Homoclinic Solutions for Autonomous Ordinary Differential Equations with Nonautonomous Perturbations
Homoclinic Solutions for Autonomous Ordinary Differential Equations with Nonautonomous Perturbations
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DOI：
10.1006/jdeq.1995.1136
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发表时间：
1995-10
影响因子：
2.4
通讯作者：
J. Gruendler
中科院分区：
文献类型：
--
作者：
J. Gruendler
考虑Rn中依赖于两个参数μ1和μ2的非自治常微分方程。假设当两个参数都为零时，微分方程是自治的，具有双曲平衡点和同宿解。相空间的维数Rn或稳定流形和不稳定流形的相交维数没有限制。利用Lyapunov-Schmidt方法构造了两个有限维空间之间的分歧函数H，其中H的零点对应于非零参数值的同宿解。H的自变量由标量μ1、μ2、μ 1和矢量β组成，其中μ 1是相位角，β对应于方向，而不是沿着原始同宿解，与稳定流形和不稳定流形相切。当λ固定时，方程H = 0在沿着μ1-μ2平面内产生若干条通过原点的分歧曲线，沿这些分歧曲线存在同宿解。当温度变化时，这些区域变成许多楔形区域。该理论适用于两个例子，一个在R6的不变流形满足三维和第二个在R4这些流形同意。
Nonautomonous ordinary differential equations, depending on two parameters μ1 and μ2, are considered in Rn. It is assumed that when both parameters are zero the differential equation is autonomous with a hyperbolic equilibrium and a homoclinic solution. No restriction is placed on the dimension of the phase space, Rn, or on the dimension of intersection of the stable and unstable manifolds. By means of the method of Lyapunov-Schmidt a bifurcation function, H, is constructed between two finite dimensional spaces where the zeros of H correspond to homoclinic solutions at nonzero parameter values. The independent variables of H consist of scalars μ1, μ2, ξ and a vector β where ξ is a phase angle and β corresponds to directions, other than along the original homoclinic solution, tangent to both the stable and unstable manifolds. When ξ is fixed the equation H = 0 yields, in general, several bifurcation curves through the origin in the μ1-μ2 plane along which there exists a homoclinic solution. When ξ is varied these become a number of wedge-shaped regions. The theory is applied to two examples, one in R6 where the invariant manifolds meet in dimension three and a second in R4 where these manifolds agree.