A note on the periodic orbits of a kind of Duffing equations

We study the periodic orbits of the modified Duffing differential equation € yþ ay ey3 1⁄4 ehðy; _ yÞ with a > 0; e a small parameter and h a C function in its variables. 2012 Published by Elsevier Inc.


Introduction
In a paper published in 1922, Hamel [7] provided the first general results for the existence of periodic solutions of the periodically forced pendulum equation (1)ÿ + a sin y = b sin t.
This equation was the main subject of a monograph published four years earlier by Duffing [5], who had restricted his study to the approximate determination of the periodic solutions for the following approximation of equation (1): which now is known as the Duffing differential equation. For more details on the history of these differential equations see the paper of Mawhin [9]. Many of the 190 references quoted in this last paper are on the periodic orbits of different kind of Duffing equations, and from its publication many new papers working on these type of periodic orbits also have been published, see for instance the papers [3,4,12] and the quoted references in there.
Here we consider the following modified Duffing differential equation (2)ÿ + ay − εy 3 = εh(y,ẏ), with a > 0, ε a small parameter and h a C 2 function in its variables. We recall thatk is a simple zero of a real function f (k) if f (k) = 0 and (df /dk)(k) = 0.
Our main result is the following.
Theorem 1. For ε = 0 sufficiently small and for every simple positive zerok of the function the modified Duffing differential equation (2) has a periodic orbit y(t) such that when ε → 0 we have that (ẏ(t), y(t)) tends to the periodic orbit given by the ellipse This is a preprint of: "A note of the periodic orbits of a kind of Duffing equations", Jaume Llibre, Ana Rodrigues, Appl. Math. Comput., vol. 219, 8358-8365, 2013. DOI: [10.1016/j.amc.2012.11.087] Theorem 1 will be proved in section 3 using the averaging theory, see section 2 where is described the result on averaging theory that we shall need to use here. In general the main difficulty for studying the existence of periodic orbits using the averaging theory is to find a change of variables which write the studied differential system into the normal form of the averaging theory (see (4)).
Now we shall do some applications of Theorem 1.

Corollary 2.
Let h(x, y) = sin x. Then for every positive integer n there exists an ε 0 > 0 sufficiently small such that for all ε ∈ (−ε 0 , ε 0 ) the modified Duffing differential equation (2) has at least n periodic orbits.
a ij x i y j be an arbitrary polynomial of degree 3.
If a 10 (a 12 + 3 a a 30 ) < 0, then the modified Duffing differential equation (2) has at least one periodic orbit y(t) such that when ε → 0 we have that (ẏ(t), y(t)) tends to the periodic orbit of the differential systemẋ = −ay,ẏ = x, given by the ellipse (3) with k = 2 − a a 10 a 12 + 3 a a 30 .
Corollaries 2 and 3 are proved in section 4.

Basic results
In this section we present the basic result from the averaging theory that we shall need for proving the main result of this paper. For a general introduction to the averaging theory and related topics see the books [2,6,8,10,11].
We consider the differential system with x ∈ U ⊂ R n , U a bounded domain and t ≥ 0. Moreover, we assume that F (t, x) and R(t, x, ε) are T -periodic in t.
The averaged system associated to system (4) is defined by The next theorem says us under which conditions the singular points of the averaged system (5) provide T -periodic orbits of system (4). For a proof see Theorem 2.6.1 of [10], Theorems 11.5 and 11.6 of [11], and Theorem 4.1.1 of [6].
Theorem 4. We consider system (4) and assume that the vector functions F , R, (a) If a ∈ U is a singular point of the averaged system (5) such that det(D x f (a)) = 0 then, for |ε| > 0 sufficiently small there exists a unique T -periodic solution x ε (t) of system (4) such that x ε (0) → a as ε → 0.
(b) If the singular point a of the averaged system (5) is hyperbolic then, for |ε| > 0 sufficiently small the corresponding periodic solution x ε (t) of system (4) is hyperbolic and of the same stability type than a.

Proof of Theorem 1
We write the second order differential equation (2) as the system of two differential equations of first order Now we shall study the solution of the differential system (7) with ε = 0.
Lemma 5. With the exception of the singular point located at the origin of coordinates, the differential system (7) with ε = 0 has all its orbits periodic leaving on the ellipses with k real and positive.
Proof. Assuming a > 0, the general solution of the system (7) with ε = 0 and with initial conditions All the orbits of the system (7) with ε = 0 with exception of the origin are periodic of period 2π/ √ a.
Proof of Theorem 1. Recall Lemma 5. The solution of system (7) with ε = 0 is the ellipsis centered at the origin with semi-major axis on the x-axis and semi-minor axis on the y-axis. The semi-major axis is equal to x 2 0 + ay 2 0 and the semi-minor axis is equal to (x 2 0 + ay 2 0 )/a. So we write the original system of two differential equations of first order (7) in elliptic coordinates. We do the change of coordinates x = x 2 0 + ay 2 0 r cos θ, y = x 2 0 /a + y 2 0 r sin θ. In these new coordinates, system (7) becomes where k = x 2 0 + ay 2 0 . Now applying Theorem 4 of the averaging theory described in Section 2 to system (10) and since 2π 0 cos θ k/r k 3 sin 3 θ dθ = 0, it follows immediately Theorem 1.

Applications
In these section we shall prove the corollaries.
Proof of Corollary 2. We apply Theorem 1 to system (7) with We write the second order differential equation as the system of two differential equations of first order (11)ẋ = −ay + ε(y 3 + b sin x), y = x.

ON THE PERIODIC ORBITS OF A KIND OF DUFFING EQUATIONS 5
Now solving f 1 = 0 with respect to θ 0 we get Substituting θ − 0 in f 2 = 0 and simplifying we get Thus if A(a − 1) < 0, we have a positive solution for r 0 .
Now substituting θ + 0 in f 2 = 0 and simplifying we get Thus if B(a−1) < 0, we have a positive solution for r 0 . Now A and B have the same sign (and consequently there are two periodic orbits) if the following expression is positive: We analyze in Figure 1 the sign of this expression. Note that it takes positive and negative values.
In fact, we must check that both solutions when AB > 0 do not correspond to different initial conditions of the same periodic orbit.
We solve equation (14) with respect to r 0 and we get two different solutions and Substituting the values for r and θ we got in (16) in the expression for x 0 and y 0 (18) x 0 = r cos θ, y 0 = r sin θ/ √ a holds (19)

ON THE PERIODIC ORBITS OF A KIND OF DUFFING EQUATIONS 7
Substituting now the values for r and θ we got in (17) in the expression for x 0 and y 0 given by (18) we get (20) Consider the semi-axis x 2 0 + ay 2 0 of the ellipsis. Restricted to the solution we found in (19) holds and restricted to the solution given by (20) we get Clearly there are two different periodic orbits when both exist and for a → 1 they tend to the same orbit, note that lim a→1 a sin 2 2π √ a 4 sin 4 π √ a + a sin 2 2π √ a = 0.
Moreover, we have a + (1 − a) sin 2 π √ a > 0 if a > 0. Indeed, we plot in figures 2, 3 and 4 the function a + (1 − a) sin 2 π √ a for different ranges of a. Assume now that a = 1. We get after a simple computation that f 1 and f 2 are given by . This finishes the proof of Corollary 2.