Publication Date:
2014-02-26
Description:
One-step discretizations of order $p$ and step size $\varepsilon$ of ordinary differential equations can be viewed as time-$\varepsilon$ maps of \begin{displaymath} \dot{x} (t) = f(\lambda ,x(t)) + \varepsilon^p g(\varepsilon,\lambda,t/\varepsilon,x(t)), x \in R^N,\lambda \in R, \end{displaymath} where $g$ has period $\varepsilon$ in $t$. This is a rapidly forced nonautonomous system. We study the behavior of a homoclinit orbit $\Gamma$ for $\varepsilon = 0, \lambda =0$, under discretization. Under generic assumptions we show that $\Gamma$ becomes transverse for positive $\in$. The transversality effects are estimated from above to be exponentially small in $\in$. For example, the length $l(\varepsilon$) of the parameter interval of $\lambda$ for which $\Gamma$ persists can be estimated by \begin{displaymath} l(\varepsilon)\le Cexp(-2\pi\eta/\varepsilon), \end{displaymath} where $C,\eta$ are positive constants. The coefficient $\eta$ is related to the minimal distance from the real axis of the poles of $\Gamma(t)$ in the complex time domain. Likewise, the region where complicated, "chaotic" dynamics prevail is estimated to be exponentially small, provided $x \in R^2$ and the saddle quantity of the associated equilibrium is nonzero. Our results are visualized by high precision numerical experiments. The experiments show that, due to exponential smallness, homoclinic transversality becomes pratically invisible under normal circumstances, already for only moderately small step size. {\bf Keywords:} Homoclinic orbit, ordinary differential equations, discretization, transversality, averaging, exponential smallness, chaos. {\bf Subject Classifications:} (AMS): 34C15, 34C35, 58F14, 65L60
Keywords:
ddc:000
Language:
English
Type:
reportzib
,
doc-type:preprint
Format:
application/pdf
Permalink
