scholarly journals Real analytic parameter dependence of solutions of differential equations

2010 ◽  
pp. 175-238 ◽  
Author(s):  
Paweł Domański
Keyword(s):  
Differential Equations ◽  
Parameter Dependence ◽  
Analytic Parameter ◽  
Real Analytic ◽  
Parameter Dependence Of Solutions

A correction to ‘Analyticity and metric transitivity on the torus’

1999 ◽  
Vol 19 (1) ◽  
pp. 259-261
Author(s):  
SOL SCHWARTZMAN
Keyword(s):  
Differential Equations ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Topological Transitivity ◽  
Topological Disc ◽  
Real Analytic

In [2], flows on the standard two-dimensional torus given by the differential equations \begin{equation*} \frac{dx}{dt}=a-Fy(x,y),\quad \frac{dv}{dt}=b+Fx(x,y) \end{equation*} were considered. It was assumed that $F(x,y)$ was real analytic and of period one in both $x$ and $y$. A key step in proving the results in [2] was to show that one could conclude topological transitivity for the flow provided one assumed: \begin{enumerate} \item[(a)] $a/b$ is irrational; \item[(b)] there does not exist a topological disc on the torus that is invariant under the flow. \end{enumerate}

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