© Hindawi Publishing Corp. EXISTENCE OF PERIODIC SOLUTIONS AND HOMOCLINIC ORBITS FOR THIRD-ORDER NONLINEAR DIFFERENTIAL EQUATIONS

The existence of periodic solutions for the third-order differential equation ˙ x + ω 2 ˙ x = µF (x, ˙ x, ¨ x) is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second-order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions if F( x,˙ x, ¨ x) has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat


Introduction.
There are many papers for the existence of periodic solutions for nonlinear second-order differential equations (e.g., see [1,2,5]) but there are few papers for nonlinear third-order differential equations. These kind of equations arise in engineering, for example, in problems related to energy and acceleration. Because of the topological characteristics of the threedimensional space, the investigation of periodic solutions for the nonlinear third-order differential equations is a difficult problem.
In what follows, we consider the third-order differential equation (1.2) and obtain some conditions for the existence of periodic solutions for it. These conditions are applicable to many third-order differential equations. Let f (x,ẋ) be C r (r ≥ 3) and F(x,ẋ,ẍ) = f x (x,ẋ)ẋ + fẋ(x,ẋ)ẍ. (1.6) We show that if f xẋ (0, 0) ≠ 0, then (1.2) has many periodic solutions. The conditions for the existence of the homoclinic orbits are also studied. For example, if f (x,ẋ) consists of some quadratic terms, then (1.2) has many homoclinic orbits which are filled up with periodic solutions. In Section 2, we present some mathematical preliminaries and show that the existence of periodic solutions for the second-order differential equation implies the existence of periodic solutions for the third-order differential equation. The existence of the fixed points for the second-order differential equation is discussed in Section 3. Section 4 is devoted to the existence of periodic solutions for the second-order differential equation. In Section 5, we give a few examples, and finally, we compare our results to that of [7]. Now we reduce the third-order differential equation to a second-order differential equation. We haveẋ = y, (2.5) where k is a real constant. If we put k = ω 2 λ andx = x − λ, then we geẗ Dropping the bars, we obtainẋ = y, Proof. It is obvious that if (x(t),ẋ(t)) is a solution for (2.9), then (x(t) + λ,ẋ(t),ẍ(t)) is a solution for (2.4). Let (x(t),ẋ(t)) be a periodic solution with period T . For each t ∈ R, we havë (2.10) Since (x 0 , 0) is a fixed point for (2.9), then (x 0 +λ, 0, 0) is a fixed point for (2.4). Hence, (x(t) + λ,ẋ(t),ẍ(t)) is a homoclinic solution.

Remark 3.3.
Assume that we have the systeṁ where g is a C 1 function. Let γ = (x(t), y(t)) be a solution of this system and let

Periodic solutions.
In this section, we obtain some conditions for (2.9) in order to have periodic solutions. We consider the set and the map x : → R defined in Section 3. (4.1) Computing the derivative with respect to λ, we get has many periodic solutions.
has many periodic solutions.
have many periodic solutions.
Proof. By Theorem 4.2 and Corollary 4.3, for λ = 0, a Hopf bifurcation occurs at the origin for the following equations: (4.16) Hence,ẋ have many periodic solutions.  both (x 1 , 0) and (x 2 , 0) are fixed points, then γ is a part of a heteroclinic.
(iii) The proof is similar to the proof of (ii).
has many periodic solutions. We consider (5.6) where ν > 0 (the case ν < 0 is similar). The equation has (0, 0) as the only fixed point. On the curve x = (ν/ω 2 )y 2 , we haveẏ = 0, in addition, the sign ofẏ is shown in Figure 5.1. Therefore, a solution γ = (x(t), y(t)) of (5.6), which intersects the positive x-axis, is as shown in Figure 5.2. Since in region (I) of Figure 5.1,ẏ < 0, so γ(t) intersects the curve x = (ν/ω 2 )y 2 transversally. After passing through x = (ν/ω 2 )y 2 , y(t) increases, hence γ(t) intersects the x-axis or converges to the origin. In the first case, by Theorem 4.6, γ(t) is a periodic solution. In the second case, since y(t) in region (II) of Figure 5.1 is increasing, therefore γ cannot intersect the negative y-axis.
x-axis y-axis ν > 0 has many homoclinic orbits and periodic solutions. The homoclinic orbits make a two-dimensional C 1 surface. In addition, each homoclinic orbit lies on a C 1 two-dimensional orientable manifold and inside it is filled up with the periodic solutions (see Figure 5.5).
Proof. By Lemma 2.4, there exist many homoclinic orbits and periodic solutions. It can be easily checked that, the graph of g hp varies C 1 with respect to (ν, λ). So, by Lemma 2.4, we conclude that the homoclinic orbits make a two-dimensional C 1 surface. Finally, since the solutions of (5.15) are distinct and varies C 1 with respect to the initial conditions, so for each (ν, λ) ∈ , the solutions of (5.15) make a C 1 two-dimensional orientable manifold in the phase space of the third-order equation. ν > 0 For convenience, we drop the bars and obtaiṅ This system has two fixed points (0, 0) and (x 0 , 0) = (− √ ω 4 − 4νλω 2 /ν, 0). We consider two cases.
Hence, by the mean value theorem, For x(T ) < x < x 0 , ( √ ω 4 − 4νλω 2 x + νx 2 )/y > 0 and lim x→x − 0 dνy(x) = 0. This result shows that dy/dx does not converge to −∞, which is a contradiction. Hence, γ(t) is a homoclinic. Similarly, a solution of (5.20), which lies in the homoclinic, does not converge to (x 0 , 0), hence, it crosses x-axis at two different points. By Theorem 4.6, this solution is a periodic solution. Therefore, the homoclinic is filled up with the periodic solutions.
(1) As a restriction, [7] guarantees only one periodic solution, also the value of |µ| may be very small; but by Theorem 4.2, we obtain many periodic solutions. Moreover, we have no restriction for |µ|. This is due to the fact that the constant M (see Theorem 3.1) can be arbitrary large, so by Corollary 4.3, if f xy (0, 0) ≠ 0, then for each µ ∈ [−M, M], (2.4) has many periodic solutions.
(3) We studied the effect of some quadratic terms for (2.4) and obtained many homoclinic orbits and periodic solutions. Also, we explained that the homoclinic orbits make two-dimensional manifolds. Furthermore, we saw that the homoclinic orbits are filled up with the periodic solutions.