Bifurcations of a Homoclinic Orbit to Saddle-Center in Reversible Systems

and Applied Analysis 3 the unstable manifold in U0 by using the method introduced in Zhu 9 . According to the invariance and symmetry of these manifolds, we can deduce that system 2.1 has the following form inU0: ẋ x ( λ ( , μ ) O 1 ) , ẏ y −λ , μ O 1 , u̇ v ( ω ( , μ ) O 1 ) u ( O x O ( y ) O u ) O ( xy ) , v̇ u −ω , μ O 1 ) − vO x Oy O v ) −Oxy, 3.1 where λ 0, μ λ, ω 0, μ ω,O 1 O x O y O u O v , the system is Cr−2, and the corresponding involution acts as R x, y, u, v y, x, v, u . In fact, by a linear transformation, system 2.1 takes the form in a small neighborhood ofU0 as follows: ẋ λ ( , μ ) x O 2 , ẏ −λ , μy O 2 , u̇ ω ( , μ ) v O 2 , v̇ −ω , μu O 2 , 3.2 and R x, y, u, v y, x, v, u . By the invariant manifold theorem, we know that there exist a local C center-stable manifold W ,μ {z x, y, u, v : x x ,μ y, u, v , x ,μ 0, 0, 0 0, Dx ,μ 0, 0, 0 0, z ∈ U0}, a local C center-unstable manifold W ,μ {z x, y, u, v : y y ,μ x, u, v , y cu ,μ 0, 0, 0 0, Dy cu ,μ 0, 0, 0 0, z ∈ U0} and RW ,μ W ,μ. By the straightening coordinate transformation which is similar to that of 1, 6, 9 , now we straighten the local manifolds W ,μ and W cu ,μ, such that W cs ,μ {z ∈ U0 : x 0}, W ,μ {z ∈ U0 : y 0}. Notice that the invariance of W ,μ and W cu ,μ implies the local invariance of {z ∈ U0 : x 0} and {z ∈ U0 : y 0}, respectively, which produces that, inU0, ẋ x ( λ ( , μ ) O 1 ) , ẏ y −λ , μ O 1 . 3.3 Now the system is Cr−1 and still reversible. By using a similar procedure to straighten the local Cr−1 stable manifold W ,μ and unstable manifold W u ,μ, and the invariance and symmetry of these two local manifolds that means the transformation is also symmetric , we get system 3.1 . Clearly, corresponding to system 3.1 , the center manifoldW ,μ is locally in the u-v plane, and the stable manifold W ,μ resp., unstable manifold W u ,μ is locally the y-axis resp., x-axis when they are confined in U0. Define r −T δ, 0, 0, 0 ∗, r T 0, δ, 0, 0 ∗ for T 1, where δ > 0 is small enough such that { x, y, u, v ∗ : |x|, |y|, |u2 v2|1/2 < 2δ} ⊂ U0. 4 Abstract and Applied Analysis Consider the linear variational system