Explaining Chaos in Constant Accelerating Systems

[LagrangeEuler]

Chaos could appear in the system if there is some nonlinearity. My question is how to explain that there is no chaos in constant acceletating system
s(t)=v_0t+\frac{at^2}{2}
when equation is nonlinear? Why is important only that difference and differential equation be nonlinear. It confusing me.

 

[fresh_42]

Chaos could appear in the system if there is some nonlinearity. My question is how to explain that there is no chaos in constant acceletating system
s(t)=v_0t+\frac{at^2}{2}
when equation is nonlinear? Why is important only that difference and differential equation be nonlinear. It confusing me.

The crucial point is probably, that $s(t)$ isn't sensitive on the initial conditions. To answer why, I'll have to ask you which definition of sensitivity do you want to use (I've found three on Wikipedia). I assume the entire trajectory will be the underlying topological space as well as its dense $s-$invariant subset in order to clear the topological assumptions.

 

[LagrangeEuler]

Exponential sensitivity. Calculation of largest Lyapunov exponent.

 

[fresh_42]

A back of the envelope calculation with the first differential as quotient of rates, the formula $\log (1+t) = - \log (1 - (\frac{t}{1+t}))$ and the power series of $\log$ gave me a Lyapunov exponent zero.

 

[hilbert2]

An anharmonic 1D oscillator with Lagrangian $\frac{p^2}{2m}-kx^4$ isn't chaotic either, even though the equation of motion is nonlinear in $x(t)$. On the other hand, you can make a harmonic oscillator chaotically sensitive to the initial condition by choosing a negative "spring constant" (a slightest deviation from the equilibrium position will start growing exponentially) . Note that in the constant accelerating system the force is an s-independent constant, which means that it's actually a mechanical system with a linear equation of motion even though the trajectory is a nonlinear function of $t$.