The Sil'nikov problem, exponential expansion, strong \(\lambda\)-lemma, \(C^ 1\)-linearization, and homoclinic bifurcation
The author introduces a new theory called exponential expansion and explores its interaction with the related theories. He studies the system of autonomous differential equations \[ (1)\quad \dot x=Ax+f(x,y),\quad \dot y=By+g(x,y)\quad (x,y)\in U\subset {\mathbb{R}}^ 2, \] assuming that there exist constants \(\lambda <0<\mu\) and \(C_ 1>1\) such that: \(| \exp At| \leq C_ 1 \exp \lambda t\) for \(t\geq 0\), \(| \exp Bt| \leq C_ 1 \exp \mu t\) for \(t\leq 0\), \(f(0,y)=0\) for (0,y)\(\in U\), \(g(x,0)=0\) for (x,0)\(\in U\), D f(0,0)\(=0\) and D g(0,0)\(=0\), where D is the differentiation operator with respect to the phase variable (x,y). The problem under consideration is so called Silnikov problem, i.e. to find solution of equations (1) satisfying conditions: \(x(0)=x_ 0\), \(y(\tau)=y_ 1\), \(\tau\geq 0.\) Let \(B=\left[ \begin{matrix} \mu \\ 0\end{matrix} \begin{matrix} 0\\ B_ 1\end{matrix} \right]\) and let max\(\{\) Re \(\sigma\) (B\({}_ 1)\}>\mu_ 1>\mu >0\). The y-component of the solution is said to admit exponential expansion if there exist a \(C^{k-1}\)-function \(\phi\) (t,\(\xi\),\(\eta)\), not depending on \(\tau\), and a constant \(\nu >0\) such that y(t,\(\tau\),\(\xi\),\(\eta)\), exp[-\(\mu\) (t-\(\tau)\)] together with all derivatives up to the order k-1 converges to \(\phi\) (t,\(\xi\),\(\eta)\) as \(\tau\) \(\to \infty\) at the rate of exp \(\nu\) (t-\(\tau)\). Using the exponential expansion the author shows the possibility of smooth linearization for equation (1). Some related problems as inclination behaviour of a given submanifold and so called strong \(\lambda\)-lemma are also discussed.
scientific article; zbMATH DE number 4103504
scientific article