Asymptotic behavior of periodic solutions in one-parameter families of Liénard equations
AAsymptotic behavior of limit cycles inone-parameter families of Li´enard systems
Pedro Toniol Cardin and Douglas Duarte Novaes
Abstract.
In this paper we consider one–parameter families of Li´enarddifferential equations. We are concerned with the study on the existenceand uniqueness of periodic solutions for all positive values of λ , andmainly on the asymptotic behavior of such periodic solution for smalland large values of λ . To prove our main result we use the relaxationoscillation theory and the averaging theory. More specifically, the firstone is appropriate for studying the periodic solutions for large values of λ and the second one for small values of λ . In particular, our hypothesesallow us to establish a link between these two theories. Mathematics Subject Classification (2010).
Primary: 34C07, 34C25, 34C26,34C29, 34D15.
Keywords.
Li´enard equation, limit cycles, relaxation oscillation theory,averaging theory.
1. Introduction and statement of the main result
In the qualitative theory of ordinary differential equations, the study of limitcycles is undoubtedly one of the main problems, which is far from trivial.Issues such as the non–existence, existence, uniqueness and other propertiesof limit cycles have been and continue to be studied extensively by mathe-maticians and physicists, as well as by chemists, biologists, economists, etc.In particular, concerning Li´enard systems, there is a considerable amountof research on limit cycles, which began with Li´enard [11]. Since the first re-sults of Li´enard, many articles giving existence and uniqueness conditions forlimit cycles of Li´enard systems have been published. See, e.g., [14, 18], thebooks [19, 20] and the references quoted therein.In this paper we consider one–parameter λ > x (cid:48)(cid:48) + λf ( x ) x (cid:48) + x = 0 . (1)In (1) the prime indicates derivative with respect to the time t . Note thatfor f ( x ) = x − a r X i v : . [ m a t h . D S ] M a y P. T. Cardin and D. D. Novaesoscillations of a triode vacuum tube. Taking x (cid:48) = y the differential equation(1) can be converted into the first order differential system x (cid:48) = y, y (cid:48) = − x − λf ( x ) y. (2)In this paper we are concerned with the study on the existence and uniquenessof limit cycles of system (2) for all positive values of λ , and mainly on theasymptotic behavior of such limit cycle for small and large values of λ .We start exposing our main hypotheses. Consider the following auxiliaryfunction F ( x ) = (cid:90) x f ( s ) ds, (3)and assume the following hypotheses: (A1) f is a C –function on R having precisely two zeros, x M and x m , suchthat x M < < x m and f (cid:48) ( x M ) < < f (cid:48) ( x m ). (A2) The straight lines y = F ( x M ) and y = F ( x m ), passing through thepoints A = ( x M , F ( x M )) and B = ( x m , F ( x m )), intersect the graphicof F , Gr( F ) = { ( x, F ( x )); x ∈ R } , at the points A (cid:48) (cid:54) = A and B (cid:48) (cid:54) = B, respectively. (A3) The differential system (2) has at most one limit cycle.We remark that hypothesis (A3) is a qualitative assumption on thedifferential system (2). It is worthwhile to say that there are many analyticalconditions for which it holds. Some of these conditions are provided in theAppendix (see Proposition 11).Before stating our main result we need to define some preliminary ob-jects. Take x < < x such that A (cid:48) = ( x , F ( x M )) and B (cid:48) = ( x , F ( x m )).From (A2) it follows that x (cid:54) = x m and x (cid:54) = x M . From (A1) and (A2) wefind exactly two zeros of F , x ∗ and x ∗ , such that x ∗ < < x ∗ . Clearly x < x ∗ < x M and x m < x ∗ < x . Denote x ∗ = min {− x ∗ , x ∗ } > r ∗ = (cid:0) x + x (cid:1)(cid:0) F ( x M ) − F ( x m ) (cid:1) x F ( x m ) + x F ( x M ) > . (4)Let AA (cid:48) and BB (cid:48) be the line segments joining A to A (cid:48) and B to B (cid:48) ,respectively. Let (cid:103) A (cid:48) B and (cid:103) B (cid:48) A be the pieces of Gr( F ) joining A (cid:48) to B and B (cid:48) to A , respectively. We define Γ ⊂ R as being the closed curve given bythe union Γ = AA (cid:48) ∪ (cid:103) A (cid:48) B ∪ BB (cid:48) ∪ (cid:103) B (cid:48) A (see Figure 1(a)). We observe that the closed curve Γ can be built fromthe hypotheses (A1) and (A2) . We note that if the hypothesis (A2) is notassumed, then it would not be possible to define Γ . Figure 1(b) illustrates acase in which the condition (A1) is satisfied but (A2) is not.In what follows S ρ denotes the circle in R centered at the origin withradius equal to ρ and d H ( · , · ) denotes the Hausdorff distance. Our main resultis the following one. A B A (cid:48) B (cid:48) Γ y = F ( x ) x M x m xy ( a ) y = F ( x ) xy ( b ) x M x m Figure 1.
Panel (a) illustrates the closed curve Γ and panel (b)shows an example where it is not possible to build the curve Γ . Theorem A.
Assume hypotheses (A1), (A2) , and (A3) . Then the followingstatements hold: (i)
For every λ > , the differential system (2) has a unique stable limitcycle Φ( λ ) depending continuously on λ . (ii) Let P λ ( x, y ) = ( x, F ( x ) + y/λ ) be an one–parameter family of diffeo-morphims. Then lim λ →∞ d H (cid:0) Φ( λ ) , P − λ (Γ ) (cid:1) = 0 . (5)(iii) There exists ρ ∈ ( x ∗ , r ∗ ) such that lim λ → d H (cid:0) Φ( λ ) , S ρ (cid:1) = 0 . (6) Furthermore ρ is the unique positive zero of the function F ( r ) = (cid:90) − (cid:112) − s f ( r s ) ds. (7)(iv) There exists x > max { ρ, − x , x } such that the limit cycle Φ( λ ) iscontained in the strip { ( x, y ) ∈ R : − x < x < x } for every λ > .Moreover, the cycle Φ( λ ) must intersect one of the straight lines x = x M or x = x m . (v) Let γ : [ − x , x ] × R + → R + be the following continuous piecewisefunction γ ( x ; λ ) = (cid:114) ( x + x )( x − x − mλ ( x + x ))1 + mλ , < λ ≤ − m , (cid:112) ( x + x )( x − x + 8 m λ x ) , λ > − m , (8) where m = min { f ( x ) : − x ≤ x ≤ x } < and, for each λ > , let R λ denote the following compact region R λ = { ( x, y ) ∈ R : − x ≤ x ≤ x , − γ ( − x ; λ ) ≤ y ≤ γ ( x ; λ ) } . Then Φ( λ ) is contained in the interior of R λ , for every λ > see Figure4 ) . P. T. Cardin and D. D. NovaesThe limit given in (5) means that, for sufficiently large values of λ , aftera change of coordinates, the limit cycle Φ( λ ) approaches the closed curve Γ .Similarly, from (6) one conclude that, for sufficiently small values of λ , thelimit cycle Φ( λ ) approaches the circle S ρ .The proof of Theorem A will be split in several results. Firstly in Sec-tion 2, we prove the existence of the limit cycle Φ( λ ), for every λ >
0, and itscontinuous dependence on the parameter λ . Item (i) of Theorem A followsfrom Proposition 1. After in Section 3, using the relaxation oscillation theory ,we study the asymptotic behavior of the limit cycle Φ( λ ) when λ takes largevalues. Item (ii) of Theorem A follows from Proposition 3 and Corollary 4.Similarly in Section 4, we study the asymptotic behavior of the limit cycleΦ( λ ) when λ takes small values, but now using mainly the averaging theory .Item (iii) of Theorem A follows from Proposition 6 and Corollary 7. Items(iv) and (v) of Theorem A are proved in Section 5, where an estimative ofthe amplitude growth of the limit cycle Φ( λ ) is provided. Item (v) followsfrom Corollary 10. Section 6 is devoted to study some polynomial and non-polynomial examples of differential systems where Theorem A can be applied,including the classical van der Pol equation and its generalization. In Sec-tion 7 we present our conclusions emphasizing the link established betweenthe relaxation oscillation theory and the averaging theory. Finally, sufficientanalytical conditions in order to guarantee hypothesis (A3) are presented inthe Appendix section.
2. Existence of the limit cycle and continuous dependence on λ This section is devoted to prove item (i) of Theorem A. We shall see thathypotheses (A1) and (A2) are sufficient to guarantee the existence of the limitcycle Φ( λ ) while hypothesis (A3) will ensure its uniqueness and continuousdependence on λ . Proposition 1.
Assume hypotheses (A1), (A2) , and (A3) . Then, for every λ > , the differential system (2) has a unique stable limit cycle Φ( λ ) whichdepends continuously on λ . To prove Proposition 1 we need the next result due to Dragil¨ev [7]. Fora proof see Theorem 5.1 of [19]. It is worthwhile to mention also that, in [4],the authors provide an extension of Dragilev’s theorem.
Theorem 2 (Dragil¨ev [7] ). Consider the differential system x (cid:48) = y, y (cid:48) = − g ( x ) − f ( x ) y. (9) Suppose that: (B1)
The functions F ( x ) and g ( x ) are locally Lipschitz, where F ( x ) = (cid:82) x f ( s ) ds . (B2) xg ( x ) > for x (cid:54) = 0 , G ( ±∞ ) = + ∞ , where G ( x ) = (cid:82) x g ( s ) ds ; (B3) There exist a < < a , such that F ( x ) > , for a ≤ x < , and F ( x ) < , for < x ≤ a . (B4) There exist k > max {− a , a } , and b < b such that F ( x ) ≤ b if x < − k , and F ( x ) ≥ b if x > k .Then the differential system (9) has at least one stable limit cycle.Proof of Proposition 1. First of all we shall see that all the conditions, (B1) – (B4) , of Dragil¨ev’s Theorem are fulfilled. From the hypothesis (A1) , F ( x )is a differentiable function and, for the differential system (2), g ( x ) = x .Therefore F ( x ) and g ( x ) are locally Lipschitz and then condition (B1) issatisfied. Condition (B2) is trivially true because G ( x ) = x /
2. Now taking a = x M and a = x m we see that hypothesis (A2) implies condition (B3) .Finally condition (B4) is assured by hypotheses (A1) and (A2) if we take k > max {− x , x } , b = F ( x m ), and b = F ( x M ).From Dragil¨ev’s Theorem we have assured, for each λ >
0, the existenceof a stable limit cycle Φ( λ ) of the differential system (2). Clearly hypothesis (A3) implies its uniqueness for each λ > λ > ε > δ > λ > | λ − λ | < δ it follows that d H (Φ( λ ) , Φ( λ )) <ε . To see that let U ε be an ε -neighborhood of Φ( λ ), that is d ( p, q ) < ε forevery p, q ∈ U ε . Here d ( · , · ) denotes the usual distance in R . Since Φ( λ ) isa stable limit cycle we can assume that the flow of system (2), for λ = λ ,is transversal to the boundary ∂U ε of U ε and crosses ∂U ε in the exterior-to-interior direction. From the continuous dependence of the solutions of (2)on the parameter λ we can find δ > ∂U ε for every λ > | λ − λ | < δ . Therefore, forevery λ ∈ ( λ − δ, λ + δ ), U ε is a positively invariant compact set for the flowof system (2). Applying now the Poincar´e–Bendixon Theorem we concludethat the differential system (2) admits a stable limit cycle (cid:101) Φ( λ ) ⊂ U ε forevery λ ∈ (0 , λ + δ ). Consequently d H ( (cid:101) Φ( λ ) , Φ( λ )) < ε . Finally hypothesis (A3) implies that Φ( λ ) = (cid:101) Φ( λ ), and then we conclude that Φ is continuousat λ = λ . Since λ > λ > (cid:3)
3. Asymptotic behavior of the limit cycle for large values of λ In this section we will study the asymptotic behavior of the limit cycle Φ( λ )obtained from Theorem A for sufficiently large values of λ . For doing thiswe will use the relaxation oscillation theory occurring in slow–fast singularlyperturbed systems. We start this section describing this theory.Consider a two–dimensional slow–fast differential system of the form µ dxds = g ( x, y, µ ) , dyds = h ( x, y, µ ) , (10)where g and h are C r –functions with r ≥
3. For positive values of µ , system(10) is mutually equivalent to dxdτ = g ( x, y, µ ) , dydτ = µh ( x, y, µ ) , (11) P. T. Cardin and D. D. Novaeswhich is obtained after the time rescaling τ = s/µ . Systems (10) and (11)are referred to as slow system and fast system , respectively. A usual way totreat with slow–fast systems is through the geometric singular perturbationtheory (GSPT). The idea is to study the (limiting) fast and slow dynamicsseparately and then combine results on these two limiting behaviours in orderto obtain information on the dynamics of the full system (10) (or (11)) forsmall values of µ .The limiting behaviours for µ → g ( x, y, , dyds = h ( x, y, , (12)which will be referred to as reduced problem , and dxdτ = g ( x, y, , dydτ = 0 , (13)which we will be referred to as layer problem . The phase space of (12) isthe so–called critical manifold defined by S = { ( x, y ) ∈ R : g ( x, y,
0) = 0 } .On the other hand, S is the set of equilibrium points for the layer problem(13). Among other things, Fenichel theory [3, 9] guarantees the persistenceof a normally hyperbolic subset S ⊆ S as a slow manifold S µ of (10) (or(11)) for small enough values of µ >
0. Moreover, the flow on S µ is a smallperturbation of the flow of (12) on S . Normal hyperbolicity of S meansthat ( ∂g/∂x )( x, y ) (cid:54) = 0 for all ( x, y ) ∈ S . That is, S is normally hyperbolicif for each (¯ x, ¯ y ) ∈ S , we have that ¯ x is a hyperbolic equilibrium point of( dx/dτ ) = g ( x, ¯ y, S . Inthe case when the critical manifold S has non–normally hyperbolic points,interesting global phenomena can occur. For instance, an interesting kind ofglobal phenomenon is the relaxation oscillations. A relaxation oscillation is aperiodic solution Γ µ of the slow–fast system (10) that converges to a singulartrajectory Γ , when µ →
0, with respect to Hausdorff distance. A singulartrajectory means a curve obtained as concatenations of trajectories of thereduced and layer problems (with a consistent orientation) forming a closedcurve.In [10] the authors described a prototypical situation where a relaxationoscillation exists, see Theorem 2.1 of [10]. In the following we describe thenecessary hypotheses for the existence of a relaxation oscillator. For moredetails see Section 2 of [10]. (C1)
The critical manifold S can be written in the form y = φ ( x ) and thefunction φ has precisely two critical points, one minimum x m and onemaximum x M , both non–degenerate (folds). (C2) The fold points are generic, i.e. ∂ g∂x ( x m , φ ( x m ) , (cid:54) = 0 , ∂g∂y ( x m , φ ( x m ) , (cid:54) = 0 , h ( x m , φ ( x m ) , (cid:54) = 0 , y = φ ( x ) xy ( a ) D (cid:48) C (cid:48) DC Γ y = φ ( x ) x m x M xy ( b ) Figure 2.
Assuming the hypotheses (C1) – (C4) , panel (a) illustratesthe phase portraits of the reduced and layer problems (12) and (13).Note that double and single arrows indicate the direction of the fastand slow flows, respectively. Panel (b) illustrates the singular trajectoryΓ. ∂ g∂x ( x M , φ ( x M ) , (cid:54) = 0 , ∂g∂y ( x M , φ ( x M ) , (cid:54) = 0 , h ( x M , φ ( x M ) , (cid:54) = 0 . (C3) ( ∂g/∂x ) < S l = { ( x, φ ( x )) : x < x m } and on S r = { ( x, φ ( x )) : x > x M } , and ( ∂g/∂x ) > S m = { ( x, φ ( x )) : x m < x < x M } .This means that for the layer problem (13) the branches S l and S r areattracting while S m is repelling. (C4) The slow (reduced) flow on S l and S r satisfies that ( dy/ds ) < dy/ds ) >
0, respectively, and on S m it has a repeller equilibrium.Figure 2(a) illustrates the phase portraits of the reduced and layer problems(12) and (13) (assuming the hypotheses (C1) – (C4) ).Let be C = ( x m , φ ( x m )) and D = ( x M , φ ( x M )). Let C (cid:48) and D (cid:48) be thepoints of intersection of the straight lines y = φ ( x m ) and y = φ ( x M ) with S r and S l , respectively. Consider Γ the singular trajectory defined as the unionof the fast fibers joining C to C (cid:48) and D to D (cid:48) and of the two pieces of thecritical manifold S joining C (cid:48) to D and D (cid:48) to C . See Figure 2(b). Let V bea small tubular neighborhood of Γ. Then, under the assumptions (C1) – (C4) ,Theorem 2.1 of [10] states that, for µ small enough, system (10) admits aunique stable limit cycle Γ µ ⊂ V which converges to the singular trajectoryΓ in the Hausdorff distance as µ → (A1) and (A2) , the differ-ential system (2) has, after a change of variables, a unique stable limit cycleapproaching to the singular trajectory Γ as λ → ∞ . Proposition 3.
Assume that the hypotheses (A1) and (A2) are fulfilled. Thenthere exists λ > such that the differential system (2) admits a stable limitcycle Φ ( λ ) for every λ ∈ ( λ , + ∞ ) . Moreover lim λ →∞ d (cid:0) Φ ( λ ) , P − λ (Γ ) (cid:1) = 0 where P λ ( x, y ) = ( x, F ( x ) + y/λ ) is an one–parameter family of diffeomor-phims, and Γ is the singular trajectory defined in Section 1, see Figure 3(a). P. T. Cardin and D. D. NovaesAs a trivial consequence of Proposition 3 and hypothesis (A3) we obtainthe following result.
Corollary 4.
Assume that the hypotheses (A1) , (A2) , and (A3) are fulfilled.Then Φ( λ ) = Φ ( λ ) for every λ ∈ ( λ , + ∞ ) .Proof of Proposition 3. Firstly we transform system (2) into a slow–fast sys-tem. To do this we first define a new independent variable s setting s := t/λ .This leads to system ˙ x = λy, ˙ y = − λx − λ f ( x ) y, where the dot means de-rivative with respect to s . After that, we apply the change of coordinates( x, u ) = P λ ( x, y ) = ( x, F ( x ) + y/λ ), where F is given in (3). Then we obtainthe system (1 /λ ) ˙ x = u − F ( x ) , ˙ u = − x. Finally, setting µ := 1 /λ and con-sidering λ large enough, we get the following slow–fast singularly perturbedsystem with small perturbation parameter µ : µ ˙ x = u − F ( x ) , ˙ u = − x. (14)Comparing system (14) with the general form (10), we have that g ( x, u, µ ) = u − F ( x ) and h ( x, u, µ ) = − x . The critical manifold is given by S = { ( x, u ) ∈ R : u = F ( x ) } . From the hypothesis (A1) , it follows that F has preciselytwo critical points, x M and x m , since F (cid:48) ( x ) = f ( x ) for all x ∈ R , and x M and x m are the only zeros of f . Moreover, as F (cid:48)(cid:48) ( x M ) = f (cid:48) ( x M ) < F (cid:48)(cid:48) ( x m ) = f (cid:48) ( x m ) >
0, then A = ( x M , F ( x M )) and B = ( x m , F ( x m )) aremaximum and minimum points of F , respectively, both non–degenerate. Alsothese two fold points are generic, since ∂ g∂x ( A ) = − F (cid:48)(cid:48) ( x M ) (cid:54) = 0 , ∂g∂u ( A ) = 1 (cid:54) = 0 , h ( A ) = − x M (cid:54) = 0 ,∂ g∂x ( B ) = − F (cid:48)(cid:48) ( x m ) (cid:54) = 0 , ∂g∂u ( B ) = 1 (cid:54) = 0 , h ( B ) = − x m (cid:54) = 0 . Further, from the hypothesis (A1) , we can conclude that F (cid:48) ( x ) < x ∈ ( x M , x m ), and F (cid:48) ( x ) > x ∈ ( −∞ , x M ) ∪ ( x m , + ∞ ). Therefore, for thelayer problem given by ˙ x = u − F ( x ) , ˙ u = 0, the branches S l = { ( x, F ( x )) : x < x M } and S r = { ( x, F ( x )) : x > x m } are attracting while the branch S m = { ( x, F ( x )) : x M < x < x m } is repelling. Relative to dynamics of thereduced problem, we have that the slow flow on S l satisfies ˙ u > S r it satisfies that ˙ u <
0. On S m the reduced problem has a repeller equilibriumat the point (0 , F (0)) = (0 , (A1) , we havethat the fast and slow dynamics are as illustrated in Figure 3(b).We remark that the situation presented above involving system (14) isnot exactly the same as in prototypical situation described from the assump-tions (C1) – (C4) . In this last case the critical manifold S is S–shaped whilein our case S has the shape of a S but reflected. Mathematically speaking,the two situations coincide after a reflection ( X, Y ) = ( − x, y ) on the y -axis.Let U be a small tubular neighborhood of Γ . Then, under the hy-potheses (A1) and (A2) , from Theorem 2.1 of [10] it follows that, for µ smallenough, system (14) has a unique stable limit cycle Γ µ ⊂ U which converges A B A (cid:48) B (cid:48) Γ y = F ( x ) x M x m xy ( a ) y = F ( x ) xy ( b ) Figure 3.
Panel (a) illustrates the singular trajectory Γ and panel(b) illustrates the phase portraits of the layer and reduced problemsassociated with system (14). The double and single arrows indicate thedirection of the fast and slow flows, respectively. to the singular trajectory Γ in the Hausdorff distance as µ →
0. Conse-quently, using the reverse change of coordinates, we can conclude that, for λ > ( λ ) ⊂ U for system (2) such that lim λ →∞ d (cid:0) Φ ( λ ) , P − λ (Γ ) (cid:1) = 0 . This completes the proof of Proposition 3. (cid:3)
4. Asymptotic behavior of the limit cycle for small values of λ In this section we shall use the averaging theory to study the behavior of thelimit cycle Φ( λ ) for small values of λ >
0. For a general introduction on thissubject we refer the book of Sanders, Verhulst and Murdock [15]. The nexttheorem is a topological version of the classical Averaging Theorem to findperiodic orbits for smooth differential systems.
Theorem 5 (Averaging Theorem).
Consider the following nonautonomousdifferential equation drdθ = εF ( θ, r ) + ε R ( θ, r ; ε ) , (15) where, for r > and ε > a small parameter, F : R × (0 , r ) → R and R : R × (0 , r ) × ( − ε , ε ) → R are continuous functions, π -periodic in the firstvariable, and locally Lipschitz in the second variable. We define the averagedfunction M : (0 , r ) → R as M ( r ) = (cid:90) π F ( θ, r ) dθ. (16) Assume that for some ρ ∈ (0 , r ) with M ( ρ ) = 0 , there exists a neighborhood V of ρ such that M ( r ) (cid:54) = 0 for all r ∈ V \ { ρ } and d B ( M , V, (cid:54) = 0 . Then,for | ε | (cid:54) = 0 sufficiently small, there exists a π -periodic solution ϕ ( θ ; ε ) of thedifferential equation (15) such that ϕ (0; ε ) → ρ when ε → . d B ( M , V,
0) denotes the Brouwer degree of the function M withrespect to the domain V and the value 0 (see [1] for a general definition).When M is a C –function and the Jacobian determinant of M at r ∈ V isdistinct from zero (we denote J M ( r ) (cid:54) = 0) then the Brouwer degree of M at0 is given by d B ( M , V,
0) = (cid:88) r ∈ Z M sign ( J M ( r )) , (17)where Z M = { r ∈ V : M ( r ) = 0 } . In this case J M ( ρ ) (cid:54) = 0 implies d B ( M , V,
0) = 1 for some small neighborhood V of ρ .The main property of the Brouwer degree we shall use in this section isthe invariance under homotopy (see [1]), which says: Invariance under homotopy.
Let M s ( r ) be an homotopy between M and M for s ∈ [0 , . If / ∈ M s ( ∂V ) for every s ∈ [0 , , then d B ( M s , V, is constantin s . The proof of Theorem 5 (Averaging Theorem) is based on the fact thatthe
Poincar´e map
Π : Σ → Σ of the nonautonomous differential equation(15) defined on the
Poincar´e section
Σ = { ( θ, r ) ∈ R × (0 , r ) : θ = 0 } readsΠ( r ; ε ) = r + εM ( r ) + O ( ε ) . (18)Let φ ( θ ; ε ) be a family of periodic solutions of the differential equation (15).So r ( ε ) = φ (0; ε ) is a branch of fixed points of the Poincar´e map Π( r ; ε ).From (18) we have that M ( r (0)) = 0, that is the branch r ( ε ) approachesto the set of zeros of M . The degree theory allows us to provide sufficientconditions for which the conversely is true, assuring when a zero ρ of M ( r )will persist as branch of fixed points r ( ε ) of Π( r ; ε ), that is Π( r ( ε ); ε ) = r ( ε )and r (0) = ρ .The stability of a periodic solution φ ( t ; ε ) associated with a branch offixed points r ( ε ) = φ (0; ε ) of the Poincar´e map Π( r ; ε ) can be studied via the displacement function , which is defined as ∆( r ; ε ) = Π( r ; ε ) − r . For a fixed ε > φ ( r ; ε ) is(i) unstable (or repelling) if there exists a small neighborhood I = ( a, b )of r ( ε ) such that ∆( r ; ε ) > r ∈ ( r ( ε ) , b ) and ∆( r ; ε ) < r ∈ ( a, r ( ε ));(ii) stable (or attracting) if there exists a small neighborhood I = ( a, b ) of r ( ε ) such that ∆( r ; ε ) < r ∈ ( r ( ε ) , b ) and ∆( r ; ε ) > r ∈ ( a, r ( ε )).We shall see that the expression (18) also help us to study the stability of aperiodic solution given by Theorem 5.Next result states that, under the conditions (A1) , (A2) , and (A3) , thedifferential system (2) has a unique stable limit cycle approaching to thecircle S ρ as λ →
0. Before stating it we recall the definitions of x ∗ and r ∗ : x ∗ = min {− x ∗ , x ∗ } , and r ∗ = (cid:0) x + x (cid:1)(cid:0) F ( x M ) − F ( x m ) (cid:1) x F ( x m ) + x F ( x M ) > x ∗ , x < < x are the abscissas of A (cid:48) and B (cid:48) , respectively, and x ∗ < < x ∗ are the unique zeros of F distinct from zero. As mentioned before x < x ∗ < x M and x m < x ∗ < x . Proposition 6.
Assume that the hypotheses (A1) , (A2) , and (A3) are fulfilled.Then there exists λ > such that the differential system (2) admits a stablelimit cycle Φ ( λ ) for every λ ∈ (0 , λ ) . Moreover there exists ρ ∈ ( x ∗ , r ∗ ) such that lim λ → d (cid:0) Φ ( λ ) , S ρ (cid:1) = 0 , (19) being ρ the unique zero of the function F ( r ) defined in (7) . As a trivial consequence of Proposition 6 and hypothesis (A3) we obtainthe following result.
Corollary 7.
Assume that the hypotheses (A1) , (A2) , and (A3) are fulfilled.Then Φ( λ ) = Φ ( λ ) for every λ ∈ (0 , λ ) . In order to prove Proposition 6 we shall need the next lemma.
Lemma 8.
The following inequality holds for every u > : u < arctan (cid:18) √ u − (cid:19) < π u . (20) Proof.
First we note that the inequality (20) holds for u = 2, since 1 / <π/ < π/
4. Now consider the functions β ( u ) = 1 u − arctan (cid:18) √ u − (cid:19) ,β ( u ) = arctan (cid:18) √ u − (cid:19) − π u . (21)The following properties hold:lim u → β ( u ) = 1 − π < , lim u → + ∞ β ( u ) = 0 , (22)lim u → β ( u ) = lim u → + ∞ β ( u ) = 0 . (23)Computing the derivatives of the functions β and β given in (21) we get β (cid:48) ( u ) = u − √ u − u √ u − , β (cid:48) ( u ) = π √ u − − u u √ u − . (24)From the limiting values (22), if the function β has a zero u ∗ > u >
1. However from (24) we knowthat β (cid:48) ( u ) > u >
1. Hence we conclude that β ( u ) < u >
1, which leads to the first inequality of (20).From the limiting values (23), if the function β has a zero u ∗ > u >
1. However from (24) we knowthat β (cid:48) vanishes only for u = π/ √ π − >
1. Since β (2) < β ( u ) < u >
1, which leads to the second inequality of (20).This completes the proof of Lemma 8. (cid:3)
Proof of Proposition 6.
In order to write the differential system (2) in thestandard form (15) of the Averaging Theorem, we transform it through thecoordinates changing x = r cos θ and y = − r sin θ . The transformed systemreads r (cid:48) = − λ r f ( r cos θ ) sin θ, θ (cid:48) = 1 − λf ( r cos θ ) sin θ cos θ. (25)We note that θ (cid:48) > λ > θ as the new time variable, that is drdθ = − λ r f ( r cos θ ) sin θ − λ f ( r cos θ ) sin θ cos θ = − λ r f ( r cos θ ) sin θ + λ R ( θ, r ; λ ) . (26)Therefore, the differential system (2) is equivalent to the differential equation(26) which is written in the standard form (15) with ε = λ . Moreover, since f is continuous, F is differentiable and then the right handside of the differentialequation (26) is locally Lipschitz in the variable r .Computing the averaged function (16) for the differential equation (26)we obtain M ( r ) = − r (cid:90) π f ( r cos θ ) sin θ dθ = − rF ( r ) . (27)To get the second above equality firstly we use the change of variable x = r cos θ , restricted to the domains [0 , π/ π/ , π ], [ π, π/ π/ , π ],and then we take x = r s . Equivalent formulae for the averaged function (27)are given by M ( r ) = − (cid:90) π cos θ F ( r cos θ ) dθ = − r (cid:90) r − r x √ r − x F ( x ) dx. (28)The first above equality can be checked using integration by parts. The secondone is also obtained using the change of variable x = r cos θ restricted to thedomains [0 , π/ π/ , π ], [ π, π/ π/ , π ] . Throughout this proofthe last equality of (28) will be more conveniently for our purposes.Since xF ( x ) ≤ x ∈ ( x ∗ , x ∗ ) and √ r − x > x ∈ ( − r, r ),we obtain a first estimative for the averaged function (28): M ( r ) > r ∈ [0 , x ∗ ].From the hypotheses (A1) and (A2) we have the following properties: (p ) xF ( x ) > x F ( x m ) for every x < x ; (p ) xF ( x ) > x F ( x M ) for every x < x < (p ) xF ( x ) > x F ( x m ) for every 0 < x < x ; (p ) xF ( x ) > x F ( x M ) for every x > x .To obtain (p ) we note that F ( x ) < F ( x m ) for x < x <
0, therefore xF ( x ) > xF ( x m ). Moreover F ( x m ) < x < x imply that xF ( x m ) >x F ( x m ), which leads to (p ) . The other properties follow using similar ar-guments.If r > max {− x , x } the last integral in (28) can be split in the domains I = [ − r, x ], I = [ x , I = [0 , x ], and I = [ x , r ]. The property (p i ) I i , i = 1 , , ,
4. Forinstance, using (p ) on the domain I we get (cid:90) x − r x √ r − x F ( x ) dx > x F ( x m ) (cid:90) x − r √ r − x dx = x F ( x m )2 (cid:32) π + 2 arctan (cid:32) x (cid:112) r − x (cid:33)(cid:33) . Doing the same for i = 2 , , M ( r ) ≤ − π (cid:0) x F ( x m ) + x F ( x M ) (cid:1) r + 2 x (cid:0) F ( x m ) − F ( x M ) (cid:1) r arctan (cid:32) − x (cid:112) r − x (cid:33) − x (cid:0) F ( x m ) − F ( x M ) (cid:1) r arctan (cid:32) x (cid:112) r − x (cid:33) . (29)From Lemma 8 we know that the following inequality holds for 0 < a < r :arctan (cid:18) a √ r − a (cid:19) ≤ πa r . (30)Note that, in (29), the coefficients of the arctangents are all positive. Soapplying (30) into (29) we obtain the following second estimative for theaveraged function (28): M ( r ) ≤ πr (cid:20)(cid:0) x + x (cid:1)(cid:0) F ( x M ) − F ( x m ) (cid:1) − r (cid:0) x F ( x m ) + x F ( x M ) (cid:1)(cid:21) . Let r ∗ be the zero of the righthand side of the above inequality, that is r ∗ = (cid:0) x + x (cid:1)(cid:0) F ( x M ) − F ( x m ) (cid:1) x F ( x m ) + x F ( x M ) > x ∗ . Since x F ( x m ) + x F ( x M ) > M ( r ) < r >r ∗ . Hence we conclude that there exists ρ ∈ ( x ∗ , r ∗ ) such that M ( ρ ) = − ρF ( ρ ) = 0. Moreover M ( r ) < r > ρ and M ( r ) > < r < ρ ,and therefore we conclude that ρ is the unique zero of M and consequentlyof F .Now consider the homotopy M s ( r ) = (1 − s )( ρ − r ) + sM ( r ) , s ∈ [0 , , between M ( r ) = ρ − r and M ( r ). Clearly, for every s ∈ [0 , , M s ( ρ ) = 0, M s ( r ) < r > ρ and M s ( r ) > r < ρ . From (17) it is easy to see that d B ( M , V,
0) = −
1. Therefore from the invariance under homotopy propertywe conclude that d B ( M , V,
0) = − λ > φ ( θ ; λ )such that φ (0; λ ) → ρ when λ →
0. Since the solutions of the differentialequation (26) for λ = 0 are constant in the variable θ we conclude that4 P. T. Cardin and D. D. Novaes φ ( θ ; λ ) → ρ when λ → θ ∈ [0 , π ] . Consequently, for λ > ( λ ) = (cid:110)(cid:0) φ ( θ ; λ ) cos θ, − φ ( θ ; λ ) sin θ (cid:17) : θ ∈ [0 , π ] (cid:111) describes a limit cycle of system (2) such that lim λ → d (cid:0) Φ ( λ ) , S ρ (cid:1) = 0.Let r ( λ ) = φ (0; λ ) be the branch of fixed points of the Poincar´e mapΠ( r ; λ ). From the hypothesis (A3) this branch is unique. Therefore, from (18)∆( r ; λ ) = Π( r ; λ ) − r = λM ( r ) + O ( λ ) , and then, for λ > r ; λ ) < r > r ( λ ) and∆( r ; λ ) > < r < r ( λ ). This implies that the periodic solution φ ( θ ; λ ) and, consequently, the limit cycle Φ ( λ ) is stable. (cid:3)
5. Limit cycle amplitude growth
We observe that, from item (iii) of Theorem A, the amplitude of the limitcycle Φ( λ ) tends to ρ when λ goes to 0, and from item (ii) of Theorem A,the amplitude of Φ( λ ) is unbounded for λ >
0. In this section we aim toinvestigate the amplitude growth of Φ( λ ). To do that we build a region R λ having an increasing diameter such that Φ( λ ) ⊂ R λ , for every λ > Proof of item (iv) of Theorem A.
Firstly, consider the functions ξ − ( λ ) = min (cid:0) π x Φ( λ ) (cid:1) < ξ + ( λ ) = max (cid:0) π x Φ( λ ) (cid:1) > , where π x denotes the projection onto the axis x . Item (i) of Theorem Aimplies that ξ ± ( λ ) are continuous for every λ >
0. Furthermore, items (ii)and (iii) of Theorem A implylim λ → ξ ± ( λ ) = ± ρ, lim λ →∞ ξ − ( λ ) = x , and lim λ →∞ ξ + ( λ ) = x . Hence ξ ± are bounded functions for λ >
0, and x = sup {| ξ ± ( λ ) | : λ > } > max { ρ, − x , x } . Therefore the limit cycle Φ( λ ) is contained in thestrip { ( x, y ) ∈ R : − x < x < x } for every λ > λ ) intersects one of the straight lines x = x M or x = x m . Note that the divergent of the vector field X λ ( x, y ) = ( y, − x − λf ( x ) y ) is given by div X λ ( x, y ) = − λf ( x ), for every ( x, y ) ∈ R . Since f ( x ) < x ∈ ( x M , x m ) then div X λ ( x, y ) > x ∈ ( x M , x m ) and y ∈ R .Therefore, it follows from Bendixson’s Criterion, that the limit cycle Φ( λ )must intersect one of the straight lines x = x M or x = x m . (cid:3) In order to build the region R λ we will need the following proposition. Proposition 9.
Let h λ ( x, y ) = y − γ ( x ; λ ) , where γ is defined in (8) , and let X λ ( x, y ) = ( y, − x − λf ( x ) y ) be the vector field defined by system (2) . Then (cid:104)∇ h λ ( x, y ) , X λ ( x, y ) (cid:105) < for y = γ ( x ; λ ) , x ∈ ( − x , x ) and λ > . - - - - - Figure 4.
Phase portrait of X λ in the van der Pol case (i.e. f ( x ) = x −
1) assuming λ = 1 /
2. The red line represents the boundary of theregion R λ . In order to plot R λ it was assumed that x = 3, neverthelessthe exactly value for x is uknown. The dashed bold line represents thelimit cycle. Proof.
Firstly, let us assume that λ ≤ − / (2 m ), where m = min { f ( x ) : − x ≤ x ≤ x } . In this case γ ( x ; λ ) = (cid:114) ( x + x )( x − x − mλ ( x + x ))1 + mλ and (cid:104)∇ h λ ( x, y ) , X λ ( x, y ) (cid:105) = − λγ ( x ; λ ) (cid:32) f ( x ) − mx (cid:112) p ( x ) (cid:33) , where p ( x ) = − (1 + mλ ) x − mx λ (1 + mλ ) x + x (1 − m λ ) . Note that p ( x ) has a maximum at x = − mx λ/ (1 + mλ ) ∈ ( − x , x ) and p ( x ) = x . Therefore (cid:104)∇ h λ ( x, y ) , X λ ( x, y ) (cid:105) ≤ − λγ ( x ; λ ) (cid:32) f ( x ) − mx (cid:112) p ( x ) (cid:33) = − λγ ( x ; λ ) ( f ( x ) − m ) < . (31)Now assume that λ > − / (2 m ). In this case γ ( x ; λ ) = (cid:112) ( x + x )((1 + 8 m λ ) x − x ) , and (cid:104)∇ h λ ( x, y ) , X λ ( x, y ) (cid:105) = − λγ ( x ; λ ) (cid:32) f ( x ) + 4 m λx (cid:112) q ( x ) (cid:33) , where q ( x ) = − x + 8 m λ x x + (1 + 8 m λ ) x . q ( x ) has a maximum at ˆ x = 4 m λ x > x . Hence, for x ∈ ( − x , x ), q reaches its maximum at x and q ( x ) = (4 mλx ) . Therefore (cid:104)∇ h λ ( x, y ) , X λ ( x, y ) (cid:105) ≤ − λγ ( x ; λ ) (cid:32) f ( x ) + 4 m λx (cid:112) q ( x ) (cid:33) = − λγ ( x ; λ ) ( f ( x ) − m ) < . (32)From (31) and (32) we conclude this proof. (cid:3) The next result, which is an immediate consequence of Proposition 9and item (iv) of Theorem A, implies that Φ( λ ) is contained in the interior of R λ = { ( x, y ) ∈ R : − x ≤ x ≤ x , − γ ( − x ; λ ) ≤ y ≤ γ ( x ; λ ) } (see Figure 4).Note that the diameter of R λ is given bydim( R λ ) = d (cid:0) ( − x , − γ ( x ; λ )) , ( x , γ ( x ; λ )) (cid:1) = x √ m λ m λ , < λ ≤ − m , x √ m λ , λ > − m , where d denotes the usual metric of R . Corollary 10.
For each λ > , it holds that Φ( λ ) ∩ { ( x, γ ( x ; λ )) : − x ≤ x ≤ x } = ∅ and Φ( λ ) ∩ { ( x, − γ ( − x ; λ )) : − x ≤ x ≤ x } = ∅ .
6. Examples
In this section we illustrate the main result of the article with some examples.We start with the well known van der Pol equation. After that we will considersome examples where the function f is not polynomial. The van der Pol equation is a well known prototypical example where relax-ation oscillations occur [17, 8]. It is given by the equation (1) with f ( x ) = x −
1. For this function it is immediate to check that hypotheses (A1) and (A2) are fullfiled. In fact, we have that x M = − x m = 1 are theonly zeros of f with f (cid:48) ( −
1) = − < f (cid:48) (1) = 2 >
0. The auxiliaryfunction F is given by F ( x ) = x / − x , and we see that the straight lines y = F ( x M ) = 2 / y = F ( x m ) = − /
3, passing through the points A = ( x M , F ( x M )) = ( − , /
3) and B = ( x m , F ( x m )) = (1 , − / F at the points A (cid:48) = ( x , F ( x M )) = (2 , /
3) and B (cid:48) = ( x , F ( x m )) = ( − , − / x ∗ , x ∗ , x ∗ , and r ∗ appearing in (4) assume the values −√ √ √
3, and 4, respec-tively. Finally, it is well known that hypothesis (A3) holds for the van derPol equation.7Computing the function F given in (7) for the van der Pol equation weobtain F ( r ) = ( r − π/ . The unique positive zero of F is ρ = 2. Moreover,note that 2 ∈ ( x ∗ , r ∗ ), that is 2 ∈ ( √ , λ >
0, the van der Pol equation has a unique stablelimit cycle Φ( λ ). For sufficiently small values of λ the cycle Φ( λ ) approachesto the circle centered at the origin of radius 2. For sufficiently large values of λ , after the change of coordinates P λ ( x, y ) = (cid:0) x, x / − x + y/λ (cid:1) , the limitcycle Φ( λ ) approaches a singular trajectory Γ .The van der Pol example can be generalized as follows. Let f : R → R be given by f ( x ) = ( x − x M )( x − x m ) ¯ f ( x )where x M < < x m and ¯ f : R → R is a polynomial function satisfying¯ f ( x ) > x ∈ R . Clearly f has precisely two real zeros, x M and x m .Moreover, f (cid:48) ( x M ) = ( x M − x m ) ¯ f ( x M ) < f (cid:48) ( x m ) = ( x m − x M ) ¯ f ( x m ) >
0. So the condition (A1) is satisfied.Regarding the auxiliary function F , note that if ¯ f ( x ) = c > F is given by F ( x ) = (cid:90) x f ( s ) ds = c (cid:18) x − ( x M + x m ) x x M x m x (cid:19) . Now if ¯ f is not constant, it takes the form¯ f ( x ) = k n (cid:89) j =1 (cid:16) x − ( x j + iy j ) (cid:17)(cid:16) x − ( x j − iy j ) (cid:17) , with k >
0. In this case the function F is given by F ( x ) = (cid:90) x f ( s ) ds = k x n +1)+1 n + 3 + H ( x ) , where H is a polynomial function of degree less than or equal to 2 n + 2. Inboth cases ( ¯ f constant or not), the following limits holdlim x → + ∞ F ( x ) = + ∞ , and lim x →−∞ F ( x ) = −∞ , which imply the condition (A2) . Therefore, assuming further hypothesis (A3) ,for the function f like above, the conclusions of Theorem A apply. We empha-size that, in the Appendix section, sufficient conditions in order to guaranteehypothesis (A3) are given. Particularly for the generalized van der Pol equa-tion one could assume that x M = − x m . In this subsection we provide examples of non–polynomial functions for whichour hypotheses are fulfilled. We start with the following class of rationalfunctions that generalizes the polynomial case f ( x ) = ( x − x M )( x − x m ) p ( x ) q ( x ) , x M < < x m , p and q are polynomial functions such that p ( x ) q ( x ) > x ∈ R , and deg( q ) < p ). Clearly f has precisely two real ze-ros, x M and x m . Moreover, f (cid:48) ( x M ) = ( x M − x m ) p ( x M ) /q ( x M ) < f (cid:48) ( x m ) = ( x m − x M ) p ( x m ) /q ( x m ) >
0. So the hypothesis (A1) is satis-fied. Since deg( q ) < p ) and p ( x ) q ( x ) > x ∈ R , thenlim x →±∞ f ( x ) = + ∞ . This implies thatlim x → + ∞ F ( x ) = lim x → + ∞ (cid:90) x f ( s ) ds = (cid:90) + ∞ f ( s ) ds = + ∞ , andlim x →−∞ F ( x ) = lim x →−∞ (cid:90) x f ( s ) ds = − lim x →−∞ (cid:90) x f ( s ) ds = − (cid:90) −∞ f ( s ) ds = −∞ . In particular we can conclude that hypothesis (A2) is fulfilled. Therefore,under the qualitative assumption (A3) , Theorem A holds for the class ofrational functions given above.We also can find other examples of functions f , that are neither polyno-mial nor rational, such that our hypotheses are fulfilled. Below we list someof them: f ( x ) = exp( x ) + exp( − x ) − b, with b > ,f ( x ) = (2 x −
1) exp( − x ) + a, with 0 < a < . In fact, for the function f we have that x M = ln(( b − √ b − /
2) and x m = ln(( b + √ b − /
2) are their unique zeros. Evaluating the derivative f (cid:48) in these two zeros gives f (cid:48) ( x M ) = −√ b − < f (cid:48) ( x m ) = √ b − > F ( x ) = (cid:90) x f ( s ) ds = exp( x ) − exp( − x ) − bx satisfies lim x → + ∞ F ( x ) = + ∞ , and lim x →−∞ F ( x ) = −∞ . Therefore, the conditions (A1) and (A2) are valid for the function f .With respect to the function f , we have that x M = − (cid:112) / − W ( a √ e/ x m = (cid:112) / − W ( a √ e/
2) are their unique zeros, where W is the Lam-bert function (principal branch), see [5]. Note that since 0 < a <
1, then0 < W ( a √ e/ < /
2. Evaluating f (cid:48) in these two zeros one obtains f (cid:48) ( x M ) = − a (1 + W ( a √ e/ (cid:112) − W ( a √ e/ W ( a √ e/ < f (cid:48) ( x m ) = a (1 + W ( a √ e/ (cid:112) − W ( a √ e/ W ( a √ e/ > . The auxiliary function F is given by F ( x ) = (cid:90) x f ( s ) ds = x ( a − exp( − x )) . F satisfieslim x → + ∞ F ( x ) = + ∞ , and lim x →−∞ F ( x ) = −∞ . Therefore, the conditions (A1) and (A2) are valid for the function f .Note that for both functions f and f , one has that x M = − x m . So thecondition (D2) of Proposition 11 of the Appendix section holds. Therefore,the condition (A3) is valid for the functions f and f . Consequently, theconclusions of Theorem A apply for these functions.
7. Conclusion
In this paper we consider one-parameter λ > λ , we prove the existence of a limit cycle Φ( λ ) as well asits continuous dependence on λ . We also provide the asymptotic behavior ofsuch limit cycle for small and large values of λ > λ >
0, the differential system (2)can be seen as a regular perturbation of the linear center ( y, − x ). In thiscontext the averaging theory provides useful tools for studying the birth oflimit cycles from the periodic solutions of the center. When λ assumes largevalues, the differential system (2) can be converted into a slow–fast singularlyperturbed system which can be treated with the techniques coming from thegeometric singular perturbation theory.Initially the hypotheses (A1) and (A2) were assumed in order to guaran-tee that the manifold S was S-shaped (reflected) assuring then (Proposition3) the existence of a relaxation oscillation approaching to the singular tra-jectory Γ , when the parameter λ takes large values. Surprisingly, the samehypotheses, (A1) and (A2) , also guaranteed the existence of a zero ρ of theaveraged function, satisfying some good properties on its Brouwer degree,which assured (Proposition 6) the existence of a limit cycle approaching tothe circle S ρ , when λ > (A3) we were able to show that the limit cycle existingnearby the circle S ρ , for small values of λ >
0, is deformed continuously andincreases its amplitude, when λ becomes larger, approaching then to thesingular trajectory Γ . We also estimate the amplitude growth of the limitcycle. Appendix: Uniqueness of the limit cycle
In this appendix we provide some analytical conditions in order to fulfillhypothesis (A3) from Theorem A.
Proposition 11.
Assume hypotheses (A1) and (A2) . Then the differential sys-tem (2) has at most one periodic solution provided that at least one of thefollowing conditions holds: (D1) F ( ±∞ ) = ±∞ and x ∗ = − x ∗ . (D2) F ( ±∞ ) = ±∞ and x M = − x m . (D3) f is nonincreasing in ( −∞ , and nondecreasing in (0 , + ∞ ) . (D4) F ( x ) /x is nonincreasing in ( −∞ , and nondecreasing in [ x ∗ , + ∞ ) , and x ∗ ≤ − x ∗ . The next two theorems are due to Sansone [16]. A proof for them canalso be found in [20], see Theorems 4.2 and 4.3 of its Chapter 4.
Theorem 12 ( [16] ). Consider the differential system (2) and assume that f is continuous and that F ( ±∞ ) = ±∞ . Suppose that there exist δ − < < δ and ∆ > such that f ( x ) < for x ∈ ( δ − , δ ) , f ( x ) > for ( −∞ , δ − ) ∪ ( δ , ∞ ) , and F (∆) = F ( − ∆) = 0 . Then the differential system (2) has a unique limit cycle, which is stable. Theorem 13 ( [16] ). Consider the differential system (2) and assume that f is continuous and that F ( ±∞ ) = ±∞ . Suppose that there exists δ > suchthat f ( x ) < for | x | < δ and f ( x ) > for | x | > δ . Then the differentialsystem (2) has a unique limit cycle, which is stable. The next theorem is due to Massera [13]. A proof for it can also befound in [20], see Theorem 4.4 of its Chapter 4.
Theorem 14 ( [13] ). Consider the differential system (2) and assume that f iscontinuous. Suppose that there exist δ − < < δ such that f ( x ) < for x ∈ ( δ − , δ ) and f ( x ) > for ( −∞ , δ − ) ∪ ( δ , ∞ ) , and that f is nonincreasingin ( −∞ , and nondecreasing in (0 , + ∞ ) . Then the differential system (2) has a unique limit cycle, which is stable. The next theorem is due to Figueiredo [6]. A proof for it can also befound in [19], see Theorem 6.9.
Theorem 15 ( [6] ). Consider the differential system (2) and assume that F ( x ) (cid:54)≡ in a neighborhood of the origin. Suppose that there exists δ > such that x F ( x ) ≤ for | x | ≤ δ , F ( x ) ≥ for x > δ , and F ( x ) /x is nonincreasing in ( −∞ , and nondecreasing in [ δ, + ∞ ) . Then the differential system (2) hasat most one limit cycle. For more analytical conditions on the uniqueness of limit cycles inLi´enard differential systems see, for instance, the books [19, 20]. Now weprove Proposition 11.
Proof of Proposition 11.
Item (D1) is consequence of Theorem 12 if we take δ − = x M , δ = x m , and ∆ = − x ∗ = x ∗ . Item (D2) is consequence ofTheorem 13 if we take δ = − x M = x m . Item (D3) is consequence of Theorem14 if we take δ − = x M and δ = x m . Item (D4) is consequence of Theorem15 if we take δ = x ∗ . (cid:3) Acknowledgements
The first author is supported by FAPESP grant 2013/24541-0. The secondauthor is supported by FAPESP grant 2016/11471-2. Both authors are sup-ported by CAPES grant 88881.030454/2013–01 Program CSF–PVE.
References [1] F. E. Browder. Fixed point theory and nonlinear problems.
Bull. Amer. Math.Soc. (N.S.) , 9(1):1–39, 1983.[2] A. Buic˘a and J. Llibre. Averaging methods for finding periodic orbits viaBrouwer degree.
Bull. Sci. Math. , 128(1):7–22, 2004.[3] P. T. Cardin and M. A. Teixeira. Fenichel theory for multiple time scale sin-gular perturbation problems.
Preprint , 2017.[4] M. Cioni and G. Villari. An extension of Dragilev’s theorem for the existence ofperiodic solutions of the Li´enard equation.
Nonlinear Anal. , 127:55–70, 2015.[5] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth. Onthe Lambert W function.
Advances in Computational Mathematics , 5(1):329–359, 1996.[6] R. P. de Figueiredo. Existence and uniqueness of the periodic solution of anequation for autonomous oscillations. In
Contributions to the theory of non-linear oscillations, Vol. V , pages 269–284. Princeton Univ. Press, Princeton,N.J., 1960.[7] A. V. Dragilev. Periodic solutions of the differential equation of nonlinear os-cillations.
Akad. Nauk SSSR. Prikl. Mat. Meh. , 16:85–88, 1952.[8] F. Dumortier and R. Roussarie. Canard cycles and center manifolds.
Mem.Amer. Math. Soc. , 121(577):x+100, 1996. With an appendix by Cheng Zhi Li.[9] N. Fenichel. Geometric singular perturbation theory for ordinary differentialequations.
J. Differential Equations , 31(1):53–98, 1979.[10] M. Krupa and P. Szmolyan. Relaxation oscillation and canard explosion.
J.Differential Equations , 174(2):312–368, 2001.[11] A. Li´enard. ´Etude des oscillations entretenues.
Revue G´en´erale de l’Electricit´e ,23:901–912 and 946–954, 1928.[12] J. Llibre, D. D. Novaes, and M. A. Teixeira. Higher order averaging theoryfor finding periodic solutions via Brouwer degree.
Nonlinearity , 27(3):563–583,2014.[13] J. L. Massera. Sur un th´eor`eme de G. Sansone sur l’´equation di Li´enard.
Boll.Un. Mat. Ital. (3) , 9:367–369, 1954.[14] M. Sabatini and G. Villari. On the uniqueness of limit cycles for Li´enard equa-tion: the legacy of G. Sansone.
Matematiche (Catania) , 65(2):201–214, 2010.[15] J. A. Sanders, F. Verhulst, and J. Murdock.
Averaging methods in nonlineardynamical systems , volume 59 of
Applied Mathematical Sciences . Springer, NewYork, second edition, 2007.[16] G. Sansone. Sopra l’equazione di A. Li´enard delle oscillazioni di rilassamento.
Ann. Mat. Pura Appl. (4) , 28:153–181, 1949.[17] B. van der Pol. On relaxation oscillations.
Philosophical Mag. , 7:978–992, 1926. [18] G. Villari. Periodic solutions of Li´enard’s equation.
J. Math. Anal. Appl. ,86(2):379–386, 1982.[19] Y. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang,M. S. Wang, and X. A. Yang.
Theory of limit cycles , volume 66 of
Translationsof Mathematical Monographs . American Mathematical Society, Providence, RI,second edition, 1986. Translated from the Chinese by Chi Y. Lo.[20] Z. F. Zhang, T. R. Ding, W. Z. Huang, and Z. X. Dong.
Qualitative theory ofdifferential equations , volume 101 of
Translations of Mathematical Monographs .American Mathematical Society, Providence, RI, 1992. Translated from theChinese by Anthony Wing Kwok Leung.Pedro Toniol CardinDepartamento de Matem´atica, Faculdade de Engenharia de Ilha Solteira, Univer-sidade Estadual Paulista (UNESP)Rua Rio de Janeiro, 266, CEP 15385–000, Ilha Solteira, S˜ao Paulo, Brazile-mail: [email protected]