# How are jerk equations connected to chaos theory?

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?

• At first I assumed jerk equations must be not-nice equations, but then I looked at the Wikipedia article...
– bob
Feb 14, 2020 at 0:42
• Haha! Then what to think about hyperjerk equations? Feb 14, 2020 at 10:54

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.

• "... at least 3D phase space" - this is a consequence of the Poincare-Bendixson theorem - see en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Bendixson_theorem Feb 13, 2020 at 12:46
• @gandalf61 Is this a consequence of the fractal dimensionality? Feb 13, 2020 at 14:31
• @descheleschilder Which fractal ? We have a 3D phase space and a set of three linked first order NLDEs. We don't yet have a fractal. Feb 13, 2020 at 14:53
• @gardenhead If the movement takes place in only one dimension (i.e., $x$ is an scalar), then yes, you can't have chaos. But if the particle is allowed to move in 2 or 3 dimensions, then you'll have one of these second-order equations for each dimension and the system is 4-D at least. Feb 13, 2020 at 21:04
• @Grego_gc What I don't understand is how a 3D phase space can exist. Doesn't it has to be an even number? Can you give an example for f((x,y,z) for which the system is chaotic? Feb 14, 2020 at 8:01

@Grego_gc's answer is great. I just wanted to give an example.

You do this all the time for second order equations in physics, you just don't realize it. i.e. we have defined velocity as $$v=\dot x$$. For example, take the second order equation of motion for a damped oscillator $$m\ddot x+b\dot x+kx=0$$

You can use the definition of velocity to turn this into a system of coupled first order differential equations $$\dot x=v$$ $$\dot v=-\frac bmv-\frac kmx$$

For higher order equations you just end up with more first order equations. All you are doing is just assigning a new variable to each derivative.