ + t − θε (t)) = Ω(τ, 0, νε (τ, t)) and so ẋε (τ + t − θε (t)) = ψ(Ω(τ, 0, νε (τ, t)) + Ω′ξ (τ, 0, νε (τ, t))(νε )′τ (τ, t). (32) On the other hand from (1) we have ẋε (τ + t − θε (t)) = ψ(Ω(τ, 0, νε (τ, t))) + εφ(τ + t − θε (t), Ω(τ, 0, νε (τ, t)), ε). (33) From (32) and (33) it follows
−1 φ(τ + t − θε (t), Ω(τ, 0, νε (τ, t)), ε) (νε )′τ (τ, t) = ε Ω′ξ (τ, 0, νε (τ, t)) and since νε (0, t) = xε (t − θε (t)) = xε (T + t − θε (t)) = Ω(T, 0, νε (T, t)) (34) we finally obtain νε (τ, t) = Ω(T, 0, νε (T, t)) + ε Z 0 τ
−1 Ω′ξ (s, 0, νε (s, t)) φ(s + t − θε (t), Ω(s, 0, νε (s, t)), ε)ds. 7 (35) Since νε (τ, t) → Ω(0, τ, x0 (τ + t)) = x0 (t) as ε → 0 we can write νε (τ, t) in the following form νε (τ, t) = x0 (t) + εµε (τ, t). (36) Subtract x0 (t) from both sides of (35) obtaining εµε (τ, t) = ε Ω′ξ (T, 0, x0 (t))µε (T, t) + o(εµε (T, t)) + Z τ
−1 Ω′ξ (s, 0, νε (s, t)) φ(s + t − θε (t), Ω(s, 0, νε (s, t)), ε)ds. +ε (37) 0 Since xε (t − θε (t)) ∈ I(t, [−r0 , r0 ]) then from (14) there exists rε (t) ∈ [−r0 , r0 ] such that xε (t − θε (t)) = Ω(T, 0, h(t, rε (t))) and by (13) we get εµε (T, t) = νε (T, t) − x0 (t) = Ω(0, T, xε (t − θε (t))) − x0 (t) = = Ω(0, T, Ω(T, 0, h(t, rε (t)))) − x0 (t) = h(t, rε (t)) − x0 (t) = rε (t) z0 (t)⊥ . kz0 (t)⊥ k Therefore µε (T, t) k z0 (t)⊥ and by (22) we can rewrite (37) as follows Z τ
−1 Ω′ξ (s, 0, νε (s, t)) φ(s + t − θε (t), Ω(s, 0, νε (s, t)), ε)ds. εµε (τ, t) = ερµε (T, t) + o(εµε (T, t)) + ε (38) 0 We now prove that the functions (τ, t) → µε (τ, t) are uniformly bounded with respect to ε ∈ (0, ε0 ). For this we argue by contradiction, therefore there exist sequences {εn }n∈N ⊂ (0, 1), εn → 0 as n → ∞, {τn }n∈N ⊂ [0, T ], τn → τ0 as n → ∞ and {tn }n∈N ⊂ [0, T ], tn → t0 as n → ∞, such that kµεn (τn , tn )k → ∞ as n → ∞, so µεn (τ, tn ) kµεn (·, tn )kC → ∞ as n → ∞, where k · kC is the usual sup-norm of C([0, T ], R2 ). Let qn (τ ) = , kµεn (·, tn )kC then from (38) we have qn (τ ) = o(εn µεn (T, tn )) + εn kµεn (·, tn )kC Z τ
−1 1 + φ(s + tn − θεn (tn ), Ω(s, 0, νεn (s, tn )), εn )ds. Ω′ξ (s, 0, νεn (s, tn )) kµεn (·, tn )kC 0 ρqn (T ) + (39) By definition the set of continuous functions A = {qn , n ∈ N}, is bounded and, as it is easy to see from (39), A is also equicontinuous. Therefore, by the Ascoli-Arzela Theorem, see e.g. ([4], Theorem 2.3), we may assume without loss of generality that the sequence {qn }n∈N is converging. Let q0 = limn→∞ qn , from (39) we may conclude that q0 (τ ) = ρq0 (T ). (40) By (40) it follows that q0 is a constant function, thus being ρ 6= 1 we have q0 = 0. On the other hand, by the definition of qn , we have that kq0 kC = 1. This contradiction shows the uniform boundedness of the functions µε with respect to ε ∈ (0, ε0 ). On the other hand from (34) and (36)we have that xε (t − θε (t)) − x0 (t) = εµε (0, t), (41) and thus the proof is complete. Proof of Theorem 1. We have to prove (16) with t → θε (t) as given in Lemma 3. For this, by Lemma 1, we can represent xε (τ + t − θε (t)) − x0 (τ + t) as follows xε (τ + t − θε (t)) − x0 (τ + t) = εaε (τ, t)ẋ0 (τ + t) + εbε (τ, t)y1 (τ + t), 8 (42) where εaε (τ, t) = hz0 (τ + t), xε (τ + t − θε (t)) − x0 (τ + t)i and εbε (τ, t) = hz1 (τ + t), xε (τ + t − θε (t)) − x0 (τ + t)i . By Lemma 1 we have that hẋ0 (t), z1 (t)i = 0, for any t ∈ [0, T ], and so ẋ0 (t)⊥ = k (43) kẋ0 (t)⊥ k z1 (t), where k = +1 kz1 (t)k or k = −1. Therefore ẋ0 (τ + t)⊥ , xε (τ + t − θε (t)) − x0 (τ + t) = εbε (τ, t)k kẋ0 (τ + t)⊥ k . kz1 (τ + t)k (44) We aim now at providing an explicit form for (44) by looking for a suitable formula for the function (τ, t) → bε (τ, t). To do this we substract (3) where x(τ ) is replaced by x0 (τ + t) from (1) where x(τ ) is replaced by xε (τ + t − θε (t)) to obtain ẋε (τ + t − θε (t)) − ẋ0 (τ + t) = ψ ′ (x0 (τ + t))(xε (τ + t − θε (t)) − x0 (τ + t)) + + εφ(τ + t − θε (t), xε (τ + t − θε (t)), ε) + o(xε (τ + t − θε (t)) − x0 (τ + t)). (45) By substituting (42) into (45) and taking into account that εaε (τ, t)ψ ′ (x0 (τ + t))ẋ0 (τ + t) = εaε (τ, t)ẍ0 (τ + t) and εbε (τ, t)ψ ′ (x0 (τ + t))y1 (τ + t) = εbε (τ, t)ẏ1 (τ + t) we have εẋ0 (τ + t)(aε )′τ (τ, t) + εy1 (τ + t)(bε )′τ (τ, t) = εφ(τ + t − θε (t), xε (τ + t − θε (t))) + + o(xε (τ + t − θε (t)) − x0 (τ + t)), and so ε(bε )′τ (τ, t) = ε hz1 (τ + t), φ(τ + t − θε (t), xε (τ + t − θε (t)))i + + hz1 (τ + t), o(xε (τ + t − θε (t)) − x0 (τ + t))i . (46) Moreover, since z1 (τ ) = ρ∗ z1 (τ − T ), from (43) it follows that bε (τ, t) = ρ∗ bε (τ − T, t). (47) System (46)-(47) has a unique solution which, as it is easy to verify, is given by the formula Z τ ρ∗ hz1 (s + t), φ(s + t − θε (t), xε (s + t − θε (t)), ε)i ds + bε (τ, t) = ρ∗ − 1 τ −T  Z τ  ρ∗ o(xε (s + t − θε (t)) − x0 (s + t)) + z1 (s + t), ds. ρ∗ − 1 τ −T ε By substituting this formula into (44) we obtain = ẋ0 (τ + t)⊥ , xε (τ + t − θε (t)) − x0 (τ + t) = Z τ kẋ0 (τ + t)⊥ kρ∗ hz1 (s + t), φ(s + t − θε (t), xε (s + t − θε (t)), ε)i ds + εk kz1 (τ + t)k(ρ∗ − 1) τ −T  Z τ  o(xε (s + t − θε (t)) − x0 (s + t)) kẋ0 (τ + t)⊥ kρ∗ ds, z1 (s + t), + εk kz1 (τ + t)k(ρ∗ − 1) τ −T ε 9 (48) where o(xε (s + t − θε (t)) − x0 (s + t)) → 0 as ε → 0 uniformly in s ∈ [−T, T ] in virtue of (25). On the other ε hand ẋ0 (τ + t)⊥ , xε (τ + t − θε (t)) − x0 (τ + t) = =
ẋ0 (τ + t)⊥ kxε (τ + t − θε (t)) − x0 (τ + t)k cos ∠ ẋ0 (τ + t)⊥ , xε (τ + t − θε (t)) − x0 (τ + t) and by taking into account (18) of Lemma 2, (48) and the fact that kẋ0 (t)⊥ k 6= 0 for any t ∈ [0, T ] we get kxε (t − θε (t)) − x0 (t)k Z = εg(t) 0 hz1 (s + t), φ(s + t − θε (t), xε (s + t − θε (t)), ε)i ds + −T + εg(t) Z 0 −T   o(xε (s + t − θε (t)) − x0 (s + t)) z1 (s + t), ds, ε where g(t) = kρ∗ kz1 (t)k(ρ∗ − 1) cos ∠ (ẋ0 (t)⊥ , xε (t − θε (t)) − x0 (t)) is a continuous function on [0, T ] with g(t) 6= 0 for any t ∈ [0, T ], Therefore, kxε (t − θε (t)) − x0 (t)k = εg(t) Z 0 hz1 (s + t), φ(s + t − θε (t), xε (s + t − θε (t)), ε)i ds + o(ε). (49) −T On the other hand from Lemma 3 we have that ∆ε (t) → 0 uniformly in t ∈ [0, T ], thus we can rewrite (49) as follows kxε (t − θε (t)) − x0 (t)k = εg(t) Z 0 hz1 (s + t), φ(s + t − θ0 , x0 (s + t), 0)i ds + o(ε), −T introducing the change of variable s + t = u in the integral we finally get Z t kxε (t − θε (t)) − x0 (t)k = εg(t) hz1 (u), φ(u − θ0 , xε (u − θ0 ), 0)i du + o(ε) t−T from which (16) can be directly derived recalling that, by Lemma 3, xε (t − θε (t)) ∈ I(t, [−r0 , r0 ]) for any ε ∈ (0, ε0 ) and any t ∈ [0, T ]. As a straightforward consequence of Theorem 1 we have the following result. Corollary 1. Assume all the conditions of Theorem 1, then for every t ∈ [0, T ] such that f1 (θ0 , t) = 0 we have kxε (t − θε (t)) − x0 (t)k = o(ε), where θε (t) → θ0 as ε → 0 and xε (t − θε (t)) ∈ I(t, [−r0 , r0 ]). Next result is also a consequence of Theorem 1. Corollary 2. Assume all the conditions of Theorem 1. Moreover, assume that f1 (θ0 , t) 6= 0 f or any t ∈ [0, T ]. Then there exists ε1 > 0 such that xε (s) 6= x0 (t) for any s, t ∈ [0, T ] and any ε ∈ (0, ε1 ). 10 (50) Proof. Let ε0 > 0 given by Theorem 1. From (50) we can choose ε1 ∈ (0, ε0 ) in such a way that, for any ε ∈ (0, ε1 ), we have both o(ε) < εM1 |f1 (θ0 , t)|, for any t ∈ [0, T ], (51) and the validity of (16). Moreover, ε1 can be also chosen in such a way that there exists δ0 > 0 such that the curve τ → xε (τ ) intersects I(t, [−δ0 , δ0 ]) at only one point for any ε ∈ (0, ε1 ) and t ∈ [0, T ]. Such a choice is possible, in fact, since ẋε (τ − θ0 ) → ẋ0 (τ ) as ε → 0 uniformly with respect to τ ∈ [0, T ] and the curve r → I(t, r) intersects the limit cycle x0 transversally at r = 0 for any t ∈ [0, T ], then there exists δ0 > 0 such that xε ([t − θ0 − δ0 , t − θ0 + δ0 ]) and I(t, [−δ0 , δ0 ]) have only one common point for any t ∈ [0, T ] and sufficiently   small ε > 0. On the other hand τ → xε (τ ) cannot intersect I(t, [−δ0 , δ0 ]) for τ ∈ t − θ0 − T2 , t − θ0 − δ0 ∪   t − θ0 + δ0 , t − θ0 + T2 and ε > 0 sufficiently small, otherwise there would exist sequences {εn }n∈N , εn → 0 as n → ∞, {tn }n∈N , tn → t0 ∈ [0, T ] as n → ∞, {τn }n∈N , τn ∈ [tn − θ0 − T2 , tn − θ0 − δ0 ] ∪ [tn − θ0 + δ0 , tn − θ0 + T2 ],     τn → τ0 ∈ t0 − θ0 − T2 , t0 − θ0 − δ0 ∪ t0 − θ0 + δ0 , t0 − θ0 + T2 as n → ∞ such that xεn (τn ) ∈ I(tn , [−δ0 , δ0 ]), thus x0 (τ0 + θ0 ) = x0 (t0 ), with τ0 + θ0 6= t0 and |τ0 + θ0 − t0 | < T , which contradicts the fact that T > 0 is the smallest period of x0 . To conclude the proof assume now, by contradiction, that there exist ε̃ ∈ (0, ε1 ) and s̃, t̃ ∈ [0, T ] such that xε̃ (s̃) = x0 (t̃). Since τ → xε̃ (τ ) intersects I(t̃, [−δ0 , δ0 ]) at only one point then Theorem 1 implies that s̃ = t̃ − θε̃ (t̃). In conclusion, from (16) we have ε̃M1 |f1 (θ0 , t̃)| ≤ o(ε̃) contradicting (51). The following result is crucial for the proof of our existence result Theorem 3, but it can be also considered as an independent contribution to the coincidence degree theory. Theorem 2. Assume conditions (10). For ε > 0, let Gε : C([0, T ], R2 ) → C([0, T ], R2 ) be the operator defined by (Gε x)(t) = x(T ) + Z t ψ(x(τ ))dτ + ε 0 Z t φ(τ, x(τ ), ε))dτ, t ∈ [0, T ]. 0  Let WU0 = x ∈ C([0, T ], R2 ) : x(t) ∈ U0 , for any t ∈ [0, T ]} . Assume that hẋ0 (0), z0 (0)i = hy1 (0), z1 (0)i = 1. (52) Finally, assume that for every θ0 ∈ [0, T ] such that f0 (θ0 ) = 0 we have f1 (θ0 , s + θ0 ) 6= 0, f or any s ∈ [0, T ]. (53) Then, for all ε > 0 sufficiently small, I − Gε : C([0, T ], R2 ) → C([0, T ], R2 ) is not degenerate on the boundary of WU0 . Furthermore, there exists a continuous vector field F : R2 → R2 such that d(I − Gε , WU0 ) = dB (F, U0 ), where F (x0 (θ)) = f0 (θ)ẋ0 (θ) + f1 (θ, θ)y1 (θ) for any θ ∈ [0, T ]. Some remarks are in order. Remark 1. As already observed condition (52) does not affect the generality of Theorem 2. Remark 2. In Theorem 2 we could replace dB (F, U0 ) by k · ind(x0 , F ), where k = +1 or k = −1 according with the orientation of the limit cycle x0 . Precisely, k = +1 if the set U0 is on the left side when one follows 11 ∂U0 according to the parameterization x0 (t) with t increasing from 0 to T, and k = −1 in the opposite case. Moreover, ind(x0 , F ) is the Poincaré index of the trajectory x0 with respect to the vector field F, namely the total variation of an angle function of the vector F (x0 (t)) when t increases from 0 to T, see Lefschetz ([21], Ch. IX, § 4). Remark 3. The Jordan theorem, see Lefschetz ([21], Theorem 4.7), ensures that the interior U0 of x0 does exist and it is an open set. To prove Theorem 2 we need the following preliminary lemma. Lemma 4. For any s ∈ [0, T ], let Z s Fs (ξ) = Ω′ξ (0, τ, Ω(τ, 0, ξ))φ(τ, Ω(τ, 0, ξ), 0)dτ, for any ξ ∈ R2 . (54) s−T Then hz0 (θ), Fs (x0 (θ))i = f0 (θ) f or any s, θ ∈ [0, T ], hz1 (θ), Fs (x0 (θ))i = f1 (θ, s + θ) f or any s, θ ∈ [0, T ]. (55) In particular, Fs (x0 (θ)) = 1 1 f0 (θ)ẋ0 (θ) + f1 (θ, s + θ)y1 (θ) hẋ0 (t), z0 (t)i hy1 (t), z1 (t)i f or any s, θ, t ∈ [0, T ]. (56) Proof. It can be shown, see Krasnosel’skii ([18], Theorem 2.1), that Ω′ξ (t, 0, x0 (θ)) = Y (t, θ), where Y (t, θ) is the fundamental matrix of the system ẏ(t) = ψ ′ (x0 (t + θ))y(t) (57) satisfying Y (0, θ) = I and since Ω′ξ (0, t, Ω(t, 0, x0 (θ))) · Ω′ξ (t, 0, x0 (θ)) = I we have Z s Fs (x0 (θ)) = Y −1 (τ, θ)φ(τ, x0 (τ + θ), 0)dτ. (58) s−T Let us now show that Y −1 (t, θ) = Y (θ, 0)Y −1 (t + θ, 0). (59) In fact, it is easy to see that Y (t + θ, 0) is a fundamental matrix for system (57) and so Y (t + θ, 0)Y −1 (θ, 0) is also a fundamental matrix for (57), moreover we have that Y (t + θ, 0)Y −1 (θ, 0) = I at t = 0. Therefore Y (t + θ, 0)Y −1 (θ, 0) = Y (t, θ) which is equivalent to (59). By substituting (59) into (58) and by the change of variable τ + θ = t in the integral of (58) we obtain Z s+θ Z s Fs (x0 (θ)) = Y (θ, 0) Y −1 (τ + θ, 0)φ(τ, x0 (τ + θ), 0)dτ = Y (θ, 0) Y −1 (t, 0)φ(t − θ, x0 (t), 0)dt. s−T s−T +θ Let Z(t) be the fundamental matrix of system (6) given by Z(t) = Z0 (t)Z0−1 (0), where Z0 (t) = (z0 (t) z1 (t)), t ∈ [0, T ]. Since Y −1 (t, 0) = Z ∗ (t), see Perron [32] and Demidovich ([6], Sec. III, §12), then we have Z s+θ Z s+θ −1 Fs (x0 (θ)) = Y (θ, 0) Y −1 (τ, 0)φ(τ − θ, x0 (τ ), 0)dτ = (Z0∗ (θ)) Z0∗ (τ )φ(τ − θ, x0 (τ ), 0)dτ. s−T +θ s−T +θ Let ∆(s, θ) = Z s+θ Z0∗ (τ )φ(τ − θ, x0 (τ ), 0)dτ, s+θ−T 12 we have = = = =  [zi (θ)]1   [z0 (θ)]1 [z0 (θ)]2 −1 + ,  ∆(s, θ) = [zi (θ)]2 [z1 (θ)]1 [z1 (θ)]2    + * [z1 (θ)]2 −[z0 (θ)]2 [zi (θ)]1 1  ∆(s, θ) =  ,  detZ0 (θ) −[z1 (θ)]1 [z0 (θ)]1 [zi (θ)]2 +   * [zi (θ)]1 [z1 (θ)]2 [∆(s, θ)]1 − [z0 (θ)]2 [∆(s, θ)]2 1  = ,  detZ0 (θ) [zi (θ)]2 −[z1 (θ)]1 [∆(s, θ)]1 + [z0 (θ)]1 [∆(s, θ)]2 hzi (θ), Fs (x0 (θ))i = = * 1 {[zi (θ)]1 [z1 (θ)]2 [∆(s, θ)]1 − [zi (θ)]1 [z0 (θ)]2 [∆(s, θ)]2 − detZ0 (θ) − [zi (θ)]2 [z1 (θ)]1 [∆(s, θ)]1 + [zi (θ)]2 [z0 (θ)]1 [∆(s, θ)]2 } = 1 {[z0 (θ)]1 [z1 (θ)]2 − [z0 (θ)]2 [z1 (θ)]1 } [∆(s, θ)]i+1 = detZ0 (θ) Z s+θ 1 detZ0 (θ) hzi (τ ), φ(τ − θ, x0 (τ ), 0)i dτ. detZ0 (θ) s+θ−T For i = 0, 1, we obtain (55). Furthermore, from Lemma 1 and (52) we have that Fs (x0 (θ)) = 1 1 f0 (θ)ẋ0 (θ) + f1 (θ, s + θ)y1 (θ) hẋ0 (t), z0 (t)i hy1 (t), z1 (t)i for any θ ∈ [0, T ] and any t ∈ [0, T ]. Proof of Theorem 2. Let η(t, s, ξ) be the solution of the system q̇(t) = ψ ′ (Ω(t, 0, ξ))q(t) + φ(t, Ω(t, 0, ξ)) (60) satisfying η(s, s, ξ) = 0 whenever ξ ∈ R2 . It can be shown, see ([16], Lemma 2), that Fs (ξ) = η(T, s, ξ) − η(0, s, ξ). (61) Therefore, from (52), (53) and (56) we have that η(T, s, ξ) − η(0, s, ξ) 6= 0 for any ξ ∈ ∂U0 and any s ∈ [0, T ], (62) and by applying ([15], Theorem 2) we obtain the existence of an ε0 > 0 such that fU0 ) = dB (η(T, 0, ·), U0 ) for any ε ∈ (0, ε0 ), d(I − Gε , W fU0 = WU0 , by fU0 = {x ∈ C([0, T ], R2 ) : Ω(0, t, x(t)) ∈ U0 for any t ∈ [0, T ]}. Since as it is easy to see W where W taking into account (56) and (61) we end the proof by defining F (ξ) = F0 (ξ), ξ ∈ R2 . The following existence theorem is the main result of the paper. Theorem 3. Assume (10). Assume that for every zero θ0 ∈ [0, T ] of the bifurcation function f0 we have f1 (θ0 , t) 6= 0 f or any t ∈ [0, T ]. (63) Let F ∈ C(R2 , R2 ) be a vector field such that on the boundary of U0 it has the form F (x0 (θ)) = f0 (θ)ẋ0 (θ) + f1 (θ, θ)y1 (θ) for any θ ∈ [0, T ]. Assume dB (F, U0 ) 6= 1. 13 (64) Then there exists ε0 > 0 such that for every ε ∈ (0, ε0 ) system (1) has at least two T -periodic solutions x1,ε and x2,ε satisfying xi,ε (t − θi ) → x0 (t) as ε → 0, i = 1, 2, (65) where θ1 , θ2 ∈ [0, T ]. Moreover, we have that x1,ε (t) ∈ U0 and x2,ε (t) 6∈ U0 , for any t ∈ [0, T ] and any ε ∈ (0, ε0 ). Proof. Denote by Wδ (∂U0 ) the δ-neighborhood of the boundary ∂U0 of the set U0 . Let U1,δ = U0 \Wδ (∂U0 ) and U2,δ = U0 ∪ Wδ (∂U0 ), thus the set U1,δ tends to U0 from inside as δ → 0, while U2,δ tends to U0 from outside as δ → 0. Since the limit cycle x0 is isolated then there exists δ0 > 0 such that G0 (x) 6= x for any x ∈ ∂WU1,δ ∪ ∂WU2,δ and any δ ∈ (0, δ0 ]. (66) Moreover, being T > 0, we can choose δ0 > 0 in such a way that ψ(ξ) 6= 0 for any ξ ∈ ∂U1,δ ∪ ∂U2,δ and any δ ∈ [0, δ0 ]. (67) From (67) we get dB (ψ, U1,δ0 ) = dB (ψ, U2,δ0 ) = dB (ψ, U0 ). Since U0 is the interior of the limit cycle x0 of system (3) by Poincaré theorem, see Lefschetz ([21], Theorem 11.1) or Krasnosel’skii et al. ([19], Theorem 2.3) we have dB (ψ, U0 ) = 1 and so dB (ψ, U1,δ0 ) = dB (ψ, U2,δ0 ) = 1. In virtue of (66) and the fact that WU ∩ R2 = U, ([5], Corollary 1) applies to conclude that d(I − G0 , WU1,δ0 ) = dB (ψ, U1,δ0 ) and d(I − G0 , WU2,δ0 ) = dB (ψ, U2,δ0 ), hence d(I − G0 , WU1,δ0 ) = 1 and d(I − G0 , WU2,δ0 ) = 1. Therefore, there exists ε0 > 0 such that d(I − Gε , WU1,δ0 ) = d(I − Gε , WU2,δ0 ) = 1 for any ε ∈ (0, ε0 ). (68) Since by the definition of z1 we have that z1 (t + T ) = ρ∗ z1 (t) for any t ∈ [0, T ], then for any t ∈ [0, T ] it is easily seen that f1 (θ, t + T ) = ρ∗ f1 (θ, t), whenever θ ∈ [0, T ], and thus from (63) we have also that f1 (θ0 , t + θ0 ) 6= 0 for any t ∈ [0, T ]. Therefore, all the conditions of Theorem 2 are satisfied and we can take ε0 > 0 sufficiently small to have d(I − Gε , WU0 ) = dB (F, U0 ) for any ε ∈ (0, ε0 ). (69) By (64), (68) and (69) we conclude that for any ε ∈ (0, ε0 ) there exist x1,ε ∈ WU0 \WU1,δ0 , and x2,ε ∈ WU2,δ0 \WU0 (70) such that Gε (x1,ε ) = x1,ε and Gε (x2,ε ) = x2,ε . From (70) we have that for any ε ∈ (0, ε0 ) there exist points t1,ε , t2,ε ∈ [0, T ] such that x1,ε (t1,ε ) ∈ U0 \U1,δ0 and x2,ε (t2,ε ) ∈ U2,δ0 \U0 . Thus x1,ε (t) → ∂U0 and x2,ε (t) → ∂U0 , for any t ∈ [0, T ], as ε → 0, otherwise there would exist a T -periodic solution x∗ to system (3) and a point t∗ ∈ [0, T ] such that either x∗ (t∗ ) ∈ U0 \U1,δ0 or x∗ (t∗ ) ∈ U2,δ0 \U0 contradicting (66). Therefore, see ([25], 14 Theorem p. 287) or ([23], Lemma 2), for every i ∈ {1, 2} there exists θi ∈ [0, T ] satisfying (65). The fact that x1,ε (t) ∈ U0 and x2,ε (t) 6∈ U0 for any t ∈ [0, T ] and ε > 0 sufficiently small follows from Corollary 2 and so the proof is complete. Remark 4. From the proof of Theorem 3 it results that d(I − Gε , WU0 \W U1 ,δ0 ) and d(I − Gε , W U2 ,δ0 \WU0 ) are different from zero for ε ∈ (0, ε0 ). This fact can be used to obtain stability properties of solutions x1,ε and x2,ε in the case when further information on the number of T -periodic solutions to (1) belonging to the sets WU0 \W U1 ,δ0 and W U2 ,δ0 \WU0 ) are available, see Ortega [31]. 3. An example. In this section we always assume that condition (A0 ) is satisfied. The well known formula by Poincaré, see Krasnoselskii et. al. ([19], formula 1.16) states that 1 ind(x0 , F ) = 2π Z T 0 [α(θ)]1 [α′ (θ)]2 − [α(θ)]2 [α′ (θ)]1 dθ, [α2 (θ)]1 + [α2 (θ)]2 where α(θ) = F (x0 (θ)), θ ∈ [0, T ]. The relationship between ind(x0 , F ) and dB (F, U0 ) was discussed in Remark 2. In this section we show how the representation F (x0 (θ)) = f0 (θ)ẋ0 (θ)+f1 (θ, θ)y1 (θ), θ ∈ [0, T ], of the function F on ∂U0 permits a simpler calculation of dB (F, U0 ). For this, we consider the case when φ(t, ξ) = −φ(t + T /2, ξ), which includes, in particular, the classes of perturbations φ(t, x) = sin t · φ1 (x) and φ(t, x) = cos t · φ1 (x), where φ1 ∈ C(R2 , R2 ). We can prove the following result. Proposition 1. Let Fe ∈ C(R2 , R2 ) be a vector field such that Fe (x0 (θ)) = f0 (θ)ẋ0 (θ) + f1 (θ, T )ẋ⊥ 0 (θ), θ ∈ [0, T ]. Assume that hẋ0 (0), z0 (0)i = hy1 (0), z1 (0)i = 1, f0 (θ) = −f0 (θ + T /2) f or any θ ∈ [0, T ], f1 (θ, T ) = −f1 (θ + T /2, T ) f or any θ ∈ [0, T ]. (71) (72) (73) Moreover, assume that there exists an unique θ0 ∈ [0, T /2) such that f0 (θ0 ) = 0. Finally, assume that the function f0 is strictly monotone at the point θ0 and that f1 (θ0 , T ) 6= 0. (74) Then either dB (Fe, U0 ) = 0 or dB (Fe, U0 ) = 2. The proof of the proposition is based on the following technical lemma. Lemma 5. Let U ⊂ R2 be an open set whose boundary ∂U is a Jordan curve q : [0, T ] → R2 , with q(0) = q(T ). Let Fe : R2 → R2 be a continuous vector field such that Fe (ξ) 6= 0 for every ξ ∈ ∂U. Assume that for a continuous function z : [0, T ] → R2 , z(0) = z(T ), the following conditions hold: 1) hz(θ), q̇(θ)i = 6 0 for every θ ∈ [0, T ], D E 2) the function f (θ) = z(θ), Fe (q(θ)) has exactly two zeros θ1 , θ2 ∈ [0, T ), 3) the function f is strictly monotone at θ1 and θ2 , D E D E 4) sign z(θ1 )⊥ , Fe(q(θ1 )) = −sign z(θ2 )⊥ , Fe (q(θ2 )) . Then either dB (Fe , U ) = 0 or dB (Fe, U ) = 2. 15 Proof. Assume that the parametrization q is positive, namely the set U is on the left side if one follows ∂U according to the orientation given by q when t increases from 0 to T , otherwise we consider the opposite parametrization q̃(θ) = q(−θ). For any t ∈ [0, T ] we denote by Θ(t) the angle (in radians) between the vectors q̇(0) and q̇(t) calculated in the counter-clockwise direction. Clearly Θ(t) is a multi-valued function of t. Let Γq̇ (t) be the single-valued branch of Θ(t) such that Γq̇ (0) = 0 and let Q : ∂U → R2 be the vector field defined by Q(q(t)) := q̇(t), whenever t ∈ [0, T ], hence Γq̇ (t) = ΓQ◦q (t). Following ([19], §1.2) the function t → Γq̇ (t) is called the angle function of the vector field Q on the curve q . Analogously, considering the angle between Fe (q(0)) and Fe (q(t)), we can define the angle function ΓFe◦q (t) of the vector field Fe on the curve q. By the definition of the rotation number for planar vector fields on the boundary of simply-connected sets, see ([19], § 1.3, formula 1.11) we have 1 [Γ e (T ) − ΓFe◦q (0)]. dB (Fe , U ) = 2π F ◦q (75) Therefore, in order to prove the lemma we must calculate the right hand side of (75). For this, denote by h\ 1 , h2 ∈ \ [0, 2π) the angle between the vectors h1 and h2 in the counter-clockwise direction, that is h\ 1 , h2 + h2 , h1 = 2π. Observe that \ ΓFe◦q (θ) − Γq̇ (θ) = Γq̇,Fe◦q (θ) − q̇(0), Fe(q(0)), (76) where Γq̇,Fe◦q (θ) is the single valued branch of the multi-valued angle between q̇(θ) and Fe (q(θ)) such that \ Γq̇,Fe◦q (0) = q̇(0), Fe(q(0)). To calculate Γq̇,Fe◦q (θ) we introduce the function ∠ : R2 × R2 → [−π, π] as follows   h\ 1 , h2 ∠(h1 , h2 ) =  h\ , h − 2π 1 2 as h\ 1 , h2 ∈ [0, π], as h\ 1 , h2 ∈ (π, 2π] By condition 3) we have that ind(θi , f ) = +1 or ind(θi , f ) = −1 according to whether f is increasing or decreasing at θi , i = 1, 2. Up to a shift in time, since θ2 − θ1 < T , we may assume that the zeros θ1 , θ2 of f (θ) = belong to the interval (0, T ). D E z(θ), Fe(q(θ)) Assume that hz(θ), q̇(θ)i > 0 for every θ ∈ [0, T ], otherwise we consider z̃(θ) = z(−θ) instead of z(θ). A possible way to write explicity the function Γq̇,Fe◦q (θ) is the following  D E   ∠(z(θ), q̇(θ)) + ∠(sign z(θ), Fe(q(θ)) z(θ), Fe(q(θ)))   D E    e  ∠(z(θ), q̇(θ)) + ∠(sign z(θ), F (q(θ)) z(θ), Fe(q(θ)))+    D E    +π ind(θ1 , f )sign z(θ1 )⊥ , Fe(q(θ1 )) D E Γq̇,Fe◦q (θ) =   ∠(z(θ), q̇(θ)) + ∠(sign z(θ), Fe(q(θ)) z(θ), Fe(q(θ)))+   D E    ⊥ e  +π ind(θ , f )sign z(θ ) , F (q(θ )) +  1 1 1   D E   ⊥  +π ind(θ2 , f )sign z(θ2 ) , Fe(q(θ2 )) as θ ∈ [0, θ1 ), as θ ∈ (θ1 , θ2 ), as θ ∈ (θ2 , T ]. It is easy to see that the above representation of the function θ → Γq̇,Fe◦q (θ) can be extend to θ1 and θ2 by continuity. Since θ θ → ∠(z(θ), q̇(θ)), D E → ∠(sign z(θ), Fe(q(θ)) z(θ), Fe(q(θ))) 16 are T -periodic functions from (75)-(76), taking into account that dB (Q, U ) = 1 [Γq̇ (T ) − Γq̇ (0)] = 1, 2π (77) (see e.g. [19], Theorem 2.4), we have D E D Ei 1h ind(θ1 , f )sign z(θ1 )⊥ , Fe(q(θ1 )) + ind(θ2 , f )sign z(θ2 )⊥ , Fe(q(θ2 )) . dB (Fe , U ) = 1 + 2 (78) Since the function f is T -periodic then ind(θ1 , f ) = −ind(θ2 , f ) (79) By assumption 4) and (79) the claim can be easily derived from (78). Proof of Proposition 1. ẋ0 (t) , t ∈ [0, T ], thus the function f0 turns out to be the function f defined kẋ0 (t)k2 in Lemma 5. Let us now show that all the conditions of Lemma 5 hold. In fact, we have that hẋ0 (t), z(t)i = 1 Let U = U0 , q(t) = x0 (t), z(t) = for any t ∈ [0, T ] and so condition 1) is satisfied. Our assumptions imply that the function f0 has only two zeros θ1 = θ0 and θ2 = θ0 + T /2 in the interval [0, T ] and it is strictly monotone at these points, thus conditions 2) D E and 3) of Lemma 5 are also satisfied. Finally, z(θ)⊥ , Fe(x0 (θ)) = f1 (θ, T ) and so (73) implies condition 4) of Lemma 5. Hence the proof is complete. By combining Theorem 3 and Proposition 1 we obtain the following result. Corollary 3. Assume conditions (10) and assume that φ(t, ξ) = −φ(t + T /2, ξ) f or any t ∈ [0, T ] and any ξ ∈ R2 . Moreover, assume that there exists a unique θ0 ∈ [0, T /2) such that f0 (θ0 ) = 0. Finally, assume that the function f0 is strictly monotone at the point θ0 and f1 (θ0 , t) 6= 0 f or any t ∈ [0, T ]. (80) Then there exists ε0 > 0 such that for every ε ∈ (0, ε0 ) system (1) has at least two T -periodic solutions x1,ε and x2,ε satisfying xi,ε (t − θi ) → x0 (t) as ε → 0, i = 1, 2, where θ1 , θ2 ∈ {θ0 , θ0 + T /2} . Furthermore, we have that x1,ε (t) ∈ U0 and x2,ε (t) 6∈ U0 , for every t ∈ [0, T ] and ε ∈ (0, ε0 ). Proof of Corollary 3. To apply Theorem 3 we only have to verify condition (64). For this we will make use of Proposition 1. Without loss of generality we can assume that y1 (0), ẋ⊥ (0) > 0. (81) We claim that, under the conditions of Corollary 3, the vector field Fe of Proposition 1 is homotopic on ∂U0 to the vector field F of Theorem 3. To prove the claim we show that the following homotopy joining Fe and F Dλ (x0 (θ)) = f0 (θ)ẋ0 (θ) + f1 (θ, λT + (1 − λ)θ)(λẋ0 (θ)⊥ + (1 − λ)y1 (θ)), 17 with θ ∈ [0, T ] and λ ∈ [0, 1], is admissible. Assume the contrary, therefore there exist λ0 ∈ [0, 1] and θ0 ∈ [0, T ] such that f0 (θ0 )ẋ0 (θ0 ) + f1 (θ0 , λ0 T + (1 − λ0 )θ0 )(λ0 ẋ0 (θ0 )⊥ + (1 − λ0 )y1 (θ0 )) = 0. By condition (81) we have that the vectors ẋ0 (θ0 ) and λ0 ẋ0 (θ0 )⊥ + (1 − λ0 )y1 (θ0 ) are linearly independent thus f0 (θ0 ) = 0 and f1 (θ0 , λ0 T + (1 − λ0 )θ0 ) = 0 contradicting assumption (80). Hence we have proved that Applying Proposition 1 we obtain that dB (F, U0 ) = dB (Fe , U0 ). dB (F, U0 ) ∈ {0, 2}, namely assumption (64) of Theorem 3 is satisfied and the conclusion of the corollary follows from Theorem 3. At the end of the paper we would like to stress that all the functions y1 , z0 and z1 can be easily determined both analytically and numerically once the limit cycle x0 is known. We give in the following a sketch of both approaches. 1) The analytical approach. Since ẋ0 is one of the two eigenfunctions of system (4) then by using well known formulas, see e.g. Pontrjagin ([33], p. 138), the dimension of the system (4) can be decreased by 1, thus the obtained one-dimensional system can be easily solved to determine y1 . Furthermore, by Lemma 1 the eigenfunctions z0 and z1 can be determined by the formula (z0 (t) z1 (t)) = ((ẋ0 (t) y1 (t))∗ )−1 . 2) A direct numerical approach. From Lemma 1 we have hẋ0 (0), z1 (0)i = 0, therefore as initial condition we may take z1 (0) = ẋ0 (0)⊥ and then z1 can be obtained by a numerical computation. By the definition of z1 there exists a T -periodic function a ∈ C(R, R2 ) such that z1 (t) = a(t) eρ∗ t . Assume, that ρ∗ < 0. Let us fix an arbitrary vector ξ ∈ R2 , which is linearly dependent with z1 (0) and calculate the solution z of system (6) satisfying z(0) = ξ on the interval [0, kT ] where k ∈ N. It turns out that larger is k better accuracy is obtained. Observe that z can be represent by z(t) = αa(t)eρ∗ t + z0 (t), (82) where z0 is an eigenfunction of (6), and since eρ∗ t → 0 as t → +∞ then for given k ∈ N we may take z0 (t) = z(t + (k − 1)T ) for any t ∈ [0, T ]. 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