The existence of chaos for ordinary differential equations with a center manifold
The authors consider the coupled system of ordinary equations \[ x'=f(x,y,\mu,t)=f_0(x,y)+\mu_1f_1+\mu_2f_2, \quad y'=g(x,y,\mu,t)=g_0(x,y)+\mu_1g_1+\mu_2g_2, \] where \(x\in {\mathbb R}^n\), \(y\in {\mathbb R}^m\), and \(\mu_1,\mu_2\in {\mathbb R}\) are small parameters. It is assumed that the reduced system \[ x'=f_0(x,0) \] has a hyperbolic equilibrium and a homoclinic solution \(\gamma\), while the eigenvalues of the matrix \(D_yg_0(0,0)\) are imaginary. Conditions are given under which the chaos generated by the homoclinic solution \(\gamma\) is ``shadowed'' by chaos in the full system.
scientific article; zbMATH DE number 2139737