I read in this Wikipedia article:
It has been shown that a jerk equation, which is equivalent to a system of three first-order, ordinary non-linear differential equations, is the minimal setting for solutions showing chaotic behaviour. This condition generates mathematical interest in jerk systems. Systems involving fourth-order derivatives or higher are accordingly called hyperjerk systems.
Now, I know that there is a connection between nonlinearity and chaos theory. But how has it been shown that there is an equivalence between a jerk equation and three first-order, ordinary non-linear differential equations?
Are these equations necessary, sufficient or both for chaotic behavior to occur:
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
Are there examples of what kind(s) of chaotic behavior these jerk equations represent or correspond to? All kinds of chaotic behavior?
Best Answer
A remark before. For chaos you need to have at least a 3D phase space.
A jerk equation consider a system of the form
$$\frac{\partial^3}{\partial t^3}x= f(x,\dot{x}, \ddot{x})$$
You can set auxiliar variables $y=\dot{x}$ and $z=\ddot{x}$ to obtain a three first order differential equation.
$$\dot{z}=f(x,y,z)$$ $$\dot{y}=z$$ $$\dot{x}=y$$
Depending on f, you can have chaos.