Nonlinear Dynamics

Nonlinear dynamics refers to a branch of the theory of dynamic systems where the differential equations (or difference equations ) that occur contain nonlinear functions . Under certain circumstances, these non-linear equations show interesting features and solutions, for example surfaces in phase space as attractors, self-similarities and fractal structures .

Important applications of nonlinear dynamics can be found in mechanics and astrophysics , for example .

example

An example of a nonlinear differential equation is the following equation of motion describing the behavior of a swing :

${\displaystyle {\ddot {\varphi }}+2{\frac {\dot {L}}{L}}{\dot {\varphi }}+{\frac {g}{L}}\sin {\varphi }=0}$

${\displaystyle L(t)=L_{0}-\Delta L_{0}\cos(\Omega t).}$

Since the sine function is applied to the zeroth derivative, it is a non-linear system. In this specific case, the sine function limits the instability of the parameter-excited oscillation that occurs, since the system is detuned into stable ranges at larger amplitudes. The non-linear component is the reason why the natural frequency depends on the vibration amplitude.

literature

• Demtröder: Experimentalphysik 1 , 5th edition, Springer-Verlag, ISBN 978-3540792949 , chapter 12

Individual evidence

1. ↑ Kurt Magnus: Vibrations , ISBN 978-3835101937 , Chapter 4