Sensitive addiction

The sensitive dependence on the initial values ​​is a central characteristic of chaotic dynamic systems . This is understood to mean the property of such systems to generate completely different system behavior over time with an only infinitesimally small change in the initial conditions. In this sense, one speaks of deterministic chaos in mathematics : The development of a chaotic dynamic system is unpredictable as a result of the inevitability of measurement errors when determining the initial state, not due to a stochastic behavior.

definition

Different conceptions of sensitive dependence can be found in the literature. Three common definitions should be given here. In the following always be

${\displaystyle f\colon I\subseteq \mathbb {R} \to I}$

a continuous mapping and a dynamic system . ${\displaystyle (I,f)}$

After Li / Yorke

${\displaystyle f}$has sensitive dependence on the initial values ​​according to Li and Yorke , if there is an uncountable subset , so that for all with : ${\displaystyle S\subseteq I}$ ${\displaystyle x,y\in S}$ ${\displaystyle x\neq y}$

${\displaystyle \limsup _{n\to \infty }|f^{n}(x)-f^{n}(y)|>0}$

and

${\displaystyle \liminf _{n\to \infty }|f^{n}(x)-f^{n}(y)|=0}$

According to Guckenheimer

${\displaystyle f}$has sensitive dependence on the initial values ​​according to Guckenheimer , if a subset of positive Lebesgue measure exists and a so that for all and every environment of one and one exist with ${\displaystyle S\subseteq I}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle x\in S}$ ${\displaystyle U}$ ${\displaystyle x}$ ${\displaystyle y\in U\cap S}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle |f^{n}(x)-f^{n}(y)|>\varepsilon }$

According to Ruelle

${\displaystyle f}$has sensitive dependence on the initial values ​​according to Ruelle , if an ergodic measure exists such that ${\displaystyle \mu }$

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\ln \left|{\frac {df^{n}}{dx}}(x)\right|=\lambda _{f}>0}$

for - almost all is fulfilled. is the Lyapunov exponent of . ${\displaystyle \mu }$ ${\displaystyle x\in I}$ ${\displaystyle \lambda _{f}}$ ${\displaystyle f}$

literature

• Werner Krabs: Dynamic systems: controllability and chaotic behavior . BGTeubner, Leipzig 1998, ISBN 3-519-02638-4 .
• Wolfgang Metzler: Nonlinear Dynamics and Chaos , BG Teubner, Stuttgart, Leipzig 1998, ISBN 3-519-02391-1