Differential equation example: I wanted a closed form solution of the range of the simplest possible trebuchet- just a seesaw. This is perfectly achievable, see for example http://ffden-2.phys.uaf.edu/211.fall2000.web.projects/J%20Mann%20and%20J%20James/see_saw.html. I wanted a closed form solution of the second simplest trebuchet, a seesaw with a sling. This is impossible, because even though the motion of the trebuchet with sling isn’t chaotic during the throw, it can be made chaotic by just varying the initial conditions, which rules out a simple closed form solution for non-chaotic initial conditions.
Lyapunov exponent example: for the bouncing balls, if each ball travels 1 diameter between bounces, then a change in velocity angle of 1 degree pre-bounce becomes a change in angle of 4 degrees post bounce (this number may be 6- I am bad at geometry), so the exponent is 4 if time is measured in bounces.
even though the motion of the trebuchet with sling isn’t chaotic during the throw, it can be made chaotic by just varying the initial conditions, which rules out a simple closed form solution for non-chaotic initial conditions
Do you know what theorems/whatever this is from? It seems to me that if you know that “throws” constitute a subset of phase space that isn’t chaotic, then you should be able to have a closed-form solution for those trajectories.
So this turns out to be a doozy, but it’s really fascinating. I don’t have an answer- an answer would look like “normal chaotic differential equations don’t have general exact solutions” or “there is no relationship between being chaotic and not having an exact solution” but deciding which is which won’t just require proof, it would also require good definitions of “normal differential equation” and “exact solution.” (the good definition of “general” is “initial conditions with exact solutions have nonzero measure”) I have some work.
A chaotic differential equation has to be nonlinear and at least third order- and almost all nonlinear third order differential equations don’t admit general exact solutions. So, the statement “as a heuristic, chaotic differential equations don’t have general exact solutions” seems pretty unimpressive. However, I wrongly believed the strong version of this heuristic and that belief was useful: I wanted to model trebuchet arm-sling dynamics, recognized that the true form could not be solved, and switched to a simplified model based on what simplifications would prevent chaos (no gravity, sling is wrapped around a drum instead of fixed to the tip of an arm) and then was able to find an exact solution (note that this solvable system starts as nonlinear 4th order, but can be reduced using conservation of angular momentum hacks)
Now, it is known that a chaotic difference equation can have an exact solution: the equation x(n+1) = 2x(n) mod 1 is formally chaotic and has the exact solution 2^n x mod 1. A chaotic differential equation exhibiting chaotic behaviour can have an exact solution if it has discontinuous derivatives because this difference equation can be constructed:
equation is in three variables x, y, z
dz/dt always equals 1
if 0 < z < 1: if x > 0: dx/dt = 0 dy dt = 1
if x < 0:
dx/dt = 0
dy/dt = −1
if 1 < z < 2:
if y > 0
dx/dx = -.5
dy dt = 0
if y < 0
dy dt = 0
dx dt = .5
if 2 < z < 3:
dx/dt = x ln(2)
dy/dt = -(y)/(3 - t)
and then make it periodic by gluing z=0 to z=3 in phase space. (This is pretty similar to the structure of the lorentz attractor, except that in the lorentz system, the sheets of solutions get close together but don’t actually meet.) This is an awful,weird ode: the derivative is discontinuous, and not even bounded near the point where the sheets of solutions merge.
Plenty of prototypical chaotic differential equations have a sprinkling of exact solutions: e.g, three bodies orbiting in an equilateral triangle- hence the requirement for a “general” exact solution.
The three body problem “has” an “exact” “series” “solution” but it appears to be quite effed: for one thing, no one will tell me the coefficient of the first term. I suspect that in fact the first term is calculated by solving the motion for all time, and then finding increasingly good series approximations to that motion.
I strongly suspect that the correct answer to this question can be found in one of these stack overflow posts, but I have yet to fully understand them:
There are certainly billiards with chaotic and exactly solvable components- if nothing else, place a circular billiard next to an oval. So, for the original claim to be true in any meaningful way, this may have to involve excluding all differential equations with case statements- which sounds increasingly unlike a true, fundamental theorem.
If this isn’t an open problem, then there is somewhere on the internet a chaotic, normal-looking system of odes (would have aesthetics like x‴′ = sin(x‴) - x’y‴, y’ = (1-y / x’) etc) posted next to a general exact solution, perhaps only valid for non chaotic initial conditions, or a proof that no such system exists. The solvable system is probably out there and related to billiards
Final edit: the series solution to the three body problem is legit mathematically, see page 64 here
I am suddenly unsure whether it is true! It certainly would have to be more specific than how I phrased it, as it is trivially false if the differential equation is allowed to be discontinuous between closed form regions and chaotic regions
Very interesting. Could you give more details one one of these concrete examples?
e.g. calculating a lyapunov exponent or proving a differential equation is chaotic for a DE that came up in practice for you
Differential equation example: I wanted a closed form solution of the range of the simplest possible trebuchet- just a seesaw. This is perfectly achievable, see for example http://ffden-2.phys.uaf.edu/211.fall2000.web.projects/J%20Mann%20and%20J%20James/see_saw.html. I wanted a closed form solution of the second simplest trebuchet, a seesaw with a sling. This is impossible, because even though the motion of the trebuchet with sling isn’t chaotic during the throw, it can be made chaotic by just varying the initial conditions, which rules out a simple closed form solution for non-chaotic initial conditions.
Lyapunov exponent example: for the bouncing balls, if each ball travels 1 diameter between bounces, then a change in velocity angle of 1 degree pre-bounce becomes a change in angle of 4 degrees post bounce (this number may be 6- I am bad at geometry), so the exponent is 4 if time is measured in bounces.
I’m curious about this part;
Do you know what theorems/whatever this is from? It seems to me that if you know that “throws” constitute a subset of phase space that isn’t chaotic, then you should be able to have a closed-form solution for those trajectories.
So this turns out to be a doozy, but it’s really fascinating. I don’t have an answer- an answer would look like “normal chaotic differential equations don’t have general exact solutions” or “there is no relationship between being chaotic and not having an exact solution” but deciding which is which won’t just require proof, it would also require good definitions of “normal differential equation” and “exact solution.” (the good definition of “general” is “initial conditions with exact solutions have nonzero measure”) I have some work.
A chaotic differential equation has to be nonlinear and at least third order- and almost all nonlinear third order differential equations don’t admit general exact solutions. So, the statement “as a heuristic, chaotic differential equations don’t have general exact solutions” seems pretty unimpressive. However, I wrongly believed the strong version of this heuristic and that belief was useful: I wanted to model trebuchet arm-sling dynamics, recognized that the true form could not be solved, and switched to a simplified model based on what simplifications would prevent chaos (no gravity, sling is wrapped around a drum instead of fixed to the tip of an arm) and then was able to find an exact solution (note that this solvable system starts as nonlinear 4th order, but can be reduced using conservation of angular momentum hacks)
Now, it is known that a chaotic difference equation can have an exact solution: the equation x(n+1) = 2x(n) mod 1 is formally chaotic and has the exact solution 2^n x mod 1. A chaotic differential equation exhibiting chaotic behaviour can have an exact solution if it has discontinuous derivatives because this difference equation can be constructed:
equation is in three variables x, y, z
dz/dt always equals 1
if 0 < z < 1:
if x > 0:
dx/dt = 0
dy dt = 1
if x < 0:
dx/dt = 0
dy/dt = −1
if 1 < z < 2:
if y > 0
dx/dx = -.5
dy dt = 0
if y < 0
dy dt = 0
dx dt = .5
if 2 < z < 3:
dx/dt = x ln(2)
dy/dt = -(y)/(3 - t)
and then make it periodic by gluing z=0 to z=3 in phase space. (This is pretty similar to the structure of the lorentz attractor, except that in the lorentz system, the sheets of solutions get close together but don’t actually meet.) This is an awful,weird ode: the derivative is discontinuous, and not even bounded near the point where the sheets of solutions merge.
Plenty of prototypical chaotic differential equations have a sprinkling of exact solutions: e.g, three bodies orbiting in an equilateral triangle- hence the requirement for a “general” exact solution.
The three body problem “has” an “exact” “series” “solution” but it appears to be quite effed: for one thing, no one will tell me the coefficient of the first term. I suspect that in fact the first term is calculated by solving the motion for all time, and then finding increasingly good series approximations to that motion.
I strongly suspect that the correct answer to this question can be found in one of these stack overflow posts, but I have yet to fully understand them:
https://physics.stackexchange.com/questions/340795/why-are-we-sure-that-integrals-of-motion-dont-exist-in-a-chaotic-system?rq=1
https://physics.stackexchange.com/questions/201547/chaos-and-integrability-in-classical-mechanics
There are certainly billiards with chaotic and exactly solvable components- if nothing else, place a circular billiard next to an oval. So, for the original claim to be true in any meaningful way, this may have to involve excluding all differential equations with case statements- which sounds increasingly unlike a true, fundamental theorem.
If this isn’t an open problem, then there is somewhere on the internet a chaotic, normal-looking system of odes (would have aesthetics like x‴′ = sin(x‴) - x’y‴, y’ = (1-y / x’) etc) posted next to a general exact solution, perhaps only valid for non chaotic initial conditions, or a proof that no such system exists. The solvable system is probably out there and related to billiards
Final edit: the series solution to the three body problem is legit mathematically, see page 64 here
https://ntrs.nasa.gov/citations/19670005590
So “can’t find general exact solution to chaotic differential equation” is just uncomplicatedly false
I am suddenly unsure whether it is true! It certainly would have to be more specific than how I phrased it, as it is trivially false if the differential equation is allowed to be discontinuous between closed form regions and chaotic regions
I am afraid I don’t quite understand the bouncing balls example, could you give a little more detail? Thank you in advance !